

# Statistical Relativity Elections.

Copyright © december 2019: Richard Lung.   
First edition.

* * *
"the thing that has brought discredit upon the form of democracy as it exists in Europe today is not to be laid to the door of the democratic principle as such, but to the lack of stability of governments and to the impersonal character of the electoral system."

Albert Einstein (1931): The world as I see it.

* * *
Physics, the romance of how the world works (with perhaps a clue to why we are here) interpreted by a long-time reader of popular expositions, without benefit of class instructions or personal tuition. (It was left to my intuition.) Like Don Quixote, whose head was turned by reading too many romances, I developed unconventional ideas. I never thought that I would write a book on physics. I was well aware, from the start, that the professionals are authoritative on conventional physics, which I generally accept.

My background is in social science. I did not take much to the presentation of the social part of the course, as to the lesser part, the science, especially the statistics, which is the approach I take to both relativity theory and electoral method. These two subjects have received my amateur attentions thru-out a working life-time (and beyond).

Basically, on one quite simple point do I disagree with physics tradition, the Michelson-Morley calculation, contradicted, by the famous experiment -- and by my own use of a different average.   
The calculation was patched-up with the so-called Fitzgerald-Lorentz contraction (read: correction) factor or gamma factor. (Corrections are inevitable. I must have made scores of errors in my own working. This book is open to corrections, criticisms and comments. Only the caldera chapter has been independently checked.)   
To replace the ad hoc gamma factor, I invoked the principle of Least Action. All local reference frames in high energy physics are unprivileged. They amount to a random distribution, which forms the graphical area under the path of least action as a normal curve.

Special relativity is based on a symmetry principle (so-called rotational symmetry of the Minkowski Interval) that there is no privileged view-point of events. Local observations, of a given event, take particular measures of space and time, but ultimately they are the same metric of a unified space-time.

A theme, by this amateur or naive physicist, is to extend the symmetry principle. By adding a damping factor to the Interval, and comparing the new result with the old, magnitude symmetry is added to rotational symmetry, to create vector symmetry, with an extension to its corresponding conservation law, from angular momentum to vector momentum.

Another extension, from the Michelson-Morley experiment (MMX), for instance, to the LISA project, is a sine-generalised Interval to non-perpendicular frames of reference.

The Minkowski Interval correctly predicts the Michelson-Morley experiment result of equal times, taken by the perpendicular light beams. It is conjectured that this equality of times is formally similar to the Einstein Equivalence principle of the equality of masses, gravitational and inertial. Hence, he Minkowski Michelson-Morley clock of the universe (M3) only shows absolute time in perpendicular frames of reference. Likewise for absolute mass.

In special relativity, kinematics, as of time, and dynamics, as of mass, are formally the same. Hence, the sine-generalised (All-angles) Interval should apply to an Extra Einstein Equivalence principle (E3), where non-perpendicular reference frames do not give equality of gravitational and inertial masses, just as they do not give equality of times.

A comprehensive comparison between special relativity and electoral method is enabled, once two-dimensional voting is introduced (FAB STV 2-D), because then both sciences, Physics and Electics, are on the same footing of using complex variables. A formal similarity of kinematics and dynamics, in special relativity, can be elucidated by a formal similarity between voting with ones hands, on the ballot paper, and voting with ones feet, by moving between electoral districts or constituencies.

* * *

## Table of Contents.

### Statistical Relativity Elections.

Summary findings and conjectures by this amateur.

Insight of youth carried into old age.

Special Theory of Relativity:

1: The Michelson-Morley Experiment and the Lorentz transformations.  
2: The Minkowski Interval.  
3: The Lorentz transformations.

Lorentz transformation and Minkowski Interval geometric means and their dispersions not manifest in classical physics.

Higher order Interval multi-geometry distributions.

The Minkowski Interval predicts the Michelson-Morley experiment; sub-luminal and superluminal connections (SLC)..

Interval magnitude symmetry vectors momentum conservation.

Mach principle in Math and Special Relativity, and a Caldera Cosmos.

An hour-glass universe.

The Michelson-Morley experiment and the Principle of Least Action

LISA and the Michelson-Morley Minkowski (M3) clock of the universe model Extra Einstein Equivalence principle (E3).

Relative motion models relative choice and FAB STV 2D.

Disclaimer

Moral

#### Other Works by the author.

Single-stroke English (Summary edition).

Guide to five volume collected verse.

Guide to two more book series.   
(Commentaries;   
Democracy Science.)

# Statistical Relativity Elections.

Table of contents

### Summary findings and conjectures by this amateur.

This is a summary not an explanation. The purpose of the book is to explain more fully the topics mentioned in this section.

The Michelson-Morley experiment result of the constant speed of light would have been correctly predicted, using the geometric mean, instead of the arithmetic mean, to average the return journeys of the perpendicularly split beams of light.

Consider the calculations of special relativity, the Lorentz transformations and the Minkowski Interval, as geometric mean forms, which average ranges of values. Deriving the dispersions of these ranges shows that for low velocities, relative to light speed, the dispersions disappear, so the geometric averages are no longer apparent as averages. That is why classical physics, limited to relatively low velocities, was not aware that it is implicitly statistical, rather than particularly deterministic.

The geometric mean form of the Minkowski Interval correctly predicts the equal times taken by the perpendicularly split light beams return journeys in the Michelson-Morley experiment. These journeys exhibit relative acceleration, which is why their suitable average is the geometric mean, rather than the arithmetic mean for a velocity range.

As an average, each journey cannot exceed the speed of light. Tho one leg of the journey is relatively greater than light speed, the other leg pulls it back, generally to less than light speed.

However, this property may explain the strange phenomenon of super-luminal connections (SLC).

Take a conservative system of quantum entangled particles, in a conservative state, such that rotating one particle makes the other particle reassert the system equilibrium. This readjustment takes place faster than the speed of light, so is not operated by light signal communication.

How does one explain this "spooky action at a distance", as Albert Einstein called it?

Well, the clue may be in the experimenters particle rotation, because rotation has a component of acceleration, initiating a faster than light connection, as if going backward in time to affect the other particle. But from that other particles viewpoint, it is going forward in time to establish a connection with its partner particle.

The entangled particles might be compared to a Feynman diagram, which has alternative interpretations as sets of particles, with either a forward moving particle in one set, or a backward moving particle in another set. Over greater than sub-atomic distances, faster than light fotons and slower than light fotons average out to the generally observed constant speed of light.

On our ordinary scale of observation of classical physics, individual interactions are observed. But on the high-energy physics scale, just as much as on the sub-atomic quantum mechanical scale, unitive interactions, measured as averages, prevail.

The human being grows more individual and then becomes less so, with age. Likewise, the whole of Creation.

The universe is by definition self-contained or independent (Mach principle). This principle should hold for mathematics. The complete number system should be self representational. Complex numbers can be represented or averaged by geometric means, which in turn form ranges, with their own averages. These are circular functions, of which the Minkowski Interval is an example. This mathematical circle has a physical analog in a sort of caldera universe, which can be duplicated into a matter-antimatter hour-glass universe model.

The circular function of the Minkowski Interval has rotational symmetry, which means, that whatever local observers different orientations to an event, on a point of the circumference, they all agree on the radius (which defines the Interval) to that particular point.

The Interval is a vector, which has both direction and magnitude. The constant length of the Interval radius signifies a constant magnitude. But the magnitude could be varied by adding a damping factor or an amplifying factor to the Interval equation, similarly to the damping and amplifying factors in wave equations.

Taking the geometric average of the solutions to the interval, with or without a damping factor, produces the same answer. The damping or amplifying factor varies the range of the solutions but it does not vary their average. (Remember the above conclusion that it is averages that apply in high-energy physics, rather as they do in quantum physics.) This is as much to say that the Interval has magnitude symmetry. Taken together with its rotational symmetry, the Interval can be said to have vector symmetry.

By Noether theorem, a symmetry implies a conservation principle. The rotational symmetry implies conservation of angular momentum. Presumably the corresponding conservation principle to vector symmetry is conservation of vector momentum.

The rotational symmetry of the Minkowski Interval is the technical description of the basic postulates of special relativity, which says there are no special or privileged frames of reference. In other words, whichever way an event is observed, the Interval provides a common space-time measurement, the same measure or symmetry, whatever observers local space and time measurements.

I assumed this reference frame symmetry to be a statistical principle: the locations of observers would form a random distribution. This compares with elections, which count statistical distributions of voters wishes.

I also wanted to find a less ad hoc explanation than the Fitzgerald-Lorentz contraction factor to the Michelson-Morley experiment. Applying the Least Action principle involved relating from the Interval, the super-luminal factor and the sub-luminal factor to kinetic energy and potential energy components of total energy. In turn, these components were identified with the "success" and "failure" components of total probability. These form the symmetrical halves of a normal or random distribution.

The path of Least Action has the least area under the graph of its curve. Subtracting kinetic from potential energy, and taking its integral, measures the area under the path. The area represents a distribution. In this case, the integral is an exponential function, that of the normal distribution. Thus, a normal curve is the path of Least Action, describing the non-privileged random reference frames of relativistic observers.

In a different way, the caldera model arrived at a normal distribution, whose norm is the speed of light, and whose positive and negative distribution ranges are observers local velocities plus and minus light speed.

There is a third means of arriving at this result. Traditional calculus of differentiation from first principles is implicitly statistical in terms of implicit series ultimately deriving an arithmetic mean or harmonic mean. Therefore a new form of differentiation in terms of the geometric mean should also be possible. Geometric mean differentiation of a form of the Interval results in a normal distribution exponential function.

The Interval equation can also be extended, to a sine-generalised Interval, to include other than the perpendicular beams (equal times) frame of reference to the classical Michelson-Morley experiment; for example, the LISA experiment of 60° beam orientations.

The Einstein Equivalence principle of gravitational mass to inertial mass is based on the Einstein lift thought experiment. It so happens that this can be compared to the Michelson-Morley experiment of equal times.

The Minkowski Interval has the same mathematical form, in dynamics as in kinematics. The former uses the mass variable, where the latter uses the time variable.

Just as the Minkowski Interval, with equal times, describes the Michelson-Morley experiment, so the Minkowski Interval, with equal masses, describes the Einstein lift thought experiment.

Moreover, the sine-generalised Interval should measure non-perpendicular frames of reference, when neither times nor masses are equal. Perpendicular frames of reference are the only times when it has one absolute time, on the Michelson-Morley Minkowski (M3) clock of the universe. (Or indeed the only reference frame for one absolute mass on a "weigh" of the universe.)

The isomorphism of relativistic kinematics to dynamics may have a mathematical model in two comparable forms of elections, where people (moving between districts) vote with their feet, as well as with their hands. But for the analogy to hold with relativistic physics, the elections must be with complex variables, in two dimensions. One dimension would be the usual Representation. The second dimension could be one of Arbitration, which could be considered neutral by way of being perpendicular to the first dimension.

Analogously to the sine-generalised Interval, it might be possible for two-dimensional elections to be held under reference frames with different degrees of Arbitration.

* * *

### Insight of youth carried into old age.

Table of contents

This is not a scientific book on physics, because the author is not qualified to teach it. I never thought I would write on the subject. I may attempt to state a consensus of opinion, but I am the first to admit that the professionals do it much better, as I know from an adult life-times reading of popular physics books. No doubt, my deficiencies are obvious enough to the informed. I try to distinguish my own speculations from generally accepted knowledge.

So, why this book at all? Well, because, for better or worse, that general reading popular accounts by many clever scientists, turned me into a naive physicist, with unconventional ideas. My reading was a flow of books, as they came out from the local library, and such old texts (from which I could glean any understanding at all) unearthed from jumble sales, over the decades, when there were such things, but no internet.

Not being authoritative, I would not have attempted a treatise, had I not surprised myself by developing my own speculations. This makes my work more difficult than the usual elementary physics. I have tried to make the book reasonably self-contained. The reader should not much need to seek explanations elsewhere. Tho, that certainly might help, and is to be encouraged.

At one stage, I decided not to make this work, number 5, in my Democracy Science series. Number 4, on FAB STV, tho I say it myself, was far ahead of its time, in electoral science. It was the first time a voting system was invented that measures one complete dimension of choice. But I am no physicist. These are the naive efforts of an unaided amateur.   
Given an involuntary period off from being a carer, I made my way thru most of this book in draft. And for all my untrained short-comings, I decided this was not a work to be ashamed of. It is amateur science but it is still science.

I once studied social science, including scientific method, which I saw could be used to determine voting method, one of those limited, precise problems that scientific method does well. As to this title, "Statistical Relativity Elections", an explanation of special relativity, and my personal study of its statistical nature, takes up the bulk of the treatment.   
Elections are also statistical, and a sufficiently mature development of voting method, as a complex statistic, makes possible a systematic comparison of physics and "electics": a transferable vote, in two dimensions (FAB STV 2D) makes its debut, in this work.

Over forty years ago, in my early twenties, I read JWN Sullivan review the popular account, Relativity by Albert Einstein. It was apparent to me that the Minkowski Interval resembled an election. Any number of observers choose different local orientations of their co-ordinate systems to measure an event, but they all culminate in a unified result. Similarly, any number of observers make different choices of candidates, with their co-ordinate systems of preference vote to proportion count, to arrrive at a unified result.

Anyway, in the course of the demoralising attempts at comparisons, I decided that I would have to put special relativity (SR) in statistical terms. Because, an election counts a distribution of choice. And I was thinking that SR could be considered as a distribution of choices that all observers are in an equal position to make of measuring an event.

I doubted the youthful wisdom of my comparison, up to the verge of demonstrating it, here, just a few years shy of half a century later. The great majority of those years of research were unsatisfactory. Only late in life did I happen to invent a more sophisticated version of the Single Transferable Vote, to make possible a systematic comparison with Special Relativity. Part of the process of innovation was recognising that both the Interval and election counts are statistical in nature.

From early on, I perceived that, to be universal, natural laws must be universally observable, in what ever way observers choose or elect to observe them. It was evident that general relativity, as a more general theory than special relativity, offers a more general choice of observational view-points. Special relativity applies to observers in uniform relative motion. General relativity also applies to observers in accelerated motion.

In my twenties, it seemed obvious (to a naive physicist) that an electoral equivalent, to uniform relative motion, in Special Relativity, was uniform member constituency systems. Whereas, the electoral analog of general relativity introduced a geometrical "curvature" of constituency systems, so that some constituencies or districts had more seats than others. A foremost example is a normal distribution of seats per district.   
(I will use the North American term, districts (symbol, d) instead of constituencies, because I need the letter, c, for candidates, when it comes to using mathematical symbols to model elections, in a closing chapter.)

The Sullivan summary of Einstein said that using the geometry of Euclid, which measures uniform motion in straight lines, was not an ultimate truth but just a convention. A more convenient convention for accelerated relative motion is the curved geometry of Riemann. I understood that, long ago, because a normal distribution of seats per constituency is a convenient following of conventional communities, such as counties and cities. They form natural constituencies or districts, that can be kept whole, by awarding them numbers of seats in proportion to their populations.

The normal distribution is an exponential function and acceleration exhibits an exponential series, in terms of rate of change. Thus, a mathematical link is maintained between Relativity and an electoral model of it.

So, a more general theory, in practice, implies a generalised choice of observations, or a more general electoral system.

I regret that I never mastered the maths of general relativity, especially as the character of its author has my sympathies. I should add that a modern conception of the relation between special and general relativity is not one of a generalising the choice of frames of reference. Instead, special relativity is considered as based on a symmetry principle that all inertial frames of reference are equivalent. General relativity is based on the equivalence of inertial and gravitational masses. (Even so, that may not be the last word. Close to ending this book, I was surprised to find I had something to say about it: E3)

However, it occurred to me that an improved knowledge of voting method, for generality of observation, might be a way of testing the generality of laws. Knowledge and freedom depend on each other. "Know the truth and it will make you free." (In my early twenties, I was influenced by the wisdom of the Gospels.) A quarter of a century later, this was the guiding principle of my Democracy Science web-site (since superseded by my books).

In my book, Science is Ethics as Electics (number 3 in the Democracy Science series). there is also a long chapter on Relativity of Choice. This is a fairly recent compilation. On the one hand, I was greatly struck by all the parallels between a science of motion, mechanics, and a science of choice, which might be called electics. On the other hand, I had never managed to show an isomorphism or one-to-one relation. I had reached a stalemate.

To my surprise, I wrote a further sequel (number 4 in the Democracy Science series) called FAB STV: Four Averages Binomial Single Transferable Vote.   
Basic binomial STV requires use of the geometric mean. The other three averages came later. Binomial STV is a more sophisticated mathematical structure than traditional STV (or other official elections).  
Part two of FAB STV is a specialised manual. It is nowhere really difficult but so complicated or involved, that, no doubt, it takes a lot of getting used to.

I do a brief recall of FAB STV below, so as not to leave you, the reader, in the dark about it. It is not necessary to know all about it, for the purposes of comparing special relativity to binomial STV. But a brief does put in context key concepts, like keep value and transfer value, election counts and exclusion counts, averaged by the geometric mean. Such concepts are central to my electoral comparison with special relativity.

This book introduces a two dimensional version of FAB STV: FAB STV 2D. This requires complex numbers (consisting of combinations of real and imaginary numbers) as does the Minkowski Interval formulation of Special Relativity.   
Complex numbers involve the square root of minus one, i (for imaginary). This is not such a big deal but it sounds daunting to anyone not familiar with this arithmetic rule. It mystified me for decades. I was too late for an education in the New Mathematics, beginning in the 1960s, which teaches complex numbers, as a matter of course in primary schools.   
Being one of the daunted, I could understand an ignorant twit of my own generation wanting these lessons scrapped as useless. Teachers, too, can fail to impress the use of foundation maths on their pupils, who may want less maths. But they have to find that out first. Unfortunately, a bad teacher can destroy any feeling for a subject. And maths texts are, on average, as badly written as any other genre.

A predominant mathematician of the nineteenth century, or of any century, Carl Friedrich Gauss remarked, with some surprise, that "imaginary" numbers were no more imaginary than real numbers, but simply meant a ninety degree turn from a first dimension of real numbers, into a second dimension of just as real numbers.   
What came as a revelation to a genius can now be taken as read, even by a small child.

Years ago, I noticed that the Michelson-Morley calculation didn't agree with experiment, because it used the arithmetic mean, instead of the geometric mean, to arrive at averages for the respective return journeys of split beam reflections.

The notice taken of my realisation was minimal. This was a blessing in disguise, because it allowed my thoughts on the subject to go wandering on. I showed that geometric averaging for the M-M calculation is consistent with the Minkowski Interval. And I considered that the arithmetic mean calculation does not provide solutions in terms of the Lorentz transformations.

An interesting result, for situations of classical mechanics with low velocities, is that there is no significant dispersion about the geometric mean. In other words, if your geometric mean has no apparent range of values to represent, it is not apparent that it is a geometric mean. And that is why classical mechanics appears to be about uniquely determined values. In reality, it is implicitly statistical, the statistics only becoming apparent for high energy physics.

I regarded the Fitzgerald-Lorentz contraction as an ad hoc hypothesis, that was just a way of bringing geometric averaging into the calculation. More recently, I suggested that the M-M experiment be better seen in terms of satisfying the Least Action principle, instead of some ex-post-facto, or after-the-fact, dub of a "shrinking factor."

More-over, I thought of mathematics having to be self-representational, just as the universe, by definition, must be self-contained, by a sort of Mach principle of consistency. It turned out that complex numbers could represent each other as (geometric) averages of range limits.

The Minkowski Interval, of special relativity, as an example of a circular function, lent itself to such statistical treatment of complex numbers.  
The Interval, and a fuller quadratic equation version of it, were reminiscent, respectively, of a wave equation and damped wave equation. It is the amplitude of the wave which is damped (or amplified).   
Averaging their respective solutions, both versions could be obtained with the same solution, hence there appeared to be symmetry of amplitude.

By theorem Noether, conservation laws have corresponding symmetry operations. The Interval is already known to have rotational symmetry. That is symmetry of direction. My result pointed also to symmetry of magnitude (in the amplitude). Rotational symmetry and, if I am right, magnitude symmetry would combine to give the Interval a vector symmetry. The corresponding conservation law of angular momentum, thus becoming a vector momentum conservation law.

I argued the geometric mean is key to special relativity, as well as being the basis of binomial STV. And I hadn't finished with the geometric mean, or the geometric mean had not finished with me. I set about inventing a geometric mean differentiation, as an extension of traditional calculus, which I identified, in terms of either an arithmetic mean differentiation or a harmonic mean differentiation. The arithmetic mean averages a constant change, like velocity. This graphs as a sloping straight line. Such calculus is refered to as linear analysis.

I speculated that a geometric mean differentiation might analyse non-linear motion, like acceleration, which is the basis of the non-linear equations of general relativity. At any rate, statistics may be the basis of a more versatile calculus.

* * *

## The Special Theory of Relativity.  
Part one: The Michelson-Morley Experiment and the Lorentz transformations.

Table of contents

#### Sections:

Introduction.

The Michelson-Morley Experiment.

The Michelson-Morley calculation.

The Lorentz transformations.

### Introduction

To top.

Special relativity was the out-come of many physicists work, chief among them, Michelson and Morley, Lorentz, Einstein and Minkowski.   
Three related topics, to be treated, are the Michelson-Morley experiment, the Lorentz transformations, and the Minkowski Interval. These topics give precise information, about special relativity, in terms of simple algebra and geometry.   
When I say simple, I mean simple by professional standards. I don't mean that the math was simple for me to teach myself -- far from it. I hope my efforts, such as they are, will help.

The name "the theory of special relativity" (as well as of "general relativity") comes from Albert Einstein. As far as special relativity was concerned, he gave physical meaning of what was going on, in such as the Lorentz transformations.

Minkowski taught Einstein, and supplied a neater mathematical form to the equations. At first, Einstein didn't see the point of the new formalism. But its progression from linear Euclid geometry was to become a point of departure for the curvilinear geometry of Riemann, adopted in his general theory.

As for the Michelson-Morley experiment, Einstein said he had heard of it, when he wrote his famous 1905 paper on special relativity. It was reputed that he hadn't. Anyway, this experiment is a land-mark to the origin of modern physics.

* * *

### The Michelson-Morley experiment (1887)

To top.

The Clerk-Maxwell equations showed that electro-magnetic waves moved at the speed of light, suggesting that light is an electro-magnetic wave. A wave normally _waves_ some material medium it moves thru. To talk of a water wave, without water, wouldn't make much sense. So, a light wave was held to manifest the matter of which the universe is made, the so-called universal "ether." This ether was a supposed "ocean" of material reality thru which light waves traveled.

(Light itself perhaps could be considered the medium of light waves, as water is of water waves. In Feynman quantum theory, light is all over the place or "oceanic," on the sub-atomic scale. Whether, or not, he recognised it, Feynman appears to have solved the problem of the medium of light waves. Light is its own medium, forming into waves, as the resultant vector of its universal "ocean" of pathways over sub-atomic distances. Indeed, there is a universal ocean of virtual particles, besides photons, leaping in and out of observable reality, even in the "empty" vacuum of outer space. But this is not to the point of the history of the "ether" as a hypothesis in physics.)

Michelson and Morley set out to detect the ether with an experiment. Given that the ether was the all-pervading medium of the universe, its characteristics, such as velocity, always would be the same relative to everyone else.

Concepts of Modern Physics, by Arthur Beiser (1973) begins the text with the Michelson-Morley experiment, and the analogy of ones position on Earth, as being on a bank, relative to a universal ether "stream."  
One could take two return journeys of equal distance, either across the stream or up and down stream. Taking into account the velocity of the stream, simple geometry shows which journey would take the longest and which would take the shortest time.

The return journey, which would be most slowed by stream velocity, is the one going directly up and down stream. Going down stream (like having a tail-wind) would be the fastest way for a boat to travel. But having to make a return trip (like facing a head-wind) would so out-weigh that advantage, that, on average, the combined up-stream and down-stream trip would be the slower than a back and forth cross-stream journey of equal length.

The quickest return journey, over an equal distance, is the one taken directly across-stream and back. Actually, this wouldn't follow a path at right angles to the banks, because the velocity of the stream is pulling the boat some way down-stream, while the crossing is being made. And the boat goes adrift by an equal amount, on the way back to the bank that one embarked from. So, one would land back some way down the bank, from ones embarkation point.

In the Michelson-Morley experiment, a beam of light stands in for the boat. This beam is split, to do the two different "boat" journeys described. Mirrors effect the split light beams return journeys.   
One light beam was expected to return slightly more slowly than the other, as a result of a difference in "ether drag" upon the two beams. This is analogous to the boat going up and down stream taking longer than the boat going across stream and back.

Of course, no physicist knew the direction of the supposed ether "wind" (to change the analogy from water to air). But the Michelson interferometer could be spun round in any direction, to check for interference of reflected light waves, in any direction. Moreover, this experiment, with the light beam split at right angles to itself and both split beams reflected, was repeated in all directions, and at all times in Earth annual orbit. So, the ether wind direction and velocity could presumably be infered from the experiment, in that series, which showed the maximum and minimum delays of a split beam subject to ether drag, analogous to wind drag.

As recently as 2016, in a popular work by John Gribbin and Mary Gribbin, Science. A history in 100 experiments, the historic notion is given:

"The most extreme effect would occur when the beams were travelling at right angles to each other, with one beam travelling across the direction of the Earth's motion and the other in the same direction as the Earth's motion. The first beam should be unaffected by the relative motion of the Earth and the ether, while the second beam felt the full influence."

Experiment 52: The speed of light is constant...(as Maxwell's equations said it must be).

In fact, the Michelson-Morley experiment showed always the same result: a null result. The split beams of light always returned with their waves still in step. The Michelson interferometer would have measured interference effects, if they hadn't. Therefore, the split light waves took the same time to make their two return journeys, traveling the same distance, without their different directions causing a more powerful ether drag on one beam than the other. The speed of light remained constant.

Einstein special theory of relativity would assume there is no ether but rather that the searched-for absolute velocity is to be found in the constant speed of light. (By this, physicists mean constant in a vacuum, disregarding that light is slightly slowed in transparent media, like air and water and glass.)

Sir Arthur Eddington led the post-WW1 team experimentally verifying the Einstein general relativity prediction of light bending in passing near the suns gravitational field.   
There were supposedly only three people in the world, who understood the general theory. Eddington was asked who they were.  
There was a long pause. When Eddington was prompted, he responded that he was trying to think who the third person was.   
Yet, in his popular work, The Nature of the Physical World (1928), unlike Einstein, he still has not abandoned the concept of the ether.

* * *

### The Michelson-Morley calculation.

To top.

This section gives the historic and conventional calculation. I later show that if you substitute the geometric mean for the arithmetic mean, to average the mirror-reflected light beam journeys, the correct prediction is achieved for the result of the famous Michelson-Morley experiment (MMX).

The Michelson and Morley calculation, for the experimental result, they expected, is essentially nothing more than simple arithmetic, involving the fact that light velocity, c, equals distance, d, traveled, divided by the time, t, taken.

Two different times were predicted, the shortest time, t, for a beam of light, split to pass athwart the ether stream, and the longest time, which can be designated t', for that part of a beam, aligned to the ether stream.

But experiment showed that the times, t and t', were the same. Therefore, to make the calculation agree with experiment, a factor had to be written into the calculation, to show t and t' as equal. This factor, would, in effect, contract the longest time, t', so it equalled the shortest time, t. Hence, its name, the Fitzgerald-Lorentz contraction factor, after the two physicists, who independently suggested it.

Eventually, Einstein would dispense with the assumption of ether absolute velocity and an absolute space and absolute time, presupposed since Newton. Einstein would explain this contraction factor, in terms of observers, in different situations or frames of reference, measuring distinct times, distances and velocities. Different observers could relate their spatio-temporal measurements, basically thru the so-called contraction factor. But no one observers frame of reference could be considered absolute, or prior, to any-one elses. Observations were all _relative_ to each other and no one was absolute. Hence, the theory of relativity.

(By the way, the concept of relativity has been corrupted to mean that anyones opinion is as good as another. This sloppy or inaccurate position - usually tracable to certain vested interests, seeking to keep their privileges \- in politics and philosophy, is known as moral relativism. It has no scientific basis in the theory of relativity.)

How, at first, did Michelson and Morley work out the times, t and t'? The longest time, t', is the return journey, up and down the ether stream. In other words, the Earth, at a certain point in its orbit, is supposed to be traveling in line with the stream of universal ethereal matter. Thereby, the experimenters light beam is also in line with the ether stream (besides being split off at right angles to measure the shortest delayed time, t, of light travel across the ether stream).

Going with the ether stream, the velocity of light, given the letter c, (for "celerity") should have a tail-wind or tail-stream of ether velocity added to it. The ether velocity is given by earth velocity, just as the velocity of a stream can be measured by the speed one is running beside its bank to make the stream appear to be standing still. This ether velocity may be given the letter, u.

The velocity of light, with an ether tail-wind or tail-stream, was calculated as its own velocity, c, combined with the ether velocity, u. That is (c+u). Therefore, the time light takes to travel a distance, d, down-stream was given as the length divided by the combined velocity, or, d/(c+u).

When the light is reflected, thought of as like a boat now going up-stream, it would be slowed down by the speed of an ether "head-wind," to be subtracted from the light speed: the light beam speed up-stream was calculated at (c-u). The up-stream time taken is the slower one of d/(c-u).

Therefore, the combined time, t', to travel up and down an ether "stream" (twice the length of a one-way journey) would be:

d/(c+u) plus d/(c-u). This works out as 2dc/(c-u)(c+u), or,   
2dc/(c²-u²).

[However, this assumes a true analogy, between boat and light travel, which is to be challenged, in this book.]

For working out the cross-ways return journey, over a comparable distance, d, of the light beam with respect to the ether stream, please refer to diagram, below. The light source, S, is the point where the "boat," as light-beam, sets off across the stream, reflected by a mirror, M, analogous to return from a far "bank."

#### Diagram of "ether" drift on the transverse light beam.

When the Earth velocity arrow is in accord with a hypothetical "ether stream," it is reckoned to cause light the greatest delay, when traveling up and down stream. The diagram shows the least delay, caused that part of the light beam that splits directly across stream. But it is still subject to ether drift downstream, both to and from the mirror.

By the time the light, from source, S, has reached mirror, M, the mirror has moved to position, M', in the diagram. This is like down-stream drift on a boat crossing. The same is true for the journey back. So, when the light-beam reaches M', it has completed half the journey in half the time, t/2. This is because crossing the ether stream affects journeys, both ways, equally.

By the time, that the light beam has reached the mirror, at M', the light source, S, has also traveled half way, S', to its rendezvous, S", with the reflected light-beam it emitted.

On the diagram, the light beam travels at speed, c, from S to M', given as distance, z. Meanwhile, the mirror has traveled from M to M', given as distance, y, down the ether stream, at a speed equated (as explained above) to earth velocity, u. The two times of travel are equal, actually time t/2, as mentioned already. Therefore, t/2 equals both distance, z, divided by velocity, c, and distance, y, divided by velocity, u.

Or, t/2 = z/c = y/u. The distance, z, can be found in terms of theorem Pythagoras. Namely, z² = y² + d².

Therefore, z² = (zu/c)² + d².

And: z²{1 - (u/c)²} = d².

So: z² = d².c²/(c²-u²).

Therefore: z = dc/(c²-u²)^1/2

The journey from S to M' is similar to the return from M' to S", also of distance, z. Therefore, the total distance traveled by a crossing light-speed "boat" is twice z, or:

2z = 2dc/(c²-u²)^1/2.

Dividing this distance, both ways across, by the light-beam cross-stream velocity, c, gives its total crossing time, t, as:

t = 2d/(c²-u²)^1/2.

This was how Michelson and Morley calculated the two times, t' and t, as a predicted out-come of maximal and minimal ether drag, respectively.   
The fame of their experiment rested on the fact that it showed the two times to be equal. The two times could only be made equal, in theory as well as practise, by introducing an ad hoc factor, F (the so-called Fitzgerald-Lorentz contraction factor) so that the shorter time, t, equals the longer time, t', multiplied by the contraction factor.

The contraction factor, F, equals t/t' = [2d/(c²-u²)^1/2]×{(c²-u²)/2dc}

= {(c²-u²)^1/2}/c = {1-(u²/c²)}^1/2.

Ad hoc explanations are frowned-upon in the philosophy of science. They are like props for an unsound building, that would fall-down, otherwise. And probably should be left to do so, to make way for a better building, that is to say, a better theory or explanation that makes more sense.

* * *

### The Lorentz transformations (1895).

To top.

Prof. H A Lorentz showed that Maxwell electrodynamic equations still held, whether or not subject to a prevailing ether "stream" or "ether wind." But the co-ordinate systems or frames of reference, of two observers would differ according as to the extent their measurements were affected by the ether wind.

Supposing, for the sake of argument, the measurements, taken in a laboratory on earth, were not subject to the ether wind. The measurements there are given by time, t, distance, x, (as well as two other dimensions of space that don't essentially affect the argument). Lorentz showed that a laboratory elsewhere, say on a rocket, could still discover Maxwell equations. But the measurements in the rocket would be in a different frame of reference, with different time, say, t', different distance, x' (and its two other spatial dimensions). Consequently, local observers would measure differently: x' = u't', instead of x = ut.

However, these different co-ordinates could be related by a set of equations, to become known as the Lorentz transformations. These involve (in the denominator) the Fitzgerald-Lorentz contraction factor, F, already met.

The Lorentz transformation for the times is:

t' = t(1-uv/c²)/(1-v²/c²)^1/2.

The velocity, v, stands for the relative velocity of two observers frames of reference. The velocities, u or u', are the velocities of an object, seen by observer O or observer O', respectively.   
We have to bear in mind that the speed of light is so great in comparison with our normal earth-bound movements, that it is in effect infinite. That being the case, the Lorentz transformation, for two local times, reduces to just the one ("absolute") time, we are familiar with.

A Lorentz transformation applies to mass, also previously thought an absolute. As a body approaches the speed of light, its mass increases. As this speed is outside our normal experience, no such mass increases are observed. This phenomenon, as well as time dilation effects to normal life-times, is regularly measured in laboratory experiments with atomic and sub-atomic particles, moving very close to the speed of light.

In theory, a body could never reach the speed of light, because that would involve acquiring infinite mass. Hence, the speed of light is a limiting maximum on all objects with mass.

It turns out that momentum, or mass times velocity, has a similar form of Lorentz transformation to that for distance measurements. Energy has a similar transformation to that for the different times observers measure, in high energy physics. (Einstein gave, as a post-script to his 1905 paper on special relativity, the most famous formula in science, E = mc². Henri Poincaré also derived this equation in his close anticipation of Einstein theory.)

The velocity, v, in the above Lorentz transformation of times, relates to the difference between the observers two measured velocities, u and u'. As things work out, it is generally not a simple subtraction between the two, when speeds significantly approach light speed.

If one of the frames of reference is considered at rest, so the velocity is zero, or u = 0, the velocity, measured in the other frame, u' = v. The Lorentz transformation of times then reduces to the Michelson-Morley calculation, modified by the contraction factor, F. For:

t' = t(1 + 0.v/c²)/F = t/F.

Two observers, with differences in relative motion, that are significant compared to the speed of light, would observe different speeds in their clocks. Moreover, these would be real effects, resulting in twins, say, aging at different rates.

The so-called "twin paradox" raises the question of why one twin, rather than the other, since they are both in relative motion, should be the one to age more slowly than normal. The retarded ager went off in the rocket, at a velocity, that was a large fraction of the speed of light, and returned to earth to find his twin long dead. The twins were just in relative motion -- most of the time.

The role of acceleration removes the paradox from the twin paradox. (Paul Davies explained it well, in Space and Time in the Modern Universe.) How to resolve which twin does not age relative to the other, when they are both separated by speeds significantly approaching light and then brought back together? The twins are moving relatively to each other, so the supposed paradox goes, so why should one be any more liable to age than the other?

The key is that the twin sent off in a rocket, from earth, has to return, and that means a change in velocity of the rocketeer, relative to the earth. A change in velocity is an acceleration of the rocketeer. So, the rocketeer is the one to under-go the relativistic effects of slowed time, at significant approaches to light speed.

Even if the rocketeer did not do an abrupt about-turn but circled back, circular motion still has a component of acceleration. This has been tested on the atomic scale, where a particle orbiting its twin is measured to have a lengthened life.   
By the way, the accelerated twin in the rocket should not experience having lived any longer than normal, tho he might return to an earth which was much older than when he left it, and his earth-bound twin long since dead.

# Special Relativity, part two:  
The Minkowski Interval.

Table of contents.

#### Sections:

The Minkowski Interval (1908).

The Interval: diagram.

The Interval: Phase-different frames of reference.

After-note: Linear and rotary co-ordinate systems.

### The Minkowski Interval (1908).

"The Interval" of Minkowski is an extension of Euclid geometry, which makes measurements in three dimensions of space. An example, of Euclid, is measuring the height, length and breadth of a box-room from one of its corners. Minkowski treated time as a fourth dimension of space and integrated it into a four-dimensional Euclidean space-time geometry.

The term space-time, long puzzled me, because of an elementary misunderstanding of mine. The Interval equation equals, on the one hand, any local observers distinct distance, or rod, measures, and on the other hand, their local clock or time measures, multiplied by the light-speed constant. That multiple, of time by velocity, has the dimensions of distance but it is still proper to call it a time variable, because the light-speed is constant, indeed is usually assigned a unit value (so that observers locally measured velocities are some fraction up to the value, one, of light-speed itself).

The advantage of this new mathematical formalism was that all observers locally different space and time measurements could be re-stated in a formula, known as "the Interval," a constant spatio-temporal measurement, which is the same for all observers.   
(One cannot say of Relativity that it is "only relative.")

A qualification is that the Interval only applies to observers moving in "uniform relative motion." To remove this arbitrary restriction, Einstein had to come up with his "principle of equivalence" of (non-uniform motion, namely) acceleration to gravity, which was the basis of a general theory of relativity.

The only mathematical difference between the three spatial dimensions and the one temporal dimension, of the Interval, is that time units are multiplied by the mysterious square root of minus one (given the symbol _i_ ).

We may see solid shapes changing in time. We may imagine our three dimensional world in a historical perspective. However, the three spatial dimensions are essentially the same in character and we need only one of them to show how the time dimension relates to spatial measurement. One space ordinate, x, and one time ordinate, t, can be co-ordinated, diagrammatically on a two-dimensional page.

So, we relate the spatio-temporal measurements of two observers, working in two different places and times, and therefore with two different co-ordinate systems: the x and t system, of one observer, is distinguished from an x' and t' system, of another observer, with different space and time measures, distinguished by indices.   
Of course, both systems have two other dimensions of space to consider. For simplicity, we've assumed their measurements don't differ in that respect. In other words, their space measures on the y and z dimensions are equal, or y = y' and z = z', comparing the two observers co-ordinate systems.

The Lorentz transformation can be expressed as an equation of one observers time, t', or the other observers time, t. Likewise, for their respective distance measurements, x and x'.   
The Interval points to a space-time co-ordinate position, that is both x + ict and x' + ict'. These four values are essentially the Lorentz transformations, we've just been speaking of. Here, they form two different sets of co-ordinate axes, that are the two observers frames of reference.

Using trigonometry, the Interval allows observers varying space and time measures to be drawn as the varying opposite and adjacent sides, of a right-angled triangle, inscribed in a circle, with one vertex, at the origin, subject to change of angle size. For instance, a more acute angle makes the opposite side, of the triangle, relatively short. These opposite and adjacent sides will vary according to the rotation of the hypotenuse, as the constant radius of a circle.

Any position on the circle boundary, that the radius points to, is a given space-time position, perhaps marking some mutually observed physical event. Any number of observers can choose different ninety degree or rectilinear co-ordinate systems of space and time, on the circle, given that their axes all share the same origin at the center of the circle.

With these different anglings of their rectangular co-ordinates, they can all measure any one given space-time position on the circumference. Their respective axes will all show different space-lengths and different time-lengths in relation to that common sighting. In effect, each co-ordinate system is like a triangle with varying opposite and adjacent sides for varying distances and times.

But all these different triangles have the same hypotenuse. By theorem Pythagoras, their varying lengths, on different space and time axes, add up to the same unitary space-time measurement, given by the length and direction of their shared hypotenuse, as a constant radius pointing out any given space-time event on the circle circumference.

This space-time radius vector is "the Interval." A vector means a (magnitude of) length in a pointed direction. The Interval applies not only in one dimension of space and one dimension of time. For example, making measurements in two dimensions of space, as well as one dimension of time, the radius could vector-in on a given three-dimensional space-time position, represented as a point on a sphere, instead of a circle.

Theorem Pythagoras, for finding the Interval as hypotenuse, extends to apply to three or more dimensions.

For the four-dimensional space-time treatment, the Interval would, presumably, be a radius 4-vector pointing out a position on a four-dimensional hyper-sphere, whatever that is - something beyond the imagination of our three-dimensional experience, perhaps.

### The Interval diagram.

To top.

Having explained the Interval in words, simple equations will give a more precise idea, with the help of the following diagram.

#### The Minkowski Interval

The Interval is given by the radius vector, OP, which is the hypotenuse to both triangulated co-ordinate systems. Thereby, the point, P, is located by both systems, the two sets of co-ordinate positions, x + ict and x' + ict', differently representing that same position.

The diagram shows, schematically, the essential difference between two observers measuring, in space and time units, the same event at position, P. Special relativity recognises local ruler or "rod" lengths and local clock rates. These correspond to the different co-ordinate systems the observers are working with.

On the diagram, the two observers each have a pair of ninety-degree axes. One observer measures distance and time, in units x and t.  
For time to be in the same length dimensions as x, time, t, is multiplied by c, the constant speed of light. The other multiple, i, is sometimes described as an operator, meaning the operation, turn thru ninety degrees, into a second dimension.

An other observer measures distance and time, in his units, x' and t', on his x' and ict' co-ordinates.   
The diagram shows the only real difference between this co-ordinate system, and the other, mathematically, is that it is a turn, thru angle, a, from the x and ict axes system, to the x' and ict' axes system.

Of the four angles, a, on the diagram, I refer to the angle a, at the origin, O, and marked by the up-wards curling arrow, to show angle, a, is the angle of tilt between the (x, ict) right-angled co-ordinate system and the indexed co-ordinate system (x', ict'). This tilt angle, a, is color-coded yellow in the diagram.

Position, P, marks the event, that the observers are both measuring, and is equally well located by one set of axes as another: x' + ict' marks the spot as well as x + ict. There is no privileged or prior frame of reference. The frames are "only relative" to each other but they have an agreed space-time measure of an event. This would be true for any number of observers, with their own co-ordinate frames, at varying angles, about origin, O, from each other.

They all would have in common what these two observers have. They would share the same hypotenuse, OP. This is "the Interval," a constant space-time measurement, that all observers (in uniform relative motion) can agree is the same measurement of the event at P. Yet, the observers arrive at the Interval from their different measurements of space and time, treated by each observer as separate measurements on a separate space axis and a separate time axis.

It is in the context of the Interval, that special relativity is about an integrated concept of "space-time."

The value of the Interval can be found from any co-od. system. Taking the (x,t) system, and using triangle OPJ: As shown on the diagram, the length of OJ is x = ut. The length of JP is ict. By theorem Pythagoras, the length of the hypotenuse, OP, which is the Interval, is found from: (OP)² = x² + ( ict )² = t²(u²-c²).

Usually the Interval is left in this squared form. (There is no set convention that the Interval be measured as a subtraction of light speed, c, from local velocity, u, as here, or vise versa.)

Taking an other observers system, (x',t'), the Interval can be found by the same procedure, this time using triangle OPM. The diagram shows that, as this other right-angled triangle has the same hypotenuse, OP, the Interval, as such, is obviously the same value.

In the indexed co-ods., the Interval squared equals t'(u'²-c²).

Usually, text books leave the Interval, I, in terms of the hypotenuse squared, while showing the equivalence of two different frames of reference, like so:

I² = t²(u²-c²) = t'(u'²-c²).

The formula only has significance for observers whose velocities, u or u', are a significant fraction of light speed, c.

* * *

### The Interval: Phase-different frames of reference.

To top.

The Interval is a rotatable radius. (The diagram doesn't include a full circle, showing the origin, O, at its center.) The Interval measures a generally agreed space-time location of an event at P, on the circumference.   
No mathematician, I still couldn't help notice that such a radius vector draws out waves, in either time or space, or both. Such wave equations have independent variables of up to three dimensions in space and one independent variable for time, on the other side of the equation. Both sides act on a dependent variable of wave displacement. The Interval seems rather like this classical wave displacement, acted upon by four separate components of space and time, but summarising them mathematically as a unified space-time measurement.

The Interval is a constant for any given space-time event but becomes a variable once one considers it the measure of an event, moving thru a circle or sphere of events (or 4-D space-time hyper-sphere of events).

The Interval was marked at P, on the diagram, by two observers co-ordinate systems. In the process, it was also marked by two different kinds of co-ordinate system: rectangular or rectilinear co-ordinates and polar co-ordinates.

The Interval was described as a space-time vector, giving both length (including length of time) and direction. In our two-dimensional example, any point on a plane, or sheet of paper, could be indicated by this means.

The same can be done with complex numbers. These are an extension and completion of the number system, with rules of counting somewhat different from traditional numbers. Complex numbers are taught to young children as part of the "new maths" in schools.

The two observers each describe the position, P, by a complex number or variable, z = x + ict or z = x' + ict'. (Complex variable, z, has nothing to do with the letter, z, of x, y, z axes of three dimensional space.) These complex variables, z = x + ict and z' = x' + ict', are in the system of rectilinear or rectangular co-ordinates, where the term with the coefficient, i, implies a second dimension to the dimension of the first term.   
This complex numeration works like a net. Depending on how fine the net is, you can mark any position, on a stretch of ground it covers, by which node or knot in the net lies over it.

The Interval diagram can use the polar co-ordinate system, because the two rectangular co-ordinate systems were related to each other, by sharing the same origin (or pole), and having their axes turned at a given angle to each other. They also shared the same hypotenuse, a radius vector, which is the Interval itself.

A co-ordinate system which measures, in this case, a plane, by sweeping out an angle, in a circle, with a radius, is called a polar co-ordinate system.

We can also express z and z' in polar co-ordinates. From the Interval diagram, let the radius, from the origin, O, pointing to position, P, be called r.

Then, cos(b+a) = x/r.

i.sin(b+a) = ict/r.

z = x + ict = r{cos(b+a) + i.sin(b+a)}

Thus, z is expressed both in rectangular and polar co-ordinates.

Similarly for z':

cos(b) = x'/r.

i.sin(b) = ict'/r.

And z' = x' + ict' = r{cos(b) + i.sin(b)}.

A well-known result says the trigonometrical terms can be replaced by an equivalent term, using the exponent (usually written "e" or "exp"). This is a constant number, the sum of an exponential series, an infinite series but converging, to approximately 2.718...

An exponential series of terms is essentially a geometric series, which changes by a constant multiple for each successive term. It corresponds to acceleration of motion. Whereas, velocity of motion corresponds to an arithmetic series, whose terms change by a constant addition for each successive term.

[When I was learning how to draw ellipses, for my simplified English alfabet system, Single-stroke English, I accidentally found out a relation between exponential growth and circular motion. An ellipse is a flattened circle and I found that drawing an elliptic quadrant of curve followed a simple geometric rule of thumb.

Starting with the narrow end of the ellipse quadrant, as the high point, and ending with the shallow midpoint of the ellipse broad-side on, you have a shape, like a graph of exponential decay. (Bearing in mind that the constant multiple of about 2 approximates to the exponent number of about 2.718...) Divide this quadrant into equal segments, starting with the height of the ellipse narrow end at unity. The second segment is drawn at half that height. The third segment is drawn at half the previous half, and so on, for each successive segment. This series can be represented by the exponential series: 1/2^0 = 1/1; 1/2^1 = 1/2; 1/2^2 = 1/4; and so on.]

Anyway, complex variable, z', equals radius, r, times the exponent, e, to the power of an index, which is the operator, i, times angle, b:

z' = re^(ib).  
For, z, the only difference is that the angle in the index is not angle, b, but angle, b+a:

z = re^(i[b+a]).

These various expressions are standard text book solutions for z or z' as a wave form, in a so-called wave equation. A complex variable, z, can be imagined as the point, P, in the Interval diagram, moving around the circle, whose circumference it is on. This circular motion can be charted, horizontally on a graph, as a wave motion. The horizontal axis can represent the lapse of time or distance.

The (positive part of the) graphs vertical axis represents the height of the wave. Its full height or crest is called the amplitude. It is the same length as the radius. When the radius reaches the northern vertical position, on the Interval diagram, this corresponds to a crest of the wave. From there, for every full circle, the radius sweeps out, the graph of the wave reaches another crest.

Our Interval diagram only shows the positive quadrant of the circle co-ordinates. When the radius reaches the southern vertical position, where the ict-axis has changed to -ict on the circle, this corresponds to a trof of the wave being drawn out on the graph.   
(A one-dimensional solution of the wave equation is typically given as a function of: x ± ct.)

Both z and z' mark an event, P. Presumably, the two observers could both track P, if it were to describe a circle in space-time, on the Interval diagram. In doing so, the two observers would trace out their respective z and z' wave forms on the graph. But these wave forms would not be synchronised, because there is a difference of angle, a, between their co-ordinate systems.

This difference of angle is called the phase. A difference of phase, between physical waves, is responsible for interference effects, such as the patterns made by colliding water waves.

On the Interval diagram, an angle, a (color-coded yellow) shows the two observers spatial axes, x and x', out of phase to that degree. Consequently, their temporal axes, ict and ict', are also out of phase by angle, a (color-coded green).

In general, this situation applies to the Interval as a four-dimensional space-time. So, you could characterise different observers, of relativistic effects, as having co-ordinate systems, with space-time phase differences from each other.

It is only a phase they are going thru.

* * *

### After-note: Linear and rotary co-ordinate systems.

To top.

This section was written to familiarise myself with the basic maths of circular and wave motion, being useful to know later in the book, but may be skipped.

Wave motion is related to circular motion, which is conveniently expressed in polar co-ordinates.

When I had to draw graphs at school, a long time ago, they were always rectangular shaped. That is to say in rectilinear or Cartesian co-ordinates. You have an x-axis which is the horizontal ordinate and a y-axis which is the vertical ordinate, both measured from a zero starting point or origin.

An equally important type of co-ordinate system is polar co-ordinates. Their origin is the "pole" or center of a circle. One, of their two ordinates, is the radius, which sweeps round like the minute hand of a clock. (Tho, the convention is that the radius turns anti-clockwise.) The other ordinate is the angle Q that the radius sweeps out, like an angle between two lines of longitude from pole to equator.

The two co-ordinate systems may be matched, when the x-axis stands at "3 o'clock" and the y-axis stands at "12 o'clock," so to speak, on the circular "face" of the polar co-ordinate system. This allows one to match or synchronise the two ways of drawing the same graph. Suppose the radius starts off at 3 o'clock and turns anti-clockwise to the vertical 12 o'clock position. In other words, the radius moves a ninety degree angle from an x-axis position to a y-axis position.

Placed beside this combined graph, you can have an ordinary rectilinear graph, which follows or maps the changing vertical height of the radius, where it touches the circumference, above or below the horizontal diameter, and traces a path up to the crest of a smooth hill.

The ascent shape is matched by the descent from the crest back to the "surface," level with the origin. This is the half wave-length point, corresponding to the radius having reached the "9 o'clock" position on the polar "clock face:" that is, its negative x-axis.

Then, the ordinary rectilinear graph trace descends into the trof of the wave, which is as deep as the crest is high. (Crest and trof correspond to the full height of the positive and negative y-axis positions on the polar circle, at "12 o'clock" and "six o'clock" respectively.) Finally, the rectilinear trace climbs out of the trof to the surface to complete the wave-length, which is equivalent to the radius completion of a full circle and returning to the positive x-axis or "3 o' clock" position.

The wave-length, l, such as from crest to crest or trof to trof, on the rectilinear x-axis is the same length, two pi times r, as the circumference, C, of the circle in the polar system: C = 2πr = l. As such, the wave-length is a spatial wave, measured by the circumference distance.

It is also possible to have a wave-length in time, called a period, P. This has been implicit in comparing the polar system to a "clock." Given clocks with 24 hours on the dial, one revolution of the hour hand marks out a period of revolution of the earth on its axis. The usual 12-hour clocks mark out a half period of revolution.

A radius vector has a number, n, of revolutions it completes. On the corresponding rectilinear graph, the wave-length, is repeated n times like a series of perfectly uniform ripples. This can be done for wave-lengths in space or in time. Hence, the distance, s, covered by n wave-lengths is: s = ln = 2πrn.  
And the time, t, over n temporal wave-lengths or periods is: t = Pn.

Distance, in a straight line, traveled continuously over time, equals velocity or speed in a straight line. Long ago, school made me familiar with this. But I don't remember learning the related formulas for rotary motion, as distinct from linear motion.   
The distance covered in circular motion is given by the number, n, of revolutions thru a complete circle, whose angle is 360 degrees or 2 pi. This is called the "angular distance," say, Q. So, Q = 2πn.   
Therefore, s = Qr. Or, the linear distance equals the angular distance times the radius.

Similarly, for (linear) velocity, v, and angular velocity, _w_. Angular velocity equals angular distance divided by time: _w_ = Q/t = 2πn/Pn = 2π/P.   
And: v = s/t = 2πrn/Pn = _w_ r.  
(In turn, a further similar relation exists between acceleration and angular acceleration.)

We've seen, above, that velocity also equals wave-length times frequency: v = l.f.

And: v = s/t = ln/Pn = l/P.

Therefore, the frequency is inversely equal to the period: f = 1/P.

A radius sweep which starts at "3" on the polar "clock" produces a so-called sine wave from its equivalent starting point in a separate graph of rectilinear co-ordinates at their origin, where x- and y-axis cross.   
Starting the radius sweep at "12" on the polar clock, starts a so-called cosine wave from that full height, or (maximum) amplitude, on the rectilinear y-axis, directly above the origin.

The sine wave starts at zero amplitude. The cosine wave starts at full amplitude or the crest of the wave. If you draw them both on the same graph, the x-axis always marks the sine wave 90 degrees, or one-half pi, ahead of the cosine wave. This is just as the polar graph shows the 3 o'clock position is 90 degrees ahead of the 12 o'clock position, as the radius counter-clockwise sweep is made to go by mathematical convention.

This ninety degrees out of step between the sine and cosine wave is called a difference of _phase_ angle (call it angle q). Suppose that the sine and the cosine waves have both gone the same length of wave, corresponding to an angle of size Q in the polar system. Given that the phase, q is 90 degrees, then sine (Q + q) = cosine Q.  
This follows from school trigonometry of sine Q equals the right angled triangle sides ratio of opposite over hypotenuse. Cosine Q is adjacent over hypotenuse (triangle sides).

In the polar system, a radius is considered as the hypotenuse of triangles it forms by dropping a vertex (for a y-axis reading), from the point the radius touches the circumference, onto a circles horizontal diameter (for an x-axis reading). One can check on ones scientific calculator, for example, if Q = 0 degrees and q = 90 degrees, then sine (0 + 90) = cosine 0; or, sin (30 + 90) = cos 30.

### Sources.

For the Michelson-Morley experiment, I relied on:

Sir James Jeans, The New Background Of Science. Scientific Book Club, 1945 edition.

Isaac Asimov, Asimov's Guide To Science, Vol. 1, The Physical Sciences. Penguin 1972. The appendix also contains a simple derivation of E = mc².

Arthur Beiser, Concepts Of Modern Physics. McGraw-Hill 1973. Chapter one.

For the Lorentz transformations, I cannot remember the first book that interested me in them, but I have read appreciatively, on the way:

Milton Rothman, The Laws Of Physics. Penguin 1963.

Allan M Munn, From Nought To Relativity. Allen and Unwin 1974.

Paul Davies, Space And Time In The Modern Universe. Cambridge University Press 1977. This carefully explains away the twin "paradox." Davies has written many popular physics books since then.

For the Minkowski Interval, I relied on:

George E Owen: Fundamentals of Scientific Mathematics. The John Hopkins Press 1961.

These sources remind me that I had to rely, for self-teaching, on second-hand book-stalls, jumble sales, and charity shops, as well as the modest local public library, long before the internet came into being.

## The Lorentz transformations.

### Special relativity (part three): an arithmetic example of the loss of simultaneous time between observers in relative motion comparable to light speed.

Table of contents.

#### Sections:

Orbitsville.

Lorentz addition and subtraction of velocities.

Lorentz addition and subtraction of times.

Lorentz addition and subtraction of distances.

The Interval.

Kinematics and dynamics.

Remarks.

### Orbitsville.

Suppose some advanced race created Orbitsville, an idea taken up by science fiction writers such as Bob Shaw. Freeman Dyson suggested that all but a tiny proportion of a suns energy is wasted in nurturing intelligent life on the odd planet orbiting it. A vastly more efficient way would be to engineer an egg-shaped surround at a distance of the earths elliptic orbit, about eight minutes light-speed distance from the sun. That way, life could exist on the entire inside surface of the egg (or rather an ellipsoid and not too different from a sphere). This concept is known as a Dyson sphere.

Orbitsville would have to be a mover or diverter from collisions with stray matter. It would be a mother ship to other space-ships.

Imagine Orbitsville and a space-ship passing by it (or being passed by it, depending on your point of view) with a uniform velocity, at a significant fraction of light speed. This scenario may illustrate how Einstein showed that two observers in relative motion cannot agree on the time of an event: in the special theory of relativity, simultaneity is lost.

The sun is in the center of Orbitsville (we suppose, rather than at one of the two focuses of Orbitsvilles elliptic hull). The climate on its narrower part (the minor axis) will be warmer than on the length-wise direction (the major axis). Observers living on the inside surface are stationed at each end of the major axis. Suppose the engineers of Orbitsville have created a night-simulating mechanism, of eclipsing satellites or whatever, such that dawn breaks at exactly the same time for the observers at the far ends of Orbitsville. For instance, internal "astronomers" on the minor axis of Orbitsville might see a simultaneous swathe of light upon either end of the major axis.

Say, both ends of Orbitsville are about eighty-nine and one-third million miles from the sun. The speed of light is about 670 million miles per hour. Therefore, the internal astronomers will measure the sunlight taking {(89 1/3)/670} = 2/15 of an hour or eight minutes, to reach either end of Orbitsville.  
(Light takes about eight minutes to reach the earth from the sun, and the distance between them, typically quoted, is about ninety-three million miles.)

Orbitsville might have a transparent band along the major axis of its shell. A space-ship, out-side, is moving in line with this at a fair fraction of light speed (say, one-half the speed of light). Relative to itself, it sees Orbitsville heading off at that 1/2 of light speed. But the space-ship doesn't agree with the internal astronomers that the shafts of sun-light, traveling from the middle of Orbitsville, arrive the same time at either end.  
What is, from the space-ship, the forward-moving end of Orbitsville appears to be distancing itself from the suns rays. So, the space-ship measures a time longer than eight minutes for the sun-light to reach the end moving forward, relative to the space-ship.   
Whereas, the aft end appears to be moving to meet the sun-light, so the space-ship measures it taking less than eight minutes. Mid-Orbitsville clocks show it took the same eight minutes for light to reach either end of Orbitsville. But the space-ship clock will show an inequality, dependent on how close its relative motion to light speed.

* * *

### Lorentz addition and subtraction of velocities.

To top.

Using the figures given, this inequality can be worked out by the arithmetic of the Lorentz transformations. These transformations also show that the space-ship will see Orbitsvilles sun-light has a constant speed, whether a light ray is moving from, or towards, the space-ship. This is an axiom of the special theory of relativity. We show this first.

Suppose the space ship passes the exact middle of Orbitsville, just as a night shield, covering its sun, opens before it. Light is made visible to either end of Orbitsville, at the same time. But, from the space-ship point of view, that does not mean both ends receive the light at the same time.

The common sense (of the Galilean transformations) would have us believe that if the relative motion of Orbitsville and space-ship is 1/2 light speed, then we have to add or subtract that speed, with respect to the light speed, depending on whether a light ray is moving against or with the direction of Orbitsville, with respect to the space-ship.

However, the Lorentz transformations, for the addition and subtraction of velocities, ensure that the space-ship sees light speed stay the same in both cases.

Take the case of the light ray moving against Orbitsvilles away-ward moving direction from the space-ship (parallel to it). This requires the Lorentz transformation for the addition of velocities. This is a bit more complicated than the Galilean transformation, that simply gives speed of light plus Orbitsvilles relative motion of one-half the speed of light.

Recalling my early efforts at understanding the Lorentz transformations, the notation for these formulas didn't follow the way, I expected. The variables, u, t, x refer to Orbitsville observation. In order to make the arithmetic work out, I found that I had to make the indexed variables, u', t', x' refer to (differing respective values of) the space-ships observations, of light rays moving relatively towards and away.

Say velocity u' may refer to either of the differing speeds of two Orbitsville light rays, going both ways, seen from the space-ship. And velocity u is that light ray, which is no different, to those stationed in Orbitsville, from the ray moving to the other end of its shell. Those standing in Orbitsville have no relative motion to (add to or subtract from) the light rays going to both ends. Which end is forward or backward has no meaning to them. In this case (as well as the other case, we shall deal with next), the velocity, u, of the light ray, within Orbitsvilles stationary frame-work, can be set at a "c" for celerity (or constant): u = c.  
Let v equal the relative motion, of Orbitsville to space-ship, given as one-half light speed or c/2.

The "aft" end of Orbitsville is only the view of the spaceship seeing what appears to be the back of Orbitsville moving away from it. Tho, front and back may have no meaning to the inhabitants of Orbitsville.   
The speed of the sun-light ray, moving in the opposite (aft) direction, within Orbitsville, combines with the speed of Orbitsville moving away from the spaceship. This combined speed is given by the Lorentz transformation for addition of velocities:

u' = (u + v)/{1 + (uv/c²)}.

To simplify the working, light speed is often treated as unitary, or c = 1. This makes v = 1/2, and u = c, in this example. Therefore:

u' = (1 + 1/2)/{1+(1 x 1/2)/1} = 1.

Or, u' = c.

Therefore, the space-ship measures u', the speed of the light ray, moving against the away-moving Orbitsville, as just c, the constant speed of light, irrespective of an addition of relative motion.

The same applies for the light ray traveling to the opposite end of Orbitsville. (This is the "forward" or "front" end from the spaceship view-point.) This ray is traveling with the relative motion of Orbitsville from the spaceship. This means that it is playing catch-up to the forward end of Orbitsville, at one half light speed.

The common sense or Galilean transformation is that the combined velocity would be one minus one-half equals a velocity of half unitary light speed for the forward ray, relative to the space-ship.

But the Lorentz transformation, this time, for the subtraction of velocities again ensures the space-ships relative motion does not allow it to see light move at other than its constant speed.  
In our example, the terms have the same meaning as before:

u' = (u-v)/{1-(uv/c²)}

u' = (c-v)/{1-(cv/c²)}

u' = (1 - 1/2)/{1-[(1 x 1/2)/1] }

u' = 1 = c.

Therefore, the Orbitsville sun-light moving relatively towards the space-ship is still measured at lights constant speed. This is the same velocity the space-ship measured for the Orbitsville sun-lights away-moving beam.

* * *

### Lorentz addition and subtraction of times.

To top.

The Lorentz transformations of time show why observers in relative motion, that is a fair fraction of light speed, cannot agree on the time they observed an event.   
All standing in Orbitsville are agreed that both light rays, from center to either end, took (time, t, of) 8 minutes, or 2/15 of an hour, to cross (distance, x, of) 89 1/3 million miles, at lights constant speed, c, which is about 670 million miles per hour.

But the space-ship clock will show different times, t', that it took the two light rays to reach the ends of Orbitsville. The away-moving light-ray will be timed as taking longer than eight minutes, as the Lorentz addition of times shows:

t' = {t+(xv/c²)}/{1-(v²/c²)}^1/2

= t{1+(uv/c²)}/{1-(v²/c²)}^1/2.

Distance, x, equals velocity, u, multiplied by time, t equals 8 minutes. Also u = c. If c = 1, and v = 1/2, the square of v equals 1/4. So:

t' = 8(1 + 1/2)/(1 - 1/4)^1/2.

t' = 8(3)^1/2 ~ 13.9.

That is, the space-ship clock will record the away-moving Orbitsville light ray as taking about 13.9 minutes (to one decimal place) to catch up with the "forward-moving" end of Orbitsville, instead of the 8 minutes that Orbitsville clocks register.

Now we find out the time the space-ship clock tells us it takes the light ray, that is moving to meet the aft end of Orbitsville. From the space-ship view, the aft is moving, at half the speed of light, toward this ray, and so reducing the time they take to meet.  
Using the Lorentz subtraction of times:

t' = {t-(xv/c²)}/{1-(v²/c²)}^1/2.

Following a similar procedure to the previous example for Lorentz addition of times:

t' = 8 (1 - 1/2)/(1 - 1/4)^1/2.

t' = 8/(3)^1/2 ~ 4.6.

This time, t', of about 4.6 minutes (to one decimal place) is the space-ship clocking of the light ray, whose relative motion makes the apparent aft of Orbitsville appear to come to meet the ray.   
Traditional Galilean relativity loses accuracy for calculating relative motion, at speeds approaching light. The Galilean transformation gives a simple subtraction of light speed minus half light speed. This is one minus one half, which is half light speed or four minutes travel time of the Dyson sphere aft, from the spaceship view-point, to meet its central sun ray.

By the Lorentz transformation, for that specific relative motion approaching light speed, the spaceship clock measures about 4.6 minutes, somewhat over half the 8 minutes time, t, that an Orbitsville clock shows for the same journey. With this other Orbitsville light ray, directed in the opposite direction, simultaneous time has been lost, again. Because of their high relative motion to light speed, the space-ship observers cannot agree with the Orbitsville observers when the light ray reaches from center to either end of Orbitsville.

* * *

### Lorentz addition and subtraction of distances.

To top.

Besides the disagreement over times between the space-ship and Orbitsville observers, they also disagree over distances. Distance equals velocity times time. Knowledge of any two of these variables will tell us the third. The previous two sections have already found the respective velocities and times measured by the two lots of observers. So, the respective distances, measured by their respective "rods," follows from that information:

In Orbitsville, the distance, x, from center to either end is eighty nine and one-third million miles. The space-ship sees one light ray appear to have to chase the end of Orbitsville moving away from the space-ship, thus elongating the distance, x', that the space-ship rod measures.

Or, x' = u'.t' = c.t'. This equals the light speed, of 670 million miles per hour, times (2/15)(3^1/2) hours. (Eight minutes is 2/15 of an hour.) Therefore, x' works out at about 154.7 million miles (to one decimal place).

For the sake of completeness, we can repeat this result, by the Lorentz addition of distances:

x' = (x+vt)/{1-(v²/c²)}^1/2.

x' = {(89 1/3)+([670/2] x 2/15)}/{1-(1/4)}^1/2.

x' = {(268+134)/3}(2/3^1/2)

x' = (134)(2/3^1/2) ~ 154.7.

This cross-checked working for x' again equals about 154.7 million miles for the space-ships "expanded" rod reading.

Now to the case where the space-ship rod or spatial measure appears to "shrink" to less than the Orbitsville rod register of eighty-nine and one-third million miles. Again the simplest solution of x', now as the shrunken distance, is u'.t', where u' still equals c, and where t' equals (2/15)/3^1/2 (hours or about 4.6 minutes).   
This puts distance x' as at about 51.6 million miles.

Had we not had the benefit of both velocity and time information, we might have had to find this result by starting with the Lorentz subtraction of distances:

x' = (x-vt)/{1 - (v²/c²)}^1/2.

Or,

x' = t(u-v)/{1 - (v²/c²)}^1/2.

Where u = c = 670 million miles per hour.

x' = (2/15)(670/2)2/3^1/2.

x' = 268/{3 x (3^1/2)}.

x' ~ 51.6.

This repeats the result we got directly because we happened to have the information for x' = u'.t'.

In the above examples, we illustrated the special relativity principle of the constant speed of light, despite an observer adding or subtracting the other observers relative motion to it. In general, observers respectively measured velocities, u' and u, need not be of light beams, whose constant speed they are both bound to equal, but can measure things at slower velocities, so that, like distance and time, velocities are not generally agreed by observers in relative motion, significantly approaching the velocity of light.

* * *

### The Interval.

To top.

The Interval, for our example, also illustrates that it is a particular case. The Lorentz transformations, as exemplified above, show that observers in high speed relative motion usually measure different distances and times for a given event.  
The Interval recovers a common measurement between observers of an event. But it is a combined space-time measurement, upon which they depend for agreement as to the "where-when" something happened.

The Interval measures a four-dimensional space-time common to observers. Our simplified example only dealt in one dimension of space, x, and the one time dimension, t. From the above sections, the Orbitsville co-ordinates are x and t, and the space-ship co-ordinates are x' and t'.

The square of the Interval, I, equals:

I² = x² - c².t² = x'² - c².t'².

Our example is of the Orbitsville observers and space-ship observers of light beams. In this measure of Orbitsville sun-light, both sets of observers measure an event with the constant velocity for light. That is both u = c and u' = c.

In this particular case, since x = ut and x' = u't', the Interval for both observers is zero, because the commonly measured event is at the speed of light. Slower events can be measured, in which case the observers local velocities, u and u', may differ. Then the Interval will not be zero. The general point is that the Interval is always the same value for both observers.

* * *

### Kinematics and dynamics.

To top.

My explanations of special relativity were in terms of "kinematics" or motion, not "dynamics," which further involves the concept of mass or energy.

Fortunately, the conversion of the Lorentz transformations, from their kinematic form to their dynamic form, is fairly straight-forward.

The Lorentz transformations correspond different observers differing distance and time measurements of an event. Call these observers differing distances, x and x', and differing distances , t and t', respectively. Consequently, the differing velocities of a body, they observe, will be, say, u = x/t and u' = x'/t'.

If the mass of the body is also to be considered, then an observer, who is at rest relative to the motion of the body, will measure its "rest mass."   
In classical mechanics, there only is a rest mass: the mass is deemed constant. In relativistic mechanics, the mass of a body is known to noticably increase as its motion noticably approaches the speed of light. (The relevant Lorentz transfomation shows that a body would have to be of infinite mass to reach the speed of light. This is why light speed is natures speed limit.)

Essentially, the Lorentz transformations for distance and for time are both multiplied by the same quantity, the mass divided by the time, to convert them into Lorentz transformations for momentum and energy, respectively.

Momentum, p, is mass times velocity. In the system of mechanics, distance has three dimensions, the x, y and z directions, to refer to all spatial motions. Therefore, velocity, as distance over time, also has direction with reference to the three co-ordinates. For simplicity, we have considered observers measuring events, which differ only in the x-direction, according to their respective measurements. That is to say where distances y = y' and z = z' between respective observers. But their respective measurements do not make x equal to x'. So, velocities, u and u' also need not be equal.

Consequently, the Lorentz transformation for momentum is in terms of differently circumstanced observers differing measures of both mass and velocity, or: p = mu and p' = m'u'.

The differing times, observers measure, are proportionate to the differing masses they measure: t'/t = m'/m. Note, also, that E = mc² and E' = m'c², where E and E' are the respective energies measured by the observers, according to the famous equation.

Then, multiply the Lorentz transformation for distance, x, by m/t, and x' by m'/t', for the Lorentz transformation between observers differing measures of momentum (in the x-direction):

mx/t = mu = p = m'(x' + vt')/ t'(1 - v²/c²)^1/2   
= (p' + vE'/c²)/(1 - v²/c²)^1/2.

The same procedure, of multiplying by mass over time, is followed for converting the Lorentz transformation of time into the Lorentz transformation of energy:

tE/c²t = ( t'E' + vx'E'/c² )/ c²t'(1 - v²/c²)^1/2.

This simplifies to:

E = (E' + vu'm')/(1 - v²/c²)^1/2

= (E' + vp')/(1 - v²/c²)^1/2.

The Lorentz transformation for (addition of) velocities follows from dividing the Lorentz transformation for distance by that for time:

u = (u'+v)/(1+vu'/c²).

In dynamics, compare the result of dividing momentum by energy. The denominators of both transformations are the Lorentz contraction factor, which cancels:

p/E = (p' + vE'/c²)/(E' + vp').

Or: u.m/mc² = m'(u' + v)/m'c²(1 + u'v/c²).

This reduces to the Lorentz transformation for (addition of) velocities.

Furthermore, the Minkowski Interval, as well as the Lorentz transformations, convert from kinematics to dynamics, in a straight-forward manner. The Minkowski Interval is a four-dimensional space-time geometry. It is like Euclid geometry in three dimensions. But time is treated like a fourth dimension of space. The only formal mathematical difference between the three space dimensions and the one time dimension is that the time is multiplied by the square root of minus one.

This factor gives the time, t, the opposite sign to the three space dimensions x, y and z, when the Interval extends Euclid geometry, of theorem Pythagoras in three dimensions, into a 4-D space-time version of the theorem.

The Lorentz transformations are the means by which an observer changes their measurements into those of differently situated observers of the same event. The point of the Interval is that it combines their respective measurements into a "space-time" measurement, of a given event, that is one and the same for all observers, in uniform relative motion.

As for the Lorentz transformations, the Interval, I, is simply given here for only one dimension of space and the one dimension of time. This is:

I² = t²(c²-u²) = t'²(c²-u'²)

Or:

I² = t²c²-x² = t'²c²-x'².

The space-time Interval changes to its momentum-energy Interval version, as with the Lorentz transformations, thru replacing time by mass:

(mc)²t²(c²-u²)/t²c² = (m'c)²t'²(c²-u'²)/t'²c².

Or: (E/c)² - p² = (E'/c)² - p'².

* * *

### Remarks.

To top.

I said that fortunately the conversion from special relativity kinematics to dynamics is straight-forward. I should add that this statement is only true from the practical point of view of "Shut-up and calculate." This phrase sums-up what students of quantum theory felt to be the attitude of their task-masters. Two veterans, only at the end of their working life-times, published a reminiscence. At least one, when a fresh-faced student, met the aged Einstein. Their book tried to spell-out the fantastic assumptions to quantum theory, that they had been rushed past, in their education and careers. (I believe this was: Quantum Enigma, by Bruce Rosenblum and Fred Kuttner.)

Not even special relativity is free from this aspect of "Never mind the principle, feel the practise." Richard Feynman was the inventor of Quantum Electro-Dynamics (QED) the reformulation of quantum mechanics, that has steadily gained ground in physics.

Feynman was prompted to look into this symmetry between special relativity kinematics and dynamics. Recognised as perhaps the most brilliant mind of his generation, he found out that he couldn't do it. He had the honesty to admit that he couldn't explain it.

Indeed, he told his students that they may have been expected to know by example that science depends on honesty. But he was telling them anyway, lest there should be any doubt about the matter.

# Lorentz transformation and Minkowski Interval geometric means and their dispersions, not manifest in classical physics.

Table of contents.

#### Sections:

Statistics and Relativity.

Lorentz transformations of geometric means.

The Interval geometric mean and dispersions.

### Statistics and Relativity

In the nineteenth century, Laplace produced his treatise to estimate ranges of error in astronomy, and so get nearer to some definitive measurement. Likewise, the Gaussian curve is also called the error curve. Clerk-Maxwell statistical theory, of molecular motion in the mass, was designed as a large scale approximation of molecular collisions, assumed, on their microscopic scale, to obey Newtonian laws. Einstein belonged to this tradition of assuming these laws were a definitive kind of law, tho in need of revision. Whereas statistical laws were supposed to be second-best approximations. Hence, his famous debates, with Niels Bohr, challenged (unsuccessfully) that statistics gives a complete explanation of quantum mechanics.

Two of Einsteins three famous papers, of 1905, are avowedly statistical: the study of Brownian motion, explained by random molecular motion, and the quantum statistics of the photo-electric effect. So, it is especially ironic that his third paper on Special Relativity also seems to have an underlying statistical explanation. That is contrary to the traditional view that SR is a deterministic explanation, like those of classical physics, which it modifies.

### Lorentz transformations of geometric means.

Special relativity can be expressed in terms of geometric means because geometric series are described, by tracking the motion of an object, significantly approaching light speed, by way of an exponential increase in energy driving it, with a corresponding increase in its mass, while only supplying, by diminishing returns, an increase in its velocity.

Take the Lorentz transformations of time. These are two complementary formulas by which two observers transform or translate their local times (t and t'), which will differ significantly with respect to an observed event at velocities significantly approaching the speed of light.

The observers respective velocities are u and u'. In classical physics of the Galilean relativity principle, their relative velocity, v, is simple addition or subtraction of their two velocities, as they go in opposite or the same directions, respectively. Coming to measure velocities of a significant fraction of light speed, a revised formula was needed to accomodate the fact that no combination of a velocity with light speed could ever exceed light speed, c.

If light speed was set at unit distance traveled per period of time, (c = 1) then a light beam carrier itself traveling at one-quarter that distance per time period, could no-how effect a combined speed of one and one-quarter, or any speed in excess of light.

The Lorentz transformations of velocities are usually derived from combining the transformations for space and for time. First, the time transformations, numbering the first equation, (1):

t = t'(1 - u'.v/c²)/F

where F is the so-called contraction factor:

F = (1 - v²/c²)^1/2.

Conversely, equation (2):

t' = t(1 + u.v/c²)/F.

The times, t' and t, can be shown as geometric means, that transform into each other, by combining their converse equations. An average implies a range of items or values that it is the average of. Therefore, the Lorentz forms may be rendered as such ranges.

Expressing the times, t and t', as a ratio means that the contraction factor, F, in both their equations, cancel. Hence (3):

t/t' = t'(1 - u'.v/c²)/t(1 + u.v/c²).

Therefore (4),

t²/t'² = (1 - u'.v/c² )/(1 + u.v/c²)

(4b):

= {1 - [(u'.v)^1/2]/c}{1 + [(u'.v)^1/2]/c}/{1 - i[(u.v)^1/2]/c}{1 + i[(u.v)^1/2]/c}.

The numerator and the denominator of the equation have both been factorised in terms of one plus and one minus the ratio of [(u'.v)^1/2]/c and i[(u.v)^1/2]/c, respectively.   
In the latter case, the symbol, i, means the square root of minus one, to make possible the factorisation of the denominator.

If we were taking an arithmetic mean of these two ranges plus or minus one, then their average would be estimated in both cases to be just one. This assumes a constant ("arithmetic") increase in values from the given lowest to highest terms in the range. The arithmetic mean is calculated by adding these two terms and dividing by two, resulting in an answer of one.   
This taking the arithmetic mean also works for the denominator even tho, i, the square root of minus one is involved, because the two imaginary terms cancel each other out.

In this instance, the arithmetic mean is not a suitable average to take, because we are not dealing with constant increases. Instead, the Lorentz transformations, effectively, take into account geometric distributions, one way or another.   
The cumulative decrease, in the rate of velocity increase of an object, corresponds to its cumulative increase in mass, as light speed is approached. A geometric mean measures the average, or most representative value, to a distribution of high velocities as they approach light speed.

The geometric mean can be found by multiplying the end values of a (regular) distribution range and taking their square root. In equation (4), of t²/t'², the numerator and denominator are both multiplications of range end values: the two factors in curly brackets are the lowest and the highest velocities. Hence, the square root of the numerator gives a geometric mean, as does the square root of the denominator. But these respectively, are in proportion to t and t', which are, therefore, a geometric mean times ratio.

That is (5):

t/t' = [(1 - u'.v/c²)/(1 + u.v/c²)]^1/2.

Similarly, for Lorentz transformation of geometric mean distances.

(6):

x = t'( u' - v )/F and x' = t( u + v)/F.

(7):

x/x' = t'(u' - v)/t(u + v) = ut/u't'

= t'u'(1 - v/u')/tu(1 + v/u)

= x'(1 - v/u')/tu(1 + v/u)

(x/x')^2 = (1 - v/u')/(1 + v/u).

(8):

x/x' = {(1 - v/u')/(1 + v/u)}^1/2.

Equation (8) can be considered a ratio of geometric means, as it is possible to factorise both the numerator and denominator, like so, (9):

x/x' = [{(1 - {v/u'}^1/2)(1 + {v/u'}^1/2)}/ {(1 + i{v/u}^1/2)(1 - i{v/u}^1/2)}]^1/2.

Equation (8) gives the respective ranges, averaged by those geometric means. The denominator factorises in terms of imaginary coefficients, i, meaning a turn of ninety degrees into a second dimension.

Thus, a ratio of velocities, u/u', could be expressed as the x/x' ratio of distance geometric means, divided by the ratio of time geometric means, t/t'. That is, u/u' equals equation (8) divided by equation (5).

It's possibly more interesting to derive the Minkowski Interval from the Lorentz transformation time ratios, from (4):

(t/t')² = (1 - u'v/c²)/(1 + uv/c²)

= (c² - u'v)/(c² + uv).

Therefore (10):

t'²(c² - u'v) = t²(c² + uv).

The Interval squared is:

I² = t²(c² - u²) = t'²(c² - u'²).

The two equations are equal if -v = u and v = u'. This equality is obtained from a relative velocity, v, between the two observers, that has an opposite sign for their opposite directions. But different magnitudes for the local velocities, u and u' can be compensated by suitably varying the times, t and t', as in the case of the Interval.

This derivation of the Interval, from the Lorentz transformations, dispenses with the contraction factor, or, rather, generalises it. The contraction factor just relates local observers measurements. The Interval gives them a (space-time) measure common to all.

When the Lorentz transformations were treated as ratios of one to another observers values of time (or distance) this canceled the contraction factor found in both observers versions of the Lorentz transformation for time (or distance). But the contraction factor itself has the form of a geometric mean. (This started me on the possibility of a statistical basis for special relativity.)

Historically, the contraction factor was the ad hoc gloss on the null result of the Michelson-Morley experiment, that over-looked the importance of choosing the right average, in what was implicitly a statistical calculation.

The contraction factor, F, comes into our present reckoning, when a special case for the Lorentz transformations of time is considered. That is when u' = v and therefore u = 0.

From equation (1), t = t'(1 - u'.v/c²)/F.   
When u' = v, t = t'( 1 - v²/c² )/F.   
Therefore, t = t'F, because F is the square root of (1 - v²/c²).   
Taking the ratio t/t' = F/1, if t is taken to be F as geometric mean, then the value of t' = 1.

For equation (2), t' = t(1 + u.v/c²)/F, reduces to t' = t(1 + 0)/F   
when u = 0.   
Then the ratio of equations (1) to (2) for t/t' can no longer be expressed as the ratios of two geometric means, but as only one geometric mean, in the familiar guise of the contraction factor.

* * *

### The Interval geometric mean and dispersions.

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If the Lorentz transformations and the Interval are geometric means, then they must be means or averages to distributions. A mean represents a dispersion of values about it. Each Lorentz transformation, also the Interval, can be re-written in terms of the lower and upper bounds of a dispersion, by which the mean or average is calculated.

Most simply, subtract the geometric mean lower bound (value) from its upper bound. This is equivalent to adding the dispersion above the average, to the dispersion below the average, for the total dispersion.

For a steady increase in energy applied to a body approaching light speed, the velocity only takes on a geometricly (or exponentially) decreasing increase. One such event should only represent one such geometric distribution of velocities. Why isn't its measurement represented by only one geometric mean, instead of all the geometric means related by the Lorentz transformations?

This is a statistical re-working of the original question, that Einstein asked of Special Relativity in 1905. Then, the question was why do observers have different space and time measurements of a same event in high energy physics?   
The answer came in 1908 with the Minkowski Interval. Observers do have a common measurement of the event, if it is treated as one combined four-dimensional space-time event.

The Interval is the Pythagoras theorem extended from three dimensions of space to four dimensions of so-called space-time. As is familiar from school, the theorem in two spatial dimensions may be given as r² = x² + y². That is the square of a triangle hypotenuse equals the sum of the squares of the right-angled sides. It also holds in three spatial dimensions as r² = x² + y² + z². The spatial dimension of distance equals velocity multiplied by time: ut. So, the theorem can be re-stated as: (ft)² = (ut)² + (vt)² + (wt)², with f, the three-vector velocity.  
(Please note: The velocity, v, here is not the relative velocity, v, in the Lorentz transformations. Tho, the velocities u and u' used here may be considered the same as the u and u' velocities in the Lorentz transformations without the extra two spatial dimensions taken into account.   
In other chapters, I use velocity, u, to signify velocity, either in one or up to three dimensions, rather than employ a new letter, f, for the velocity three-vector.)

Theorem Pythagoras has the property that the hypotenuse of a triangle, considered, say, as the constant radius, r, of a circle, can swing round the circle, so that the other sides of a right-angled triangle it forms, the x and y axes of a graph, whose origin is the circle center, change their lengths, while the direction-changing radius stays the same length. The same applies for a radius, r, moving in a spheres three dimensions, measured by x, y and z axes.

This means that the square of the radius equals the sum of the squares of its three trigonometric sides not just in one set of co-ordinate positions but in any set of such positions that any observer may arbitrarily take-up with respect to the sphere. (The sphere, being symmetrical, gives no indication where the x, y and z co-ordinates should be. The radius vector always expresses the magnitude and direction of any given event, on the circumference of circle or sphere.)

Thus, theorem Pythagoras not only holds for r, with respect to rectangular co-ordinates, x, y, z, but also with respect to polar co-ordinates, of a radius swung round some angle (say angle Q) from the center or origin of the sphere. Equally arbitrary alternative co-ordinate positions in the sphere may be denoted by indices, x', y', z', so that:

r² = x² + y² + z² = x'² + y'² + z'²

= (ut)² + (vt)² + (wt)² = (u't')² + (v't')² + (w't')².

We are now almost at the Minkowski Interval, needing just a fourth term to these three. Tradition describes this as time supplying a fourth dimension. This fourth term, when one looks at it, is actually another distance variable, being the velocity of light, c, multiplied by the time, t or t', depending on the co-ordinate frame of reference. As light speed is constant, only the time part, of the distance measure, ct, is variable. So, it is justifiable to call the fourth dimension a temporal dimension (essentially).

The peculiarity of the fourth dimension term is that the squares of ct or ct' are not added to the other three terms but subtracted.

Usually left as a square of its value, the Minkowski Interval, I, another distance term, is thus:

I² = (ct)² - {(ut)² + (vt)² + (wt)²} = (ct')² - {(u't')² + (v't')² + (w't')²}.

(As mass is proportional to time, in Special Relativity, an analogous form for the Interval is derived in terms of energy and momentum.)

From the point of view of a statistical interpretation of special relativity, this unexpected negative is providential, because it enables the Interval to be simply re-stated as a geometric mean. Indeed, that is really the whole point of the mysterious negative fourth dimension.

This is seen by refering the three spatial dimensions to their combined equivalent in magnitude and direction, that is to their vector, sometimes called three-vector: a distance radius, say, r = ft = f't'.

The Interval equation, in terms of velocity three-vector, f, is:

I² = (ct)² - f²t² = (ct')² - f'²t'².

This can be re-stated with the Interval in the form of a geometric mean:

I = {(ct - ft)(ct + ft)}^1/2 = {(ct' - f't')(ct' + f't)}^1/2

Or:

(I/c) = {(t - ft/c)(t + ft/c)}^1/2 = {(t' - f't'/c)(t' + f't/c)}^1/2

This answers our question about any given observed geometric distribution of velocities or masses having the same geometric mean for different observational frames of reference. That is, when observers are in uniform velocity to each other (or "uniform relative motion") as far as special relativity is concerned. The Interval provides that commonly observed geometric mean.

The dispersion about the geometric mean Interval, as for the Lorentz transformations, is a difference of an upper bound from a lower bound of dispersion, namely equation:

(ct + ft) - (ct - ft) = 2ft, for the dispersion to Interval, I

Or, since I and c are both constants, for the dispersion to I/c:

(t + ft/c) - (t - ft/c) = 2ft/c.

In classical mechanics, a low velocity and time, or a short distance, with respect to light, creates no observable dispersion, and thus does not reveal this statistical basis of physics.

With the Interval, a difference between upper and lower dispersions, about the geometric mean, also can be taken.

The dispersion above the Interval, I, is:

(ct + ft) - I.

The dispersion below the Interval is:

I - (ct - ft).

Subtracting the two dispersions:

(ct + ft) - I - {I - (ct - ft)} = 2(ct - I).

This difference, of upper and lower limits, means that when velocities become small compared to light speed, the Interval, I, approaches ct and the dispersal difference approaches zero.

Why bother to take a dispersion difference as well as a dispersion sum total? The two options may correspond to relative motion, effectively adding velocities by moving in contrary directions or subtracting velocities moving in the same direction.

This is a change from the conventional meaning of relative motion as between different observers local velocities. The relative motion, of above-average dispersions, adds observer velocities to light speed, while below-average dispersions subtract those observer velocities from light speed.

It may be that Galilean relative motion, as simple additions or subtractions of velocities is statisticly re-invented in simple additions and subtractions of high energy motion dispersions.

On this reasoning, classical physics is not more definite and precise than statistics, as was assumed. Rather it is a less than precise approximation to statistical measurement. (As is well known, this is the quantum physics view that gradually prevailed in the twentieth century.)

Neither does special relativity belong to some supposedly ultra-statistical precision of classical physics. Rather, my statistical view-point is that special relativity is precisely the condition of statistical dispersion becoming apparent.

Much the same considerations apply to mass, as time. For a steady increase in speed of a body, significantly towards light speed, the mass of the body increases geometricly (or exponentally). The dynamics Interval measure of energy and momentum is similar in form to the kinematics Interval for time and distance and can be likewise re-formulated in terms of geometric mean and dispersion.

# Higher order Interval multi-geometry distributions.

Table of Contents.

The Interval is in terms of a vector, of the three dimensions of space, subtracted from the so-called fourth dimension, in terms of light speed multiplied by observers local clocks or time measures of an event.

A geometric mean may be taken not only of the extreme values but also of any number of inter-mediate values. Whatever the number of such values, they are all multiplied together. Then a root of the multiple gives the geometric mean. For two values multiplied, the root taken is the square root. For three items, the cubic root of the multiple gives the geometric mean. The number of the root is the number of values (on the range) multiplied. The geometric mean of four values in a range multiplies them and takes the quartic root, or takes the multiple to the power of one quarter.

Instead of assuming a two-valued range of velocities, say, from (1 - u/c) to (1 + u/c), we assume a four-valued range of velocities.

Where f and g are new velocity terms, let the four-valued range of velocities from least to greatest be: (1 - f/c), (1 - g/c), (1 + g/c), (1 + f/c). Their geometric mean is their multiple to the power of one-quarter, or:

{(1 - f/c)(1 - g/c)(1 + g/c)(1 + f/c)}^1/4.

This relates to a form like the contraction factor (but can be more generally applied, as part of the Interval): (1 - r²/c²)^1/2.

Why might we do this?

Taking a geometric mean of more values may give a more detailed result. Nevertheless, when we only calculate with the two end values, we still hope the result is not far out. And in equating the simple end-valued geometric mean to the geometric mean of a fuller range of values, we may assume that the simpler calculation happens to be in accord with fuller calculations.

Hence, equation (1):

{(1 - f/c)(1 - g/c)(1 + g/c)(1 + f/c)}^1/4 = (1 - r²/c²)^1/2.

Raising both sides of the equation by the power of four, we get (2):

(1 - f²/c²)(1 - g²/c²) = (1 - r²/c²)².

Therefore (3),

1 - (f²/c² + g²/c²) + f²g²/c²c² = 1 - 2r²/c² + r²r²/c²c².

Cancel the ones and multiply thru by c². Then (4):

f²g²/c² - f² - g² = r²r²/c² - 2r².

Suppose r² = fg. Then, r = (fg)^1/2.

Thus, for velocities significantly approaching light speed, the velocity, r, is the geometric mean velocity of a given observers velocities, considered as a velocity range from f to g.

For this to hold, then the square of the velocity, r, must also equal an arithmetic mean of the squares of the observers velocities, f and g.

That is:

r² = ( f² + g² )/2.

An average, that is a geometric mean, with an arithmetic mean element in it, expands into a distribution of terms that are themselves averages, including the geometric mean in one term and an element of arithmetic mean in another term. The contraction factor type geometric mean is an average of averages.   
Expansions involving more than two velocities show a refinement of this basic feature.

Taking the geometric mean, of a range of three velocities, would involve three multiplied factors. Two new range values, (1 - h/c) and (1 + h/c), would be multiplied by those on the right side of equation (1). In this case, the geometric mean of the six multiplied values would require a root to the power of one-sixth, for equation (5):

{(1 - f/c)(1 - g/c)(1 - h/c)(1 + h/c)(1 + g/c)(1 + f/c)}^1/6 = ( 1 - r²/c² )^1/2.

This simplifies to (6):

(1 - f²/c²)(1 - g²/c²)(1 - h²/c²) = (1 - r²/c²)^3.

Using the binomial theorem to expand both sides, (7):

1 - (f² + g² + h²)/c² + (f²g² + f²h² + g²h²)/c^4 - (f²g²h²)/c^6

= 1 - 3(r²/c²) + 3(r^4/c^4) - r^6/c^6.

As for the example of two velocities, for the three velocities, we assume the terms, on each side of the equation, correspond. Taking from the last term on each side, r^6 = (f²g²h²). So, r = (fgh)^1/3. The velocity, r, is the geometric mean of the three range velocities.

That assumes the other terms correspond. So, 3(r²/c²) = (f² + g² + h²)/c². Here, the square of the velocity, r, exhibits some arithmetic mean character, as the squares of the three range velocities are added and may be divided by three, from the left side coefficient.   
It seems significant that the first term, on either side, namely one, is the arithmetic mean of the two factors that each observer contributes, such as (1 - f/c) and (1 + f/c), in the case of velocity measure f. Likewise, for the other two velocities g and h.

The term corresponding to 3(r^4/c^4), that is ( f²g² + f²h² + g²h² )/c^4, combines arithmetic mean and geometric mean elements of averaging. Three terms added and divided by three constitutes an arithmetic mean: 3r^4 is their arithmetic mean. But those three terms are multiples. And the square root of a multiplied pair is their geometric mean.

Anyone looking at 3(r²/c²) = ( f² + g² + h² )/c² might think that, apart from the coefficient, 3, it involves theorem Pythagoras in three dimensions of space, with f, g and h, being velocities in the direction of x, y and z co-ordinates. And r² appears the hypotenuse squared, moving in a sphere, like the radius vector outcome of a tug of war between the three velocities at right angles to each other.

(The term "geometric" mean as a kind of average of a "geometric" series is not to be confused with "geometric" as in the geometry of space that yields the likes of theorem Pythagoras, with its use of an arithmetic sum of squares.)

When I first wrote on-line about these ideas of the Interval as a geometric mean, I didn't appreciate what I was, in effect, doing at this point. By introducing a second observed velocity, into a geometric mean-interpreted Interval, I was describing the Interval, of any given local observer, under-going a change in velocity, in other words, an "acceleration Interval."

This takes a step away from special relativity, for relative velocities between observers, towards a new formulation for relating accelerated motion between observers with different frames of reference.

Thus, it would be possible to have two different observers, relating two different velocities. If one observer measures velocities, f and g, the other measures, say, velocities, f' and g' (distinguished by indices).

The two velocities would have their own times. For more than two velocities, some notation, like t0, t1, t2, etc would have to be used, with corresponding times for the other observer: t'0, t'1, t'2 etc. (Apologies for the clumsy notation, Normally, the numbers would be written as subscripts. Current e-book formats [mercifuly] do not allow the W3 Math code. The rule, here, is that if you see the numbers appearing after a variable, like time, t, then you know the number identifies a term in a series. Whereas, a number, before a variable, is a coefficient,)

Then a simple (two-velocities) acceleration Interval would look like equation 8:

(I/c)^4 = (t0)²(1 - f²/c²).(t1)²(1 - g²/c²) = (t'0)²(1 - f'²/c²).(t'1)²(1 - g'²/c²).

And: (I/c)^4 = [T²{1 - (r/c)²}]² = [T'²{1 - (r'/c)²}]²,

where (t0)(t1) = T² and (t'0)(t'1) = T'².

Let us remind ourselves what the conventional (Special Relativity) Interval (of one velocity and time for each observer) looks like in this notation:

(I/c)² = (t0)²(1 - f²/c²) = (t'0)²(1 - f'²/c²).

This is identical to:

(I/c)² = T²{1 - (r/c)²} = T'²{1 - (r'/c)²}.

By putting times into equation 6, consider the Interval for three velocities and times per observer (equation 9):

(I/c)^6 = (t0)²(1 - f²/c²).(t1)²(1 - g²/c²).(t2)²(1 - h²/c²) = (t'0)²(1 - f'²/c²).(t'1)²(1 - g'²/c²).(t'2)²(1 - h'²/c²).

And: (I/c)^6 = [T²{1 - (r/c)²}]^4 = [T'²{1 - (r'/c)²}]^4,

where [(t0)(t1)(t2)] = T^3 and (t'0)(t'1)(t'2) = T'^3.

Consider the binomial expansion of equation 6, to equation 7, with the times inserted, to complete the Interval formula, for one observer (for brevity leaving out the other side of the Interval equation to a second observers related values with indexed times and velocities). Equation 10:

(I/c)^6 = (t0)²(t1)²(t2)²{1 - (f² + g² + h²)/c² + (f²g² + f²h² + g²h²)/c^4 - (f²g²h²)/c^6}

= T^6{1 - 3(r²/c²) + 3(r^4/c^4) - r^6/c^6}.

There is nothing special about using three velocities, f, g, h, to mark out a range of velocities. One could use any number. This means that change in velocity, or acceleration, could be measured to any degree of accuracy.

In equation 10, it happens that three velocities, f, g, h, forming a geometric series, on a one-dimensional range, that can be averaged by the geometric mean, also partly resemble velocities, say: u, v, w, on the three dimensions of space. To approximate the conventional Interval, expressed in terms of u, v, w, (rather than a three-vector, say, r) the two last terms of the expansion have to be small enough to be ignored.

Looking at the composite term from equation (10), (f²g² + f²h² + g²h²)/c^4, note that if two of the terms are small, compared to light speed, then the whole factor is small and probably can be ignored. That applies to the fourth term, (f²g²h²)/c^6, if only one of the terms is small.

The remaining first two terms (in the curly brackets) resemble the conventional Interval. There are still the two times associated with the two velocity terms, negligible under certain conditions. It might be contrived that the two redundant times cancel out the Interval term from (I/c)^6 to the conventional (I/c)².

Thus, supposing {(t1)(t2)}² = (I/c)^4, then approximately:

(I/c)² = (t0)²{1 - (f² + g² + h²)/c²},

where one dimensional range velocities, f, g, h are approximately equivalent to the conventional Interval, in three-dimensions of velocities, u, v, w. Variables, f, g, h, could themselves be three-vectors, and together have their own three-vector, presumably in variable, r, on the other side of eqn. 10.

Altho f, g, h, are, at intervals on a (geometric or non-linear) scale of one dimension, each term can itself be a three-vector resultant of velocities in three (linear) dimensions, like u, v, w.  
Thus velocity f could have co-ordinates u1, v1, w1.   
Velocity g has co-ods. u2, v2, w2.  
Velocity h has co-ods. u3, v3, w3.

This could be continued indefinitely. Say the next term in the geometric series scale is velocity j with 3-D co-ods. u4, v4, w4.

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An interesting feature of this statistical derivation of an acceleration Interval is that relativistic time appears in a probability distribution.  
In thermo-dynamics, the direction of time, or times arrow, is attributed to the fact that if you saw a broken bottle re-assembling itself, you would know it was a film being played in reverse and not actually happening. The reversed film shows how the re-assembly might happen in principle. In all probability, you may safely assume it never happened.

Classical physics only accounts for time being reversible in principle, without explaining the probability of time having one direction. Philosophers have reasoned that time no more has direction than space.

Also, an explicit statistical treatment of relativity should be more compatible with quantum theory.  
It makes sense to consider the probabilities different space-times occur, as exemplified in a binomial or other statistical distribution of geometries. The break-down of continuous space-time, at the Planck scale, into a quantum foam of abruptly changing curvatures, or indeed massive astronomic distortions of the fabric of space-time, might so be represented as probabilities of the incidence of different space-time curvatures.

The famous equation, E = mc², is essentially a binomial expansion of the contraction factor, for two locally observed masses of a body, taken to the first two terms, as later terms are insignificant.

Mass is proportional to time, respectively in the dynamic and kinematic versions of the formulas for special relativity. It may be that the conventional Interval is also an approximation of a statistical expansion, such as equation 10.

The binomial distribution here is a distribution of logicly possible combinations of variables. These form kinds of geometrical structure or "geometries" such as Euclid geometry. Also, another term plus the Euclidean term formed Minkowski Interval, a further geometry. The distribution clusters about an average geometry, such that its structure is about half arithmetic mean and half geometric mean, considered as an average as well as a geometry.

Different geometries can be considered as different averages. And the binomial theorem, on which the expansion is based, can be considered as an average of averages, which is also a geometry of geometries. In other words, geometries may be considered as averages, which can be expanded from second order averages or second order geometries.

Special relativity postulates no privileged frames of reference within the context of Minkowski space-time. Minkowski geometry is itself to some extent a confining frame of reference to observers only in relative velocities to each other, excluding consideration of accelerated reference frames. Minkowski Interval is a privileged reference frame, in Special Relativity, which only works for the conditions, it exacts for observers, of uniform motion in a straight line.

Euclidean space is in three linear dimensions of flat space. The Interval does not add, but subtracts these three positive dimensions from a fourth dimension. This subtraction is enough to turn the Euclidean form into the Interval, as a geometric mean form, which is to say, a curved space. But one that always curves in the same way, according to Brian Cox and Jeff Forshaw (Why does E = mc² ?)

The conventional Minkowski Interval may be an approximation of some statistical expansion, that gives both an Euclidean treatment for uniform motion and a non-Euclidean or geometric mean treatment for non-uniform or accelerated motion.

This treatment may also imply no privileged classes of geometry frames of reference, (whether of Euclid, Minkowski, or presumably others). Different geometries (different classes of reference frame) may appear as terms in a random distribution represented by, and expanded from, a geometry of geometries, as an average of averages.

Suppose the binomial distribution that derives the Interval in its first two terms has its conditions changed. Suppose we multiply equation (10), by c², for equation (11):

(I^6)/c^4) = (t0)²(t1)²(t2)²{c² - (f² + g² + h²) + (f²g² + f²h² + g²h²)/c² - (f²g²h²)/c^4}

Suppose also that (f² + g² + h²) is as close as we like to c². This still allows a great deal of possible variation in the three velocities individually. But it effectively cancels out the first two terms, eliminating the Interval, as we know it. This still leaves the subsequent two terms which might be considered as an Interval, whose more or less geometric mean terms measure more or less curved space-times.

This final, fgh, term is a function of a geometric mean. This is an average of a series associated with curvature, the graphic description in geometry of acceleration or deceleration, characteristic of a geometric series. General Relativity, dealing with observers in accelerated reference frames, uses the geometry of curvature.   
The third term in the distribution, before the geometric mean term, is the next closest to a geometric mean, but is not merely a multiple of all the velocities. Different velocities are partly multiplied and partly added.   
So, the binomial distribution of terms progresses from arithmetic mean to geometric mean, with more or less one or the other mixed in, as the series progresses. When the binomial theorem is expanded, with more than three velocities, further inter-mediate terms are introduced, suggesting a more and more refined distinction between the original contrast of a conventional Interval and an acceleration Interval of curved space-time.

This statistical approach to an acceleration Interval might have the advantage of applying finite mathematics to problems in General Relativity, whose laws break down when its equations produce infinities, encountered as singularities in its solutions, such as at the origin of the Big Bang and at the destination of Black Holes.

Instead of a singularity, an infinitely dense point of zero spatial dimensions, statistics may come up with alternative, discrete formulations.

Classical mechanics is not the master science, it was once thought to be, not even with regard to the progress of mechanics itself. But it does seem a bit surprising that special relativity does not follow classical mechanics, in that the concept of acceleration follows in a straight-forward manner from space, time and velocity.

Special relativity deals in space, time and velocity, but acceleration, as a gravitational reference frame, seems to come out of no-where, based on a completely new theory, the general theory of relativity, on the basis of a principle of equivalence between acceleration and gravitation.

# The Minkowski Interval predicts the Michelson-Morley experiment; sub-luminal and superluminal connections (SLC).

Table of contents.

#### Sections:

The traditional Michelson-Morley calculation.

Galileo principles of relativity and force.

The geometric mean approach.

Michelson-Morley times adapted to masses.

The Interval demonstrates the Michelson-Morley experiment.

SR non-solutions of the M-M calculation.

Transformations between affected and unaffected light beam times.

Sub-luminal and superluminal connections (SLC).

* * *

### The traditional Michelson-Morley calculation.

The Michelson-Morley experiment (MMX) measured the speed of light on the assumption that the universe had an absolute motion, which light could go with or against, like a boat down or up stream. Alternatively, the light could go back and forth across the universal "stream" called the "ether". The experiment split a light beam into two journeys. These were reflected with mirrors, so that one journey was supposed to go up and down stream, the other journey across the stream, from bank to bank and back.

For both split beams, the distance traveled, one way, is d; the light speed is c, and the ether velocity is v, which is assumed to be relative to earth velocity, rather as someone in a boat sees the bank (of Earth) passing by, tho it is the (Ether) stream that is moving. The combined time, t', for light to travel up and down an ether "stream" would be d/(c+ v) plus d/(c - v).

This works out as 2dc/(c - v)(c + v), or,   
2dc/(c² - v²).

Unlike the up and down stream journeys, the back and forth journeys take the same time. The total distance traveled by a crossing light-speed "boat" is:

2dc/(c² - v²)^1/2.

(The chapter on the Michelson-Morley experiment showed how to calculate this, using theorem Pythagoras.)

Dividing this distance, both ways across, by light-beam cross-stream velocity, c, gives its total crossing time, t, as:

2d/(c² - v²)^1/2.

Thus, Michelson and Morley calculated the two times, t' and t, as a predicted out-come of maximal and minimal ether drag, respectively. But the fame of their experiment (repeated in all directions, at all seasons) rested on the fact that it showed the two times to be equal. The two times could only be made equal, in theory as well as practise, by introducing a factor, F (the so-called Fitzgerald-Lorentz contraction factor) so that the shorter time, t, equals the longer time, t', multiplied by the contraction factor.

Thus, F = t/t' = {2d/(c² - v²)^1/2} × {(c² - v²)/2dc}

= {(c² - v²)}^1/2/c = (1 - v²/c²)^1/2.

* * *

### Galileo principles of relativity and force.

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Galileo relativity principle derived from the experience of someone at rest on a moving boat relative to the bank. In the heyday of the train, we experienced this in the surprise felt, when sitting in a railway carriage, at the start of a journey, that the station platform appeared to lurch away in the opposite direction to the way taken.

The point of this odd feeling is that there is no way we can decide whether the train or the platform is the one that is really moving or really at rest. Neither situation is a privileged frame of reference.

The contraction factor was introduced to explain the unexpected result of the Michelson-Morley experiment that light always measured at the same speed.   
Consider two observers moving uniformly, at a significant fraction of light speed, in opposite directions, like passing trains. "Einstein proposed ...no measurement could determine which train was stationary and which was moving. That being the case, the equations of electricity and magnetism would have to appear the same on the two trains, and thus the speed of light must also be the same." (Robert Laughlin, "A Different Universe.")

In the Michelson-Morley experiment, I talk about light beams, from reading the old popular accounts of physics. Einstein was a formative influence on the idea of lasers, which took over, as the light beams of choice. White light is light mixed at different wave-lengths. Sound, at all different wave-lengths, is noise. A laser coheres light into a much stronger beam all of one wavelength and phase.

The Michelson and Morley calculation didn't agree with their experimental result of equal times for the split light beams on the two journeys. This was despite certain "ether wind resistance" considerations with respect to a back and forth journey directly into the ether wind and a back and forth journey across the wind.

In terms of the Galileo relativity principle, the relative motion of two observers velocities is straight-forward addition or subtraction. That won't do for the Michelson-Morley experiment.

The point may be that the Michelson and Morley calculation was based on the Galileo relativity principle, when it should have been based on the Galileo force principle, which became Newton law two of motion. Light is a force.

The Einstein photo-electric effect demonstrates this. Light of shorter wave-lengths has more energy and acts like more powerful bullets to knock electrons out of the surface of a metal. Only a few of them are enough to do this trick, whereas it doesn't matter how much of the lower energy light is played on the metal, its photons don't have the energy to dislodge the electrons.

Force is mass multiplied by acceleration. More generally, force is change of momentum over time. Change of momentum leaves the door open to special relativity recognising a possible change of mass as well as change of velocity. Einstein and Infeld, in The Evolution of Physics, say that Galileo opened the door to modern physics, when he recognised that force is not a function of motion but of change in motion, or acceleration.

When air-craft approach the speed of sound, they encounter a sound barrier. When bodies approach the speed of light, they encounter a light barrier. This becomes an ever increasing deterent force. Therefore, when motion significantly approaches light speed, it should be measured not in terms of Galilean relative motion, by adding (or from the other point of view, subtracting) observers velocities with respect to each other, but in terms of Galilean law of force, in this case light force, which is measured not in terms of velocity but change in velocity, which is acceleration or deceleration.

For a steadily increasing input of energy into a bodys motion, approaching light speed, there is a deceleration in the extra velocity, that the body achieves. The energy input increases the bodys mass, which would have to become infinite for the body to reach light speed.

The Michelson-Morley calculation takes an average time for a light beams back and forth journey but it takes the wrong average, an arithmetic mean, which deals in terms appropriate to the adding and subtracting of velocities. The average should be the geometric mean, dealing with change in velocity, in this case, deceleration with respect to a body significantly approaching light speed.

The Michelson-Morley experiment, with the averaging calculation based on Galileo acceleration principle of "force", means the geometric mean applies, instead of Galileo velocity principle of relative motion applying an arithmetic mean.   
Can the Einstein principle of equivalence of acceleration to gravity (the foundation of General Relativity) be related to the experiment?

* * *

### The geometric mean approach.

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However, averaging, by the geometric mean, instead of the arithmetic mean, of the light beam, split into two perpendicular journeys, predicts that the two beams will take the same time on their journeys, just as the Michelson-Morley experiment found. There is no need for the ad hoc explanation of a contraction factor to match the Michelson-Morley calculation to the experimental finding.

An average is the measure of the most typical or representative item in a range of values. The two MMX light beams, on their return journeys are two such ranges. The times calculated for light to travel down stream and to travel up stream are, respectively the quickest and slowest times, d/(c + v) and d/(c - v). In the earth-aligned return journey, these quickest and slowest times form the limits of a range of times for the down and up stream journey.

A range of values may form an arithmetic series, in which case its average is measured by the arithmetic mean, the most familiar kind of average. A range, that is in a geometric series, would be averaged by the geometric mean.

To obtain the geometric mean time, T', for the longer, down and up stream (or down-wind and up-wind) journey, multiply the quickest and slowest times and take the square root of the product:

T' = {d/(c + v) x d/(c - v)}^1/2 = d/(c² - v²)^1/2.

Now for the geometric mean time, T, for the shorter, back and forth light-beam journey. Well, the time is the same for the back crossing as the forth crossing. So, this time must also be the average time taken for the whole back and forth crossing. (Any kind of average, used, arithmetic mean or geometric mean, is bound to come out the same as the two identical crossing times.) That is:

T = d/(c² - v²)^1/2.

The average time taken, to cover one leg of the journey, is half the time taken (given above) to cover the whole distance. It so happens that this time for the back and forth journey is the same as the geometric mean time for the up and down stream journey.

Hence, T' = T = d/(c² - v²)^1/2

That implies that taking the geometric mean times for the two light beam journeys correctly predicts that light will take the same time on both journeys, in the Michelson-Morley experiment. This does away with the need for a supposed contraction factor.

If the geometric mean time is multiplied by the speed of light, c, then a distance value is obtained, which is a function of the familiar contraction factor, F:

c x d/(c² - v²)^1/2 = d/(1 - v²/c² )^1/2 = d/F.

The Michelson-Morley experiment sets the two perpendicular light beam distances as equal. The light beams go at constant speed, c. So, multiplying the geometric mean (GM) time by velocity, c, derives a geometric mean distance, which, like the GM times, T & T', is the same for both light beams.

Or,

cT =D = cT' = D' = d/F.

The contraction factor derives the geometric mean distance, as a coefficient to both beams, given one-way light-travel distance, d.

* * *

### Michelson-Morley times adapted to masses.

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E = mc² is the most famous equation in science, derived by Henri Poincaré, and independently by Albert Einstein. Einstein gave the name, Special Relativity, but it is as much the work of Michelson and Morley, Lorentz, Poincaré and Minkowski.

E = Mc² applies to an increase in mass with motion, significantly approaching light speed, which is to say, in high energy physics. (The following explanation is taken from the Asimov Guide to Science.) The mass, M, in the equation can be regarded as a difference in motion of mass m' from mass, m. The energy, E relates to the energy of motion or kinetic energy. In classical physics, this is recognised as: mv²/2.

The divisor, of two, means that one of the velocity multiples is an average of a velocity starting from zero to a final velocity. The other velocity multiple is just the final velocity. Such calculations apply when the kinetic energy is equated to a classical quantity, called the Work, meaning the work done to set a body in motion.

The multiple of the mass, m, by c² signifies the enormous energy, E, in any given mass.

The kinematic and dynamic forms of the Lorentz transformations, respectively for time and mass, are similar. So the Lorentz transformation for mass can be simplified as shown for the time transformation:

t = t'(1 - u'v/c²){(1 - v²/c²)^1/2}.

Reduced to its Michelson-Morley special case:

t' = t/(1 - v²/c²)^1/2.

Substituting m' for t' and m for t:

m' = m/{(1 - v²/c²)^1/2 = m(1 - v²/c²)^-1/2.

The far right side of the equation merely restates the square root denominator as the power (denoted by a circumflex) of minus one-half. This may be expanded or put in a series of terms, by the binomial theorem. Thus:

(1 - v²/c²)^-1/2 = 1 + v²/2c² + 3v^4/8c^4 + 5v^6/16c^6 +....

The series has an infinite number of terms but only the first two terms count for much, because of the one-sided weight of the two terms in the factor to be expanded. If both terms were unity, instead of unity and a fraction, then the series would form a balanced distribution, gradually increasing and symetricly decreasing in importance of terms. Instead, the one-sided expansion results in a steep falling-off in importance of terms from the start, or a skewed distribution like exponential decay.

Consequently, the energy equation is derived as an approximation that only uses the first two terms of the series. Hence:

m' = m(1 + v²/2c²).

At this point, Energy is equated to mv²/2. And M is equated to the difference between mass m, and mass m'. Hence:

M = m' - m = E/c². That is: E = Mc².

It wouldn't trouble Einstein that the equation is an approximation, because its actual value can be calculated to any degree of accuracy. The equation isn't an approximation for want of knowing a better value. So, this appears to be no concession of the determinist philosophy of classical physics to statistical approximation.

The real state of affairs is obscured by a biased use of language. The statistical "approximation" is actually an approximation to a more exact determination of a value, afforded by the complete expansion, of the binomial theorem into its series, which forms a statistical distribution.

In fact, special relativity is not a classical theory of physics, more exactly determined than statistical measurement (as physicists once believed - and may still believe, for all I know). On the contrary, it is a statistical measure, to which classical theory is the approximation. This primacy of statistics has long been known of quantum theory, in relation to classical physics (having been contested so fruitfully, in the Einstein, Podolsky, Rosen experiment). This statistical basis of physics also can be made explicit for relativity in relation to classical physics. At least, it is here so contested.

Moreover, it can be shown that the energy equation boils down to an approximate expression of the contraction factor expanded to its first two terms.

One can show this with the equivalent equation for times, to the energy equation for masses. Thus:

t' - t = tv²/2c².

From above, t' = d/c(1 - v²/c²). And t = d/c(1 - v²/c²)^1/2 .

Therefore:

d/c(1 - v²/c²) - d/c{(1 - v²/c²)^1/2 } =

{d/c(1 -v²/c²)^1/2}v²/2c².

Dividing thru by the value for t:

{(1 - v²/c²)^1/2} - 1 = v²/2c².

The binomial expansion of the right side factor produces the first two terms, on the left, below:  
F = 1 + v²/2c² +...

This is back to the approximate expansion of the contraction factor. It doesn't matter whether we do this reduction, for times or for masses. The only difference is that masses use a different constant to the time constant of d/c. Whichever constant is used, it cancels out, to reduce to the above equation.

This re-appearance of the form of the "contraction factor" in the dynamics context of the relativist energy equation, emphasises the statistical nature of special relativity. Tho, not (yet) physics conventional wisdom, the contraction factor was just an ad hoc substitute for geometric averaging, in special relativity kinematics.

A Michelson-Morley type experiment should be possible, in principle, not just for the speed of light but for the mass of light. A geometric mean time, for reflected light beams in relative motion, suggests relative acceleration of the light with and against earth motion. This should produce light of drawn-out and contracted wavelengths changing the frequency and therefore the light energy.

Altho light has no rest mass, it always has energy in motion, which must have a mass equal to the energy divided by the square of the speed of light. The more energetic photon or light particle associated with a gamma ray has about the mass of an electron.

* * *

### The Interval demonstrates the Michelson-Morley experiment.

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The equation for the Minkowski Interval is (1):

I² = (ct)² - u²t² = (ct')² - u'²t'².

The Interval, I, is a common space-time measure, shared by observers of a given event, at speeds approaching light speed, c, which is constant. Observer O has a frame of reference which measures time, t, and velocity, u, and distance x = ut. Observer O' in a different frame of reference has the different (hence indexed) measures of time, t', and velocity, u', and space, x'.   
The distances can be considered in one dimension or as a combination of two or three spatial dimensions. Distance becomes a vector, which means with different directions as well as magnitudes.  
Velocity is defined as speed in a certain direction. Displacement is defined as distance in a certain direction. The terms, speed and distance are often used, loosely, tho velocity and displacement are tacitly meant, because direction usually has to be taken into account.

The Michelson-Morley experiment showed that light takes the same time to travel over the same distance, either at right angles to, or aligned with (say) earth motion.

The Interval shows this, too, in terms of two observations that take the same time. That is when t = t'. Moreover, the Interval, I, can be considered as the common distance that both the split beams have to travel.

Also the velocity has a counterpart velocity, at right angles, because the M-M experiment has split light beams, one aligned, and the other at right angles, to earth velocity, u.

The right-angled velocity is given by an "imaginary" velocity, iu. The misleading term, i, for imaginary, simply means that the velocity is multiplied by the square root of minus one, i. Using more jargon, term, i, is an "operator," which places the velocity, u, at right angles to light velocity, c.

Therefore from equation (1), u' = iu.  
(If ut = x and as t = t', then x' = ix.)

Firstly, considering the Interval in terms of longitudinal motion or the light beam aligned with and against earth motion. This can be expressed in terms of equation (2):

I² = (ct)² - u²t² = t²(c + u)(c - u).

The factorised values, on the right side of the equation, represent a light beam, c, moving with or against earth velocity, u. This is the scenario for the part of the light beam in the Michelson-Morley experiment that is aligned with or against earth motion. The beam is reflected, so an out-going journey, with earth motion, means the return journey must be against it. And vise versa.   
The part of the beam that is split off to travel cross-ways, to earth motion, will be considered afterwards.

Re equation (2), if earth velocity and the beam velocity are moving in the same direction, the light has to catch up with the reflecting mirror, that the earth is carrying away from it. And the light velocity is that much reduced by the earth velocity.   
That is: (c - u).

If the beam is moving in the opposite direction to the earth, the beam arrives that much sooner, and earth velocity must be added, as speeding the arrival. That is (c + u).

The journey takes the same over-all time, whether the beam, moving forth and back to source, from the mirror, starts by traveling with or against earth motion.

Thus, the Interval as equation (2) may represent the longitudinal journey of the light beam in the Michelson-Morley experiment (MMX).   
No, this isn't the way Michelson and Morley calculated the time taken by the light beam. It was over twenty years before Minkowski formulated the Interval.

Reformulate equation (2) to equation (3):

t² = I²/(c + u)(c - u) = {I/(c + u)}{I/(c - u)}.

The right side of the equation is the multiple of two ratios. A constant distance, I, is divided either by the faster velocity (c + u) or the slower velocity, (c - u), representing relative motion of the light beam, respectively, against and with earth motion.  
The arithmetic says these ratios measure respectively faster and slower times.

Using the geometric mean to average these times means multiplying the ratios, just as on the right side of equation (3). Then taking the square root, for the geometric mean time, t, as in equation (4):

t = ({I/(c + u)}{I/(c - u)})^1/2.

Anyone of the thousands of books on relativity (or indeed my own first chapter) will show that this result for the average time taken by the reflected light beam in alignment with earth motion is NOT the result calculated by Michelson and Morley. (This book ends with a Disclaimer, which confirms this, all the more emphaticly, by quoting a hostile censor of my alternative MMX calculation.)

Later, the Fitzgerald-Lorentz contraction was cooked-up to make their calculation agree with the result of their epochal experiment that light takes the same time to return when reflected cross-ways and in alignment to earth motion (supposedly relative to some "ether" motion).

Equation (4) for the earth-aligned light beam DOES agree with the Michelson-Morley calculation for the cross-ways light beam to earth motion.

It makes no difference which average, arithmetic mean or geometric mean, is used for the cross-ways journeys time. But to get agreement with the light beams earth-aligned journey time, the geometric mean must be used, as explained above.

Now, consider the MMX cross-ways relative motion of light beam to earth motion. Velocity, u, stands for earth motion that is aligned to the light beams direction of motion. As explained, a motion at right angles to aligned velocity, u, is given by iu, the same magnitude of velocity but re-directed ninety degrees in direction.

Substituting in the Interval as equation (5):

I² = (ct)² - u²t² = t²{c² + (iu²)} = t²{c² + (-iu²)}.

Notice about equation (5) that the Interval squared can be a plus sign or minus sign with respect to transverse velocity, iu or -iu, whose opposite signs may stand for the opposite cross-wise directions, to the alleged ether stream, of the beam reflected from stream "bank" to "bank", so to speak. In other words, the conventional (negative) form of the Interval is just with respect to aligned motion. Equation (5) is the same Interval merely re-stated for transverse motion.

It goes half way to resembling a pure four-dimensional Euclid geometry, as given by theorem Pythagoras.   
This theorem says the sum of the squares of the sides of a right-angled triangle equal the square of the hypotenuse. This theorem extends to the sum of the squares of three sides of a cube equaling the hypotenuse in three-dimensions.

The Interval is a break, from extending the Pythagoras rule, in that the fourth dimension is a subtraction of its square from up to three spatial dimensions.   
The present treatment simply considers the velocity in one spatial dimension. It could be considered as a distance vector, vt = x, of up to three dimensions, with ct as the fourth dimension, and of opposite sign. The intrusion of the negative value, in the Pythagorean formula, represents aligned motion, back and forth, with respect to the usual one, two or three spatial dimensions. Or, as just shown in equation (5) it also represents the intrusion of cross-ways motion back and forth.

In short, the Interval is the intrusion of change of direction into theorem Pythagoras. This change of direction implies change in velocity, which is acceleration, and a geometric series of motion in progress. A geometric series is averaged by the geometric mean, which is the form of the Interval.

It may need to be explained why the geometric mean is suitable to average the transverse motion, as well as to average the aligned motion. The transverse motion was put in the form of complex variables.

A "complex variable" is, say, z = c + iu, where light speed, c, is the so-called "real" part and earth velocity, iu, the "imaginary" part. The point about these forms is that they involve rotation, as implied in the use of the operator, i, as meaning: turn thru ninety degrees. Acceleration, which is non-uniform motion, is a component of rotation, and this requires the use of the geometric mean for averaging (as shown in below chapter on the Mach principle).

Two things. Firstly that the Minkowski Interval calculates the true time for the Michelson-Morley experiment. And secondly that the Interval is a geometric mean. So, special relativity is a theory of statistical averages.

As is well known, the irony is that Albert Einstein, who used statistics, in so much of his research, baulked at the idea of special relativity being a statistical theory. And how creatively and usefully he thought against the statistical basis of quantum theory! He regarded the statistical perspective, as an infringement of his ideal of relativity as a deterministic progression from the classical physics of Newton.

* * *

### SR non-solutions of the M-M calculation.

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Here, I hope to show that the more obvious choice of average, the arithmetic mean, used by Michelson and Morley, is mathematicly meaningless, when applied to the Lorentz transformation of Special Relativity. Likewise, for application to the Minkowski Interval.

Above, I gave proof positive that the geometric mean is the suitable calculation in the M-M experiment (MMX). Now to give proof negative, that the arithmetic mean is actually the wrong average to use for that experiment.

You might think I am over-doing my contention, but I found that, when it could no longer be ignored, as usual, it was denied.

What I called the proof positive also implies a negation and refutation of the Michelson-Morley calculation. This is for the following reason. The transverse light reflection and the longitudinal light reflection are both with respect to earth velocity. But when you put the two different Michelson-Morley time calculations into the Minkowski Interval, I, equation, the answer only works for the earth-perpendicular beam time side of the equation having a velocity zero, which is inconsistent with the earth velocity that both sides of the equation must have, to represent the Michelson-Morley experiment.

Hence the Interval is:

I² = t'²(c² - u'²) = t²(c² - u²).

Substituting in the Michelson-Morley values for the times, t' and t, and bearing in mind that both sides must share the earth velocity, u' = u, then:

{cd/(c² - u²)}²(c² - u²) = {d/(c² - u²)^1/2}²(c² - u²).

So:

c²/(c² - u²) = 1.

Or:

(1 - u²/c²)^1/2 = 1.

In this special case of applying the M-M calculation, the Interval reduces to the form of the contraction factor, because the arithmetically averaged earth-aligned beam time is out by that amount. This error was the reason, in the first place, for the ad hoc "contraction factor," really a correction factor.

According to this (mis)calculation, u = 0. Some will say, as I once thought, that the right-angled beam has zero relative motion to the earth. But this is inconsistent with the beam journey being perpendicular to an actual earth velocity, u, which is not zero.

A similar result is found using the equivalent Lorentz transformations between the two times of two different local observers:

t' = t(1 + uv/c²)/(1 - v²/c²)^1/2.

In general, the velocity, v, as in the contraction factor, is the relative velocity between two observers, not the local observers velocities, u and u' with respect to an event, which they are measuring with their local co-ordinates or frames of reference. (There is no difference, in the particular case of MMX.)

The observer who locally measures the faster time, t, of the two times also has a local velocity, u. This must be zero, so that the Fitzgerald-Lorentz contraction factor, the square root factor in the denominator, relates time t' to time t.   
Therefore, like the Interval, the Lorentz transformation also makes the M-M calculation inconsistent. And once again shows the two times out by the "contraction" (correction) factor coefficient.

Hendrik Lorentz didn't start with the full Lorentz transformations, named after him. Historicly, the simple contraction factor is how Fitzgerald and Lorentz independently explained the discrepancy of the Michelson and Morley experimental result from their calculation. They came-up with this ad hoc hypothesis of a contraction factor, as if there was some physical effect of the supposed "universal ether wind" that balanced out the Michelson and Morley calculation.

A much simpler explanation is that Michelson and Morley should have used the geometric mean, instead of the arithmetic mean, and then the times come out equal.   
The Lorentz transformations were built on the contraction factor, with its implicit assumption that this was a physical correcting or accounting for the M-M experiment rather than that the M-M calculation was mathematicly or statisticly incorrect. This assumption prevented Lorentz and his successors from seeing that his full transformation rendered the M-M calculation inconsistent.

This explanation in terms of the two M-M experiment times, t and t', is admittedly simplistic.

* * *

### Transformations between affected and unaffected light beam times.

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A further time, to be taken into consideration, is the unaffected, or non-relativist light beam journey time, T. This is simply distance, d, divided by the light speed, c, both ways, which gives the same value for the average time. That is: T = d/c.

The Minkowski Interval would give a different time for T, compared to identical light beam times for perpendicular beam time, t, and earth-aligned beam time, t'. In fact, the Interval, I is the equal distance, d, of the light beams. And both have the same earth-relative velocities, u.

Hence,

I² = d² = t²{c² + (iu)²} = t'²(c - u)(c + u).

The factor with the imaginary earth velocity, iu, represents a transverse light beam traveling at 90 degrees to the earth. The two multiplied factors, on the left side, represent the earth-aligned beam traveling, at speed, c, with and against earth velocity and therefore, relatively speaking, added and subtracted to and from earth velocity, u.

These Michelson-Morley experiment times would not be the same for a light beam traveling the same distance but not relative to earth velocity. In the scenario of the two boats, one goes back and forth or up and down stream. The other boat goes across from bank to bank and back.

The analogy, with a light beam not moving relative to earth velocity, would be for a third boat to be an amfibian that would not cross the stream but move the same distance, of the other two boats, on land (perhaps on some ideal frictionless rail), which is to say unaffected by the motion of the stream.

In that case the earth-relative velocity of the "amfibian" beam would be zero. That is to say that the non-relative time, T, implies an insignificant velocity compared to light speed. In other words, this is the classical physics case.   
In terms of the Interval, this beam taking time, T, would relate to either of the earth-relative beams, taking time, t, as:

I² = t²(c² - u²) = T²(c² - 0).

Therefore:

T = t(1 - u²/c²)^1/2.

In effect, the contraction factor form has to be applied to the classical Time, T, to derive the relativist time. But this simply reflects the changed (geometric) conditions of high energy physics compared to classical physics.

Also, the Lorentz transformation of times can be applied to relate the two times, earth-perpendicular time, t and earth-unaffected or non-relativist time, T.

Hence:

t = T(1 + 0.v/c²)/(1 - v²/c²)^1/2.

There again, the earth-unaffected or non-relativist light beam time has earth-relative velocity zero.

Using (the geometric mean form of) the Interval to relate T to t', with the M-M calculation of an arithmetic mean longitudinal time, t', produces the incorrect result that T has the same value as t.

Using the Lorentz transformation to relate M-M calculation of t' to T = d/c gives the result that the contraction factor is equal to unity, which is only true when relative velocity is insignificant compared to light speed. (That is the classical, not the relativistic, case.)

The MMX is sometimes called a null result. This is with reference to the defeated expectation that the earth-aligned light journey would take longer than the perpendicular journey. The irony is that the real null result is in the inconsistency of the M-M arithmetic mean calculation with the equations of special relativity.

The minor calculation short-coming of the Michelson-Morley experiment, the technical triumph of its day, is rather as if the NASA astronauts got to the Moon, in 1969, without remembering to bring back some rock samples, and everyone was too excited by the magnitude of the achievement, to heed the over-sight. Without the experimenters, there is no science.   
I learnt perhaps as much from Albert Michelson as I did from Albert Einstein. I have nothing but admiration for this modern Archimedes.

* * *

### Sub-luminal and superluminal connections (SLC)

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The Michelson-Morley Experiment transverse beam has a magnitude-neutral acceleration, being only a change in a change of direction, with respect to earth motion: forth and back from a reflecting mirror placed, at right angles to earth velocity (around the sun, say).   
The earth-aligned beam has a change in the magnitude of its velocity with respect to earth motion, but direction-wise remains in alignment, one way or the other, confined to one dimension.  
MMX shows an equal time taken by light, under both transverse and longitudinal kinds of relative acceleration.

Gary (of the email group Special Relativity) wouldn't have it, that using the arithmetic mean is the wrong average for MMX. His argument was that the Michelson-Morley (arithmetic mean) calculation was a legitimate way of analysing or breaking-up the calculation into steps.

This begs the question of why mathematicians may break-up a measurement of a curve into finite segments. Archimedes treated the area of a circle as the progressive measurement of inscribed and outscribed polygons, with an increasing number of sides. The circle may be considered as an infinite polygon, beyond a finite number of sides, which would only be an approximation. (The recently rediscovered palimpsest of a lost Archimedes classic may have taken the integral calculus, further than previously thought.)

Likewise, measuring the MMX light beams, as straight lines, which graph velocity or rate of change of distance over time, ignores that their reflection is a bend, however sharp. That is technically a curving motion, which is to say, of changing velocity, or, acceleration: change with time, in the change of distance over time, or, a second order change in distance over time.  
Consequently, an arithmetic mean measure of the reflections is only an approximation, just as polygons only approximate a circle.

The "contraction factor" corrected the MMX calculation against experiment. And that is what the contraction factor really was: a correction factor. Its geometric mean form is actually a clue to the correct average that should be used: the geometric mean.

The acceleration, measured as a (geometric) averaging phenomenon of the MMX longitudinal beams whole journey under reflection, is a unitary time.   
Traditional MMX arithmetic averaging assumes two distinct times, that the beam takes, before and after reflection, from a differing relative velocity between beam and earth, in each case.  
Perhaps in high energy physics, relativistic acceleration time-unifies distinct velocity times, perhaps frees from times divisions or schedule. Approaching light speed, time slows and at the speed of light, time comes to a stop. The realm of light is itself timeless. There are no divisions into times, only one unitary time which amounts to timelessness. It is the Buddhist transcendance of the opposites.

if you take the trouble to read what the Minkowski Interval equation says, then you will see it takes the form of two factors. One factor adds a locally observed velocity to the speed of light, which amounts to a super-luminal speed. The other factor subtracts the same locally observed velocity from the speed of light, which amounts to a sub-luminal speed.

This fact no doubt spurred the search for tachyons, or fotons moving faster than the generally observed speed of light. Groups of fotons in waves have been infered to show faster than light speed behavior, not so any individual fotons. Tachyons have long since been compared to unicorns as figments of the imagination, with at best a mathematical semblance of reality.

What gets no mention, in this context, is that it would be equally logical to search for tardyons, as well as tachyons. Tardyons would be fotons moving slower than the generally observed speed of light.

The point about the Minkowski Interval is that the multiple of super-luminal speeds with sub-luminal speeds may approach but never surpass a limit of the generally observed constant speed of light. So, the super-luminal and sub-luminal factors do point to the empirical observation of the constant speed of light. Therefore, it is reasonable to suppose that those super-luminal and sub-luminal factors have a reality of their own, which contributes to the more manifest constant speed of light.

If it is accepted that in one direction local velocities add to the speed of light, and in another direction they subtract from the speed of light, the question arises: which direction is the superluminal direction and which direction is subluminal?

Light is the same speed in all directions. There is no individual direction that will add or subtract to light speed. But the Minkowski Interval is a collective direction that both adds and subtracts to light speed.

This situation in large-scale high-energy physics corresponds to the situation in subatomic scale quantum physics. There, super-luminal and sub-luminal tracks cancel each other out over observable distances, leaving the recognised constant speed of light. Quantum physics makes precise predictions of probability, or the average likelihoods of events. Likewise, the Minkowski Interval is a geometric average. Just as individual tachyons, or indeed tardyons, do not exist as observable entities, in quantum experiments, also they are not observed in relativistic high-energy physics. What is observed is just their average, taken by the Interval.

There is no absolute reference frame that can show observed velocity to be moving against or with a light beam, respectively adding to, or subtracting from, its speed. This issue of absolute or universal direction is a rerun of the old hypothesis of the ether as absolute velocity, of the universe, or its speed in a certain direction.

Likewise, there is no absolute time, as a standard measure by which there might be motion back in time or forward in time. There is only an arbitrary before and after other arbitrary local times.

Super-luminal and sub-luminal speeds are locked together as factors in the Minkowski Interval.

The Interval factors are of relative acceleration; they are directional: This is the loop-hole for light acceleration, given there is no increase in velocity magnitude of light. It appears that this kind of acceleration is associated with super-luminal connections (SLC).

Acceleration is in the rotation of one of the quantum-entangled pairs, which conservatively produces an equal and opposite reaction in the other pair. The rotation might be a 90° turn of polarised light or 180° turn of electron spin.  
(This reminds of the perpendicular and aligned light beams, whose two-way journeys could be geometrically averaged.)

I guess that the clue to understanding the super-luminal connection is the experimenters rotation of one of the entangled particles. The rotation produces directional acceleration on one of a pair of fotons (or, say, an electron, which emits a foton) which produces a faster than light response from the other particle. This is not a mystical instantaneous response, or "spooky action at a distance" as Einstein called it.

From the point of view of the opposite direction, relative motion is not added to the light velocity but subtracted, for a subluminal connection. Consequently, on average, there is no faster than light communication.

This situation may perhaps be compared to a Feynman diagram, with its alternative interpretations of a reaction between subatomic particles. One reaction may include a particle going forward in time. An alternative reaction, which is just as valid, may involve a particle going back in time.

Super-luminal connections are perhaps like this. Tho, in their case, the most obvious explanation is that altering an entangled particles state goes back in time, to instantly change the other particle, to bring the conservative system back into equilibrium.

Alternatively, the partner particles changed state could be considered as a going forward in time to balance the experimenters changing a particle state.

We have to divest ourselves of the prejudice that individual explanations are always more fundamental than unitary explanations.

Human individuals themselves start life without much sense of identity, gradually acquiring individuality, becoming individuals. In our youth and strength, we are more concerned with developing our personal contributions to society. In our decline, we become helpless to do much ourselves as individuals, and see in ourselves more the universal qualities, that will survive in other people, after our death. Old age divests us of ourselves as individuals and opens our eyes to our-selves as universals.

The whole universe is like an individual, in this respect. It starts elemental, with little individuality, gradually differentiating. Ultimately this gathered individuality loses energy and organisation, like an elderly person, dissipating into a universal sameness. Or so it would seem by the second law of thermodynamics.

There are so many physical parameters, influencing the exact structure of the universe, it seems improbable that the many other possible adjustments to nature do not exist as hosts of other universes. It seems as unnatural to assume there is only one highly individual universe, as it would be to assume that Robinson Crusoe was the only manifestation of his kind.

In short, the individual and the universal are just two sides of the same coin, neither more basic than the other. In certain circumstances, the prominence of one over the other may lead or mislead one to believe in the greater general importance of one rather than the other.

The statistics of quantum physics, this book contends are matched by the statistics of special relativity, offering explanations in terms of unitive averages.   
Modern physics does have a lesson to teach us as living beings: We may be both individuals and universals.

# Interval magnitude symmetry vectors momentum conservation.

Table of contents.

#### Sections:

Linear and angular acceleration in the M-M expt.

Conservation of linear and angular momentum.

Quadratic damping of the Interval.

The Interval and Symmetry.

Amplitude Symmetry of the GM Interval.

After-note: Interval exponential decay for higher dimensional unification of forces.

References.

### Linear and angular acceleration in the M-M expt.

The so-called null result of the Michelson-Morley experiment (MMX) is that the beams take the same time to return the same distance, irrespective of their different relations to earth velocity thru space (or any supposed universal ether velocity, with which it might be in relative motion).

Of course, the future of physics was based on the experimental result, and not on the misleading calculation. So, it didn't appear to matter. My own first chapter on the M-M experiment, when I was trying to teach myself special relativity, faithfully followed the calculation error -- as this error-prone amateur dared to call the calculation used, of the arithmetic mean, of each two-way journey.

The geometric mean gives the correct prediction that the two light beams will take the same time to return to source. This measurement was made possible by the legendary Michelson interferometer to synchronise light waves.

The below diagram (not to scale) attempts to justify the simultaneity of MMX in terms of vectors: arrows give directions, as well as having lengths, which give their magnitudes, labeled in terms of the speed of light, c, and earth speed, u.

### Vector analysis of MMX.

On the diagram, the light beam is split at source, so that part of the beam is deflected at right angles. Both beams go the same distance before they are reflected back to source.

The experiment is arranged so that the deflected beam goes at right angles to earth motion, while the original beam continues in line with earth motion. The original beam going against earth motion, before reflection, increases (to c+u) the velocity with which the beam reaches its reflecting mirror. On reflection, going with earth motion decreases, to (c-u), the velocity with which the beam goes the same distance back to its source.

Subtracting the earth-aligned beam return vector from its out-going vector, or (c+u) - (c-u), gives a net vector of two units in the direction of earth motion, or 2u.

By the time that the earth-perpendicular beam has reached its reflector, the Earth has moved on slightly (with velocity u), which position marked on the diagram is now perpendicular with the deflected beams reflector.

The same thing happens again, by the time the deflected beam reflects back to its source, which the earth motion has moved on, by the same amount (at velocity u). So, the deflected light beam has merely moved by two units of earth velocity (2u). These two units are in fact two resultant vectors of the outgoing and return journeys.

The component vectors of the resultant vector, are for the outgoing journey, the slanting arrowed line of velocity, I/t, pointing towards the reflector, and the perpendicular line, of value c, the speed of light, pointing towards the reflector.  
The value, (I/t)², is, by theorem Pythagoras, the square of the Minkowski Interval divided by time, which equals: (c² + [iu]²) = (c² - u²).   
The operator, i, which is the square root of minus one, signifies a turn thru 90 degrees. Just as the minus sign can be an operator for a turn thru 180 degrees, or going in the opposite direction.   
A complex number (composed of an ordinary or real number combined with a so-called imaginary number, i, like a vector, is a way of expressing direction as well as magnitude.

Don't be put off by the fact that the longer length hypotenuse, I/t, in the diagram, has a lesser magnitude than the shorter vertical value, c. This is a common-place peculiarity of the representation of the Minkowski Interval radius vector on a plane, defined by a complex variable, t(c + iu). The coefficient term, t, for time, shown here, changes the dimension from velocity to time multiplied by velocity equals displacement or distance (in a given direction).

The perpendicular vector, c, when pointing in the opposite (return) direction, is one of the two component vectors to the resultant vector for the return journey. The other component vector is the slanting arrowed line, marked, pointing to the destination, which is the source moved-on two units of earth velocity, 2u.  
This return perpendicular beam destination coincides with the resultant vector of the earth-aligned beam, indicating that the two beams have taken the same time, by different paths.   
In other words, this vector analysis of the Michelson-Morley experiment shows that the split light beams have the same resultant velocity from different component velocities.

Two different kinds of change in velocity affect the split beams. A vector consists of magnitude and direction. The earth-aligned beam velocity was changed in magnitude by being reflected in line with earth motion. Change in velocity is also called acceleration (or deceleration, if a slowing down of velocity, as in this case). This beam undergoes magnitude deceleration. But the direction of the earth-aligned beam remains on the same one dimension.

The ninety-degrees deflected beam was changed in direction relative to earth motion, as shown on the diagram. By convention, angular rotation is anti-clockwise. First, the split beam is rotated by 90 degrees or π/2 radians. The angle of drift with earth motion on reaching the reflector, from source, is minus Q (or positive Q clockwise).

As the diagram shows, the initial angle of reflection of the perpendicular beam is π/2 - Q. Its final angle of reflection is π/2 + Q. On average, the perpendicular beam is indeed perpendicular, at π/2 radians, as between the source and perpendicular reflector, at the mid-point of the right-angle deflected beam journey.

* * *

### Conservation of linear and angular momentum.

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There is no reason why magnitude acceleration should be exactly compensated by directional acceleration. It is just the result of the perpendicular set-up of the Michelson-Morley experiment.

Directional acceleration and magnitude acceleration are found as the vector components of a resultant acceleration involved in circular motion. A component of acceleration vectors from the circumference to the center of the circle, and is called normal acceleration. Tangential to that point, there is tangential acceleration.

When multiplied by mass, normal and tangential acceleration, respectively, are centripetal and centrifugal force.

The Michelson-Morley experiment was meant to tell time differences in differently oriented light beams over the same distance. But this vector analysis suggests that the set-up does exactly the opposite. It synchronises the split light beams. That is, it creates a time symmetry.

There must be constants at work that constrain this result. The split beams have the same distance to travel. Both beams are subject to the same constant velocity of the earth. And the light beams themselves must have constant velocity.

The Minkowski Interval appears to neatly express the simultaneity of the Michelson-Morley experiment, because the velocity terms can be interpreted as an identity of reflected light, either moving with and against earth motion, or, light moving crossways to it.

Velocity is usually multiplied by mass, to make momentum. Physicists speak of conservation of linear momentum and conservation of angular momentum.

The Michelson-Morley experiment seems to be a set-up involving the equation of linear and angular momentum, as a condition of time symmetry.

My own understanding (or lack of understanding) is that linear motion, or motion in a straight line, is a special case of angular motion. Free motion in one dimension must allow movement back and forth. And that implies an angular turn of 180 degrees. If the motion is only one-way, that would seem to imply, to my way of thinking, a turn of zero degrees. In trigonometry (of sines and cosines) these angles either produce a multiple of one, which leaves a term as it is, or produce a multiple of zero, which makes a term disappear.

* * *

### Quadratic damping of the Interval.

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The Interval, I, is the commonly observed space-time measurement, for two observers, O and O', of a given event, from their differently measured local view-points, at velocities significantly approaching light speed, and where velocities, u and u', can be in one dimension or a resultant vector of up to three dimensions.  
Equation 1:

I² = t²(c² - u²) = t'²(c² - u'²).

Where t and t' are two observers different local times, and c is the commonly observed speed of light. For convenience, consider only one observer, eqn. 2:

I² = t²(c² - u²) = {(tc)² - x²}.

To factorise the equation, add the terms, 2x² - 2xct, to both sides. Eqn. 3:

I² + 2x² - 2xct = (tc)² - x²) + 2x² - 2xct.

Factorising for eqn. 4:

I² - 2x(ct - x) = {(tc) - x}²

Or (5):

{(ct) - x}² + 2x(ct - x) - I² = 0.

This equation can be solved for (ct - x) by the formula for solving quadratic equations.

Given that:

ay² + by + c = 0,

y = {-b ± (b² - 4ac)^1/2}/2a.

Substituting in the quadratic formula for eqn. 6:

(ct - x) = {-2x ±(4x² + 4I²)^1/2}/2

= {x ±(x² + I²)^1/2}

There are two alternative solutions, which form a range, from which an average value solution can be found, in the form of their geometric mean [GM].

Eqn. 7:

GM =

[{x +(x² + I²)^1/2}{x -(x² + I²)^1/2}]^1/2

= {x² - (x² + I²)}^1/2

= {(-I²)^1/2} = ±iI.

Therefore, the geometric mean, of the positive and negative range solutions to factor, (ct-x), is itself a range or scale, of a positive and negative Interval, on an imaginary axis, which means on a second dimension, at ninety degrees, to some first dimension (real axis) Interval.

Thus it is possible to take a geometric mean of this new Interval range. That means taking a geometric mean of the geometric mean value of factor, (ct-x), which is to say a second order geometric mean (signified as [GM]²).

Eqn. 8)

[GM]²(ct-x) = {(-iI)(iI)}^1/2

= (I²)^1/2 = ±I.

Thus, the second order GM of the factor, (ct-x), represents a first dimension (real axis) of the Interval, whereas the first order GM represents the second dimension (imaginary axis) of the Interval.

The Interval equation, when transformed into an equation between the second order geometric means of observers O and O' looks like this:

[GM]²(ct-x) = [GM]²(ct'-x') = ±I.

This draws on eqn. 8, only stated for one observer, so as to include another observer, both of whose locally distinct observations share the Interval value, ±I, which is an oscillation of the Interval amplitude from positive "crest" to negative "trof" making a "wave."

Suppose this quadratic equation solution exercise with the factorised Interval is repeated for the conventional Interval.

Re-write eqn, 2:

I² = (ct)² - x²), as:

{(ct)² - x²} -I² = 0.

This equation has a second order term, {(ct)² - x²}, and a constant, I², but no first order term, 2x(ct-x). and therefore zero coefficient, b, in the quadratic equation solutions formula.

So, eqn. 9:

(ct-x) = {0 ±(0 + 4I²)^1/2}/2

= ±2I/2 = ±I.

Eqn. 10:

[GM] (ct-x) = {I(-I)}^1/2

= (-I²)^1/2 = ±iI.

Thus, eqn. 10 agrees with eqn. 7.

The second order Geometric Mean, of factor (ct-x), is eqn 11:

[GM]²(ct-x) = {(+iI)(-iI)}^1/2 = ±I.

Thus, eqn. 11 agrees with eqn. 9.

Of course, they should agree, because the equations 2 and 4, they derive from, are equivalent.

Successively taking the geometric mean of the factor, (ct-x), is equivalent to successively multiplying the Interval by the operator, i, which signifies repeating the operation of a ninety degrees turn of the Interval.

Adding the second and first order geometric mean Intervals derives a complex Interval form:

GM]²(ct-x) ± [GM = ±I ± ±i/I = ±(1±i)I.

But equation 4 is no longer an equation of the (conventional) Interval. It is the Interval minus an extra first order term, in 2x(ct-x), reminiscent of a damping factor. Or, it might have been added as an amplifying factor.

To see how a first order term in a quadratic equation can be a damping (or amplifying) term, the algebra for the Fibonacci series is derived from this form. In the formula for solving quadratic equations, a Fibonacci series can be obtained by one, of all the possible adjustments to the coefficient of the first order term, to come out with the square root of five. With this in the formula, the Fibonacci numbers, approximately 0.618 and 1.618, (with the unique property of being each others inverse) may be derived.

These numbers represent the ratios of the successive terms of the Fibonacci series, which increasingly accurately approximate the Fibonacci numbers. When these are graphed, they produce a zig-zag "curve" in which the zig-zags get successively smaller and more flattened out, like the subsidance of angular "ripples." In other words, the Fibonacci series graph is like a discrete form of wave damping.

Compare the pair of second order quadratic equations, with or without a first order term, with a likewise pair of second order differential equations. This compares, respectively, algebra with calculus. Apparently, the difference between the two branches of maths is the difference between the discrete and the continuous treatment of variables.

An ordinary wave equation, whose algebraic equivalent is eqn. 2, is a second order derivative and a constant. It expresses a wave, in simple harmonic motion, that is a uniform wave with a constant amplitude of crests and trofs. The Minkowski Interval is a constant radius vector, pin-pointing a given event, on the circumference of a circle, whose turns can roll out a wave. The magnitude of the Interval is traced by the maximum and minimum amplitudes, or crests and trofs of the wave. The same Interval amplitude is measured by all observers, whose local viewpoints are merely differences of phase, or different starting points on the circle with their respective local co-ordinates.

It sounds more complicated than it looks: a picture is worth a thousand words. So, I have drawn it. (It is the same, with minor modifications, as the color-coded diagram in chapter two.)

### Interval radius vector.

The damped wave equation, whose algebraic equivalent is equation 5, includes a first order derivative, as the damping factor. It is usually solved in terms of the equation for a sine wave, multiplied by an exponent to a negative power, which makes the otherwise uniform wave under-go a diminution like successive waves in decreasing ripples, on a pond, and known as exponential decay.

In fact, sine waves themselves can be expressed in terms of exponential functions, known as Euler equations, which simplify trigonometry. Circular motion is more conveniently expressed in polar co-ordinates than rectilinear (Cartesian) co-ordinates.

The complex variable in rectilinear co-ordinates, z = x + iy, maps any point (by convention) on the upper right quadrant of a plane. And converts to polar co-ordinates. For example, with respect to the black co-ordinate system of observer, O: (ct+iut) = r(cos[Q+q] + i.sin[Q+q]), with circle radius, r, and angle [Q+q]. The red co-ordinate system of observer, O' is given by: (ct'+iu't') = r(cos q + i.sin q).

Then, the observers respective Euler relations are: z = r.e^i[Q+q] and z' = r.e^iq, where the exponent, e, is an infinite number constant of about 2.718...  
So, the damped wave equation could be expressed as a multiple of two exponents, one with a negative power and the other with an imaginary power. As multiplied exponents have their powers added, this would result in an exponent with a power that is a complex variable (or a real and an imaginary variable, in other words, a two-dimensional variable).

Likewise, the Interval can be pictured as a constant radius, located by any arbitrarily placed rectilinear co-ordinates. These stand for all the possible different locations by which observers might see a given event, occuring at the position, where the Interval radius meets the circumference. The Interval is the common magnitude and direction, in space and time (space-time) measured by all observers in uniform relative motion, significantly approaching light speed.

That is a pictorial representation of the Interval as expressed in eqn. 2. But equation 5 includes a damping factor which means that the equation is of a damped or decreasing Interval.

One could imagine the Interval following a succession of events moving round the circumference of the circle. Observers could follow these events from their differently positioned co-ordinate systems. (We would say that the x-axis of each co-ordinate system is at a different angle, with respect to, say, an x-axis, assigned by convention as the base axis, common to all observer axes. Those different angles between the base axis and observers x-axes are known as their respective phases.)

We could draw a separate but corresponding rectilinear or Cartesian graph to a regular orbit of events on the Interval circle (in polar co-ordinates). Instead of a rotation, its track, on the rectilinear graph, would now look like a simple harmonic wave. The equal amplitude, of its crests and trofs, which is the magnitude of the Interval (constant), are measured by the positive and negative vertical rectilinear axis. At the same level as the origin or center of the circle, the zero point is at the equilibrium level of the wave, marked out by the rectilinear horizontal axis.

Suppose that the events, on the Interval circle, do not follow a constant revolution but begin to spiral inwards. Then the corresponding rectilinear graph would trace out a damped wave. This would be the graph of a decreasing Interval.

An after-note refers to physicist Lisa Randall. She used an exponential damping factor on the Interval, in connection with work on a full force of gravity, being wrapped-up in hidden dimensions, to explain why it appears much weaker than the other three known forces of nature.

* * *

### The Interval and Symmetry

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In the conventional Interval, the observers are inter-changable, without changing the value of the Interval. The conventional Interval has the symmetry of uniform velocity in a straight line. Thus, the Interval measure stays unchanged, as the common space-time measure, for all observers with different local space and time measures.  
Measurement inter-changes, which do not affect the result, are known as symmetry transformations.

The geometric mean of two polar limits is either of the other two polar limits, at right angles to them. The average values are inter-changable with the limit values, or the representative values with the represented values. (This is explained, in below chapter, of a caldera model.) The real and imaginary dimensions have statistical symmetry.

The Interval is a complex variable circular function, in perpendicular and bi-polar real and imaginary variables, like the four points of the compass. Whereas electro-magnetism, consists of positive and negative electricity, which is bi-polar and magnetism, which is monopolar.

An analogy with democratic representation might illustrate the difference. A representative democracy consists of a parliament and people, that is representatives and represented. The representatives are the averages of popular opinion that cover a political spread from Left to Right. The representatives may be positive or negative, perhaps analogous to government and opposition.

While the People cannot represent the Parliament, tho the Parliament can represent the People, the latter are in a one-way or monopolar relation of being represented but not representative. Representative democracy only is analogous to magnetism.

Suppose the People are also allowed to be representative and Parliament in turn becomes represented. This can happen when politicians cannot agree (say, are split or polarised between Left and Right) on some issue, and leave a decision to the public in a referendum. (This actually happened in 1975, when the UK Labour government couldn't agree with itself over staying in the Common Market, and so tossed the decision to the British people. This happened again with the Tories 2016 referendum on European membership.)  
In that case, with a combination of direct democracy, and representative democracy, there is a symmetry, of representation and represented, between People and Parliament.

The monopolar nature of magnetism may be compared to the monopolar state of Representative democracy. The bipolar nature of electricity may be compared to Representative democracy combined with Direct democracy.

* * *

### Amplitude Symmetry of the GM Interval.

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In physics jargon, the Interval possesses "invariance." The Interval is a measure, in the dimensions of directional distance, or displacement, common to all observers. (It is often said to be a four-dimensional measure of space-time.)

The Interval shows that local observers give different space and time measures from their different co-ordinate systems, which are merely out of phase with each other. Their only difference is an angle of rotation between the different co-ordinate systems, about a common origin. Thus, the jargon describes the Interval as having "space-time rotational symmetry."

The Interval invariance is despite any rotations between local co-ordinate systems of measurement. The change of angle between these local co-ordinate systems, leaves the conventional Interval the same. A change or transformation, that leaves something the same, is called a symmetry transformation.

The Interval is rotation-invariant or phase-invariant. The Interval is the same measure, it has symmetry, despite turning round local observers different space and time co-ordinate systems - like a circle or sphere is symmetrical, because it looks the same from turning about its center.

When the geometric mean is taken of the positive and negative solutions of a quadratic equation, any first order term is automaticly canceled in the working. That is why it does not matter, on average, whether the Interval is kept in its conventional form, analogous to a simple harmonic wave form, or whether it is qualified, with an extra term, analogous to a damping or amplification factor, in its quadratic equation, changing the (Interval) amplitude of the wave form.

A first-order term means a variable which is not multiplied by itself, unlike a second order term, which is a variable squared. A first-order term is analogous to a damping term in a wave equation that reduces the amplitude of the wave. The damping term would not change the geometric mean of the oscillation, it would just change the amplitude range limits of the oscillation.

The geometric mean of a damped or amplified Interval remains unchanged. The amplitude is the height of a wave or corresponding depth of a trof. Such a wave can be plotted from the rotation of a circle, whose radius (the Interval) is the same length as the amplitude.   
So, I suggest that just as the Interval is invariant, under rotation, or by phase difference between local observers co-ordinate systems, it might also be considered as invariant by amplitude change, or equivalently by radius magnitude change. But it is a (geometric mean) average Interval amplitude invariance of magnitude.

This seems logical. If one changing property of a wave, that is phase, leaves a given law unchanged, why can not its other property, amplitude? Alternatively, if one property of a circle, rotation, may leave some law invariant, why should not the other property, radius?

The Interval is equal to a multiple of velocity factors, which are greater than and less than light speed, by an equal amount, which is a given local velocity. The important point is that the Interval magnitude may vary on a positive and negative scale of values but it does so, such that their geometric mean value remains invariant. The Interval is a geometric mean form. The property is one of geometric mean Interval magnitude invariance. A damping or amplifying factor affects both positive and negative values of a range of Interval values, while its average value may be unchanged.

The positive and negative values are the upper and lower values of a scale from superluminal velocities to equally subluminal velocities. Superluminal velocities are not a mere mathematical contrivance, in Quantum ElectroDynamics (QED). GM invariance of the Interval presupposes, as quantum physics also presupposes, their counter-balancing existence to subluminal velocities. Classical physics only recognises the latter.

The Interval, having rotation symmetry is only part of the picture. This is just a direction symmetry. A vector, like the Interval, has both direction and magnitude. The Interval rotation symmetry implies conservation of angular momentum. Therefore, (and this is my qualification, not that of conventional physics) the Interval has vector symmetry of radius magnitude, as well as rotation direction, which implies a conservation law, not only of angular momentum, but of vector momentum.

(Momentum is mass times velocity. And a damping factor or amplifying factor is changing the energy put in the system.)

A symmetry operation is one that leaves something, it works on, unchanged. It displays an invariance in the world, which implies conservation of something. The Noether theorem derives conservation laws from symmetry.

* * *

### After-note: Interval exponential decay for higher dimensional unification of forces.

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Lisa Randall popularised her work, in Warped Passages. The subtitle is "Unravelling the Universe's Hidden Dimensions."

In the Mathematical Notes, at the end of the book, which she makes look a lot easier than they are, note 36 gives a modified formula for Minkowski Interval of space-time (Euclid geometry of three spatial dimensions minus a fourth dimension of time). Randall and her colleague, Raman Sundrum modified the Interval into five dimensions, by adding a fourth spatial dimension to the other four dimensions. They also multiplied the Interval by a damping factor.

A damping factor is the coefficient used to describe things like the diminishing trofs and crests of a series of water waves, such as from a stone thrown in a pond, or the decrease in vibrations of a spring after it has been stretched and let go.

The damping factor is the inverse of an exponent to some power. In this case, the power is in terms of a fifth dimension (as a fourth spatial dimension) multiplied by a constant. This factor makes the strength of gravitational interaction fall off exponentially in the fifth dimension, traveling between areas ("branes") bounding that higher dimensional space. Randall refers to the factor as a "warp factor" which measures the warping of space by the presence of highly concentrated gravitational mass.

Gravity is extraordinarily weak compared to the other three forces of nature. They have become known, not too helpfully, as the strong, weak, and electromagnetic forces. This disparity might be explained, if gravity is as strong as the other forces on one brane, but being confined there, only interacts weakly with the other three forces on another brane, the area of our own experience of nature.   
Randall explains:

"...extra dimensions can be hidden either because they are curled up and small, or because spacetime is warped and gravity so concentrated in a small region that even an infinite dimension is invisible. Either way, whether dimensions are compact or localised, spacetime would appear to be four-dimensional everywhere, no matter where you are."

There are many variations on this possible scenario, including the possibility of unification of the four forces of nature at comparable strengths in higher dimensional space.   
Also Randall has developed the idea of different numbers of dimensions too far away to see, on the scale of the universe, in contrast to the older idea of extra dimensions being rolled up too small to see.

#### References.

This chapter does not pretend to be an authoritative account of symmetry and conservation law. Victor J Stenger, in The Comprehensible Cosmos (2006) supplies that very well. I thought I understood some of the earlier part of this popular exposition fairly well at the time but had long since forgotten reading it.

Heinz Pagels: two popularisations, regardable as modern classics: Perfect Symmetry (1982); The Cosmic Code (1985).

Richard Feynman: The Character of Physical Laws. (1965).

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# Mach principle in Mathematics and Special Relativity, and a Caldera Cosmos.

Table of contents.

#### Sections:

Mach principle in physics and its meaning for mathematics.

Vectors as complex numbers.

Imaginary numbers as averages of their dimensional ranges.

Means and dispersions of complex numbers.

Mach principle in maths applied to the Interval, and a Caldera cosmos.

Foot-note on circular functions.

References and comments.

### Mach principle in physics and its meaning for mathematics.

Ernst Mach was a distinguished physicist. He was also a philosopher of science, who formulated the doctrine of Positivism. (I briefly discussed this and Max Planck taking exception to it, in the chapter, The Moral Sciences as the Ethics of Scientific Method, of my book, Science is Ethics as Electics.) Crudely speaking, Mach believed that scientific concepts should be tied closely to reality.

Max Planck believed that sometimes a concept could be useful for explaining a process, even if you couldn't identify something to which it corresponded. In 1900, Planck didn't believe in the reality of his quantum concept. But in 1905, Einstein refered to light quanta or photons, as they are now called, in explaining the photo-electric effect.

One can see that Mach has the experimental point of view and Planck has more of a theorist view-point. This Positivist emphasis of Mach is evident also in the Mach principle. This was a program for physics, that aimed to remove preconceived ways of looking at the physical universe.

The classic case is Newton making a picture frame of absolute space and time, within which to see the universe in motion. This was the best technique available at the time. But a complete picture of the universe has no outside picture gazer. It is like a Velazquez, where you see a small image of the painter reflected in a mirror in the painting.

By definition, the universe contains everything within itself. The whole picture shows the universe entirely in terms of its own motions, and not with respect to some outwardly imposed frame. After all, anyone can come up with their own point of view. The theory of Relativity relates different points of view so that all observers have a common basis for precisely agreeing in their measurements of any event.

Julian Barbour, author of The End Of Time, describes his collaborative work to fully apply Mach principle to modern physics. He found that General Relativity is consistent with Mach principle. Einstein was influenced by the principle but he didn't know that he had succeeded in incorporating it. That just goes to show how involved is his theory.

Barbour illustrates mathematical structures. His attempts are severely tried on the mathematics of General Relativity. This essay sets a much simpler problem, indeed a childishly simple problem. That is because the mathematics, to follow, comes from the so-called New Mathematics taught to primary school children. Here, we treat the new math in a newer way. But this Newer Mathematics merely applies very simple forms of statistics to the foundation course.

Some elementary math, I was doing (the only sort I know how to do) made me realise that mathematics itself can be treated as self-representative. I mean that numbers can represent each other. This compares to Mach principle, in which he envisages any given mass in motion as gravitationly measurable solely in relation to the distribution of all other masses in the universe.

An elementary statistical treatment of basic mathematics shows that numbers can represent each other. The function of statistics is to represent or measure for typical items, in terms of suitable averages of any given range of items. Abstract numbers are inter-changable, as range and average, which makes statistics self-representative mathematics. Saying mathematics is itself statistical effectively applies Mach principle to mathematics. This may be the abstract basis for Mach principle of a self-representative physical universe.

A statistical meaning to complex numbers is just about enough to demonstrate that mathematics is self-representative, in the statistical sense. This is because it has been demonstrated that, after imaginary numbers, no more extensions to the number system are needed. At least, algebraic equations are always solvable within the complex number system, of real numbers combined with imaginary numbers. This was the fundamental theorem of algebra, proved by Carl Friedrich Gauss.

Without going into any detail, real numbers can be represented on a measurement scale, like two rulers, end to end, at a point marked zero. From zero, one ruler is conventionly assigned to be going in the direction of the positive numbers, +1, +2, +3,... and the other ruler is marked off as going in an equal and opposite direction, -1, -2, -3,...

Mathematicians have proved that rational numbers, ratios of whole numbers, or fractions, can be placed on this scale. And so also can be irrational numbers, which do not have an exact fraction. For example, the square root of two roughly equals 7/5. The ancient Greeks proved it does not have an exact fraction.

Mathematicians thought all numbers were accounted for by the real number scale, until the emergence of so-called imaginary numbers.

### Vectors as complex numbers.

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In trying to solve algebraic equations, mathematicians sometimes came across solutions in terms of the square root of minus one. This didn't make sense as a number, so it was called an imaginary number and was given the symbol, _i_.

Thus i² = -1. So, this imaginary number, to the power of four, must equal minus one squared, which equals plus one.   
i, to the sixth power, equals minus one again; i, to the eighth power, equals plus one. In short, powers of i produce a cycle of alternating negative ones and positive ones.

Consider this cycle in polar co-ordinates. Going from plus one to minus one and back involves two consecutive turns of 180 degrees or pi (the Greek letter for pi being the usual symbol, π). The square of i is in effect two repeated turns by 90 degrees or π/2. The imaginary number came to be called "operator", i, or the operation of one ninety degree turn. i³ means a 3 times π/2, or 270 degree turn. And so forth.

Imaginary numbers are not really imaginary. This is just a historical name that has stuck. They are used in algebraic representations of vectors. A vector like wind velocity has speed magnitude and compass direction. Wind vectors might be represented on a map by length of arrows for their strength, and pointed with reference to global co-ordinates, for their direction.

A complex number is a familiar or "real" number and an imaginary number. The way they are added together on a graph (see figure 1) is like the way two vectors, as arrows with their lengths and orientations, add up to a combined length and orientation.

#### Figure 1: Complex numbers add like vectors.

Take the complex number, z = x + iy, which means real axis, x, and right-anles axis, y, the imaginary axis. Most simply, this could be: 1 + i1, that is 1 + i. The meaning of this complex number, z, is in the operation it performs. Refering to figure 1, suppose a marker, z is at position P, on the x-axis of a co-ordinate system. The operation 1, on z, leaves it where it is. (If x were 2, the marker would be sent twice its present position along the x-axis.) The operation i, on z, turns it thru 90 degrees, to position N on the y-axis. If x were zero, so that only the latter operation were to be considered, then z would simply be moved from its original position on the real axis to the position N, at 1 on the imaginary axis.

This two-part operation of a complex number is performed like the addition rule for vectors. The imaginary number, i, is like a vertical arrow that must be added to the end of the horizontal arrow on the real axis, resulting in the new marker position at S.

The diagonal line, r, corresponds to the so-called resultant vector, OS, of OP and ON, the two vectors combined effect. So far, the complex variable has been described in rectilinear co-ordinates. It can be put in polar co-ordinates of angle Q and radius r. Here, the term "radius vector" would apply. The equivalent values are x = r cos Q and y = r sin Q.   
r is given, in terms of x and y, by Pythagoras theorem.

In polar co-ordinates, the stretching (or shrinking) and turning, of complex number operations, is given to the radius and the angle, respectively.

In rectilinear co-ordinates, any complex number is given the convention in algebra of z = x +iy.   
Complex variable, z, is not to be confused with a third co-ordinate of three traditional quantities or real numbers. Complex z is any position on the plane marked out by the two x and y ordinates, where y is refered to as the imaginary axis, as distinct from the real axis, x.

This differs from traditional geometry. Suppose there is a second complex variable, w = u + iv. Traditional relationships between variables, in effect, make y and v equal zero, so that we have only to deal with functions of real variables, which here would be variable, u, as a function of variable, x. On a graph, the real function can be two correlated sets of values plotted from the linear vertical axis for the dependent variable in relation to the linear horizontal axis as the independent variable.

But a complex function involves the correlation of two planar functions on two distinct graphs.

The complex variable, z = x + iy, can have a change of sign into Z = x - iy. On a graph (see figure 2) the latter is the mirror image, across the x-axis, of the former. The mirror variable, on the negative side of the y-axis, is called the "complex conjugate," of the complex variable on the positive side of the y-axis. (It has its own special symbol in the text-books: the letter, z, with a tilde or wavy bar on top. Here we have to use "Z".) The multiplication, of a complex number by its conjugate, is: (x+iy)(x-iy) = x² + y² = r².

#### Figure 2: Graph of circle with example of relative positions of a complex variable and its conjugate.

### Imaginary numbers as averages of their dimensional ranges.

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There seems to be no obvious reason why any real number cannot always be an average that is representative of other numbers. Rather it is the obviousness of this assertion that makes it seem trivial and unimportant. Less immediately obvious is that any imaginary number can always be considered as an average of other imaginary numbers.

In figure 2, the two axes with their four poles can be represented in terms of imaginary numbers. Thus, the x-axis at 1 becomes i^0, which reads: the imaginary number i to the power of zero. This equals one. Turning 90 degrees or π/2 degrees, we come to the positive y-axis, already designated i, which is short for i^1, or i to the power of one. Ninety degrees further, on the negative x-axis, -1 = i² (or i^2, in different notation). Another π/2 degrees turn brings to the negative y-axis at -i = i^3.

Turning another π/2 degrees brings us back to where we started, the positive x-axis. But to signify that, we have been thru 4 turns of the circle, the imaginary number becomes i^4. This is the same position as i^0. The difference is that i^0 means that there have been zero turns of the π/2 or 90 degree turn operation. When the letter, i, is used to signify this operation, turn π/2 degrees, it is sometimes called the operator, i.

The degree, of the power of operator i, signifies how many 90 degree turns it goes thru. To show that an imaginary number can always be represented as an average, we remember that an average or mean represents, or is a typical item in, a range of values. We can show that each operator is the operational mean of the range of its adjacent 90 degree turns. It could be the average of a range of indefinite length.

The mean, to show this relationship, is the geometric mean. For the range, we take the ends of either the x or y axis. Thus, a dimension has a statistical interpretation as a range. It doesn't matter how many number of turns are taken. But we'll start at the beginning with x-axis range from i^0 to i^2, that is from one to minus one in real number terms. (We actually will be using an imaginary number as the average of a real number scale. Except that the real numbers have been transformed into turn-operators along with the imaginary number dimension.)

When powers are multiplied, an addition rule applies to their indices. Multiplying i^0 by i^2, you simply add indices zero and two: i^(0+2) = i². Taking the square root of two multiplied end-points of a range gives the geometric mean of the range. The square root of i², which is: (i²)^1/2 = (-1)^1/2 = i.

This interpretation modifies the famous expression that the imaginary number is the square root of minus one. Its statistical meaning is that the imaginary number is the geometric mean of the range from minus one to one. The geometric mean imaginary number is found by applying the arithmetic mean to the indices or powers of the range values. In the above example, take indices, zero plus two, then divide by two (for the two range values) which gives average index, one. And i, to the power of one, is just i.

To take the next example, if the y-axis is the range of values from i (or i^1) to -i (or i^3), simply add the powers one and three, which equals four, and divide by two, which results in two. Thus, i to the power of two, or i^2 = -1.

In principle, one needn't stop at end-points of two number ranges. For instance, a geometric mean of a range of three imaginary numbers would be the cubic root of the three numbers multiplied.

### Means and dispersions of complex numbers.

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Having considered imaginary numbers as rotation operators, what of complex numbers, that combine real numbers with imaginary numbers? How are they to be considered statisticly, as averages with ranges?

We can find the geometric mean of a complex number by multiplying a range of complex numbers and taking the root corresponding to the number of items in the range. For a range of two complex numbers, we take the square root. With regard to figure 2, there are four quadrants, each with their own complex number variable. Upper right quadrant is the complex number z = x + iy. This maps any position in that quadrant of the two-dimensional plane.

Lower right quadrant is the complex conjugate of z, usually symbolised as z with tilde (or wavy line on top), here as Z = x - iy. The top left quadrant is -x + iy = -(x - iy) = -Z. That leaves bottom left quadrant, -x - iy = -(x + iy) = -z.

We can take the square roots of all the combinations of z, Z, -Z and -z, to find their respective geometric means (GM). The GM of z(-z) = (-z²)^1/2 = iz = i(x+iy). The GM of Z(-Z) = iZ = i(x-iy).

We now multiply, in turn, the adjacent complex numbers for each quadrant. This gives us a result that can be equated, by theorem Pythagoras, in terms of the radius squared. The square root gives the geometric mean, which is in terms of the radius, r.   
The GM of zZ = (x²+y²)^1/2 = r.   
The GM of z(-Z) = i(x²+y²)^1/2 = ir.   
The GM of (-z)(-Z) = -r.   
The GM of (-z)Z looks the same as of z(-Z) but we assign it the value -ir. This answers to the possibility that the square root of minus the radius squared can be plus or minus ir.

But this assignment does require a general sign convention regarding the non-equivalence of z(-Z), whose square root equals ir, and (-z)Z, whose square root equals -ir.   
The same applies to the next step, where (-z){-(-Z)} is also not equivalent to {-(-z)}(-Z). Likewise for successive negations, with the complex number or its conjugate having one more negative sign, before it, than the other.

To use the jargon, the previous paragraph is a non-commutation rule, where the two multiples have the same number of the same signs but their different positions in the brackets render a different result, when their respective roots are taken.

We can now relate each geometric mean to the two bounds of its range. I assumed, without really knowing, that this can be done for a complex number range, as it is done for a real number range. For example, this involved taking GM, iz, of -zz, then subtract lower bound -z from iz: -z - iz = -z(1+i). Now subtract the upper bound, so: z - iz = z(1-i). If you add the two differences from the mean, you should get the total dispersion (that the GM represents as an average or typical item): -2iz.

However, by trial and error, I found it was slightly simpler, to find the dispersion about the geometric mean, by subtracting each bound from the geometric mean, and then adding each difference, for the over-all dispersion. Changing the way round the difference is taken, between the mean and its bounds, changes the signs. The dispersion, about iz, changes from -2iz to 2iz. The dispersion about iZ, GM of -ZZ, is similar: 2iZ.   
Thus, in both cases of the geometric means, iz and iZ, their dispersions are twice their magnitude.

The meaning of a complex number multiplied by an imaginary number is that the x and y axes are inter-changed: iz = i(x+iy) = -y+ix = -(y-ix). This is like -Z = -(x-iy), except the axes have been changed round. And iZ = i(x-iy) = y+ix. This is like z = x+iy, except that the axes have been changed round.

More revealing are the cases of multiples of complex numbers by their conjugates, including under various signs. The GM, of zZ, is r. Its dispersion is the sum of the two bound differences, from the mean, of: r - (x+iy) and r - (x-iy). Adding the two sides of the dispersion gives: 2r-2x = 2(r-x).

The two sides of the dispersion about the geometric mean are unequal. We would expect this from the geometric mean on a real number range. But this is a complex number range and there is a twist, that when you add the two parts of the dispersion, their mutual cancellations are such that each side is left with an equal contribution to the dispersion.

This turns out to be the case for all the other dispersions. In table 1, the dispersions are all multiples of 2, for the two sides of the dispersion making an equal contribution to the over-all dispersion. (See foot-note.)

Table 1 is for the geometric means and their dispersions of the successive multiple pairs of complex numbers, where each complex number maps a quadrant of a circle.

Table 1: Geometric means and dispersions of multiplied pairs of quadrant complex numbers. Complex multiple | Geometric mean | Dispersion  
---|---|---  
zZ | r = r.i^0 | 2(r-x)  
z(-Z) | ir = r.i^1 | 2i(r-y)  
-z(-Z) | -r = r.i² | -2(r-x)  
(-z)Z | -ir = r.i^3 | -2i(r-y)  
-(-z){-(-Z)} | r = r.i^4 | 2(r-x)  
-(-z){-(-(-Z))} | ir = r.i^5 | 2i(r-y)  
-(-(-z)){-(-(-Z))} | -r = r.i^6 | -2(r-x)  
-(-z)(-Z) | -ir = r.i^7 | -2i(r-y)

The geometric means of complex numbers multiplied by their conjugates, under various signatures, all appear to be geometric means of rotation of the radius. The GM of zZ equals r, or (i^0)r. This stands for a zero turn of the radius line, resting on the positive x-axis.

In the next row, ir would be the radius line turned π/2 degrees or 90 degrees to rest on the y-axis as the imaginary axis.   
i².r is then two 90 degree turns to the negative x-axis, and -ir = i^3.r is three such 90 degree turns.

i^4.r is four such turns, starting to repeat the cycle all over again.

The previous section showed how imaginary numbers can be geometric means. The only difference was that their magnitude was implicitly unitary. That is they were all set at r = 1, such as: i^0 = 1.i^0 = 1.1 = 1; i = i^1 = 1.i^1; i² = 1i² etc. By replacing 1 with r, any positive number, all we've done here is to lift the restriction on the magnitude of the imaginary number. These imaginary numbers represent in increasing powers, the successive axes that divide the circle into quadrants. Multiplying the imaginary numbers by r, merely represents the circle as not being confined to a radius of unity, in considering rotation operations.

The dispersions from geometric means of complex numbers multiplied by their conjugates all relate to the difference between the radius r of a circle and values on the x or y axes, as formed by dropping perpendicular lines, from where the radius line meets the circumference, to the x or y axis. That is, as in figure 1, if the radius r were inscribed in a circle.

Table 1 further signifies not only a circle of radius r, but an outer circle of twice the radius or 2r. The inner circle of radius r, represents the geometric mean. The distances, from zero to r and from r to 2r, represent the respective dispersions with their equal distance contributions to the over-all dispersion. The dispersion varies, according to the values of x or y. The maximum values of x or y, at equal to r, reduce the dispersion to zero. Minimum values of x or y, at zero, maximise the size of the dispersion to 2r. That is the first row on table 1, similarly for corresponding positions on the outer circle, 2ir, -2r, etc, shown in the successive rows of the table.

### Mach principle in maths applied to the Interval, and a Caldera cosmos.

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The Minkowski Interval, unifies different view-points of an event, approaching light speed, into one shared space-time observation. It is essentially the circular function just described. The Interval, I, relates to the radius, r, which is fixed, like the shared space-time observation.

Below is the equation of the Interval, I. Light speed is c. Two observers of an event, measure different local times, t and t', and local velocities, u and u'. The velocities can be, but (being vectors) don't have to be, considered in just one dimension.

I² = (ct)² - u²t² = (ct')² - u'²t'².

Suppose we subtract the below-average dispersion (which is the Interval, as the geometric mean, minus the lower dispersion bound) from the above-average dispersion (which is the upper dispersion bound minus the Interval):

I - ( ct - ut ) - {(ct + ut ) - I} = 2I - 2ct.

This difference between the dispersions, below the (Interval) average and above the (Interval) average, for any given observer, may be compared to classical relative motion which subtracts the velocities of two observers going in the same direction.

Thus statistical special relativity seems to re-create a statistical version of Galileos relativity principle in classical physics.   
The sum total dispersion of the below average dispersion plus the above average dispersion may likewise correspond to the classical relative motion for two observers moving in opposite directions, so that they have to add their respective velocities to get their relative motion to each other.

The Intervals dispersion difference, 2(I - ct), is like table 1, on the first row: 2(r-x). Conversely, the circular function table values were derived by adding dispersions but with complex number bounds.   
Also conversely, the circular functions dispersion difference is: r - (x-iy) - {r - (x+iy)} = 2iy. While, this corresponds to the Interval dispersions sum total of:   
I - ( ct - ut ) + {(ct + ut ) - I} = 2ut.

From table 1, the radius becomes the Interval; ct is on the x-axis; ut = iy, where y is distance, is on the y-axis. In special relativity, using Minkowski Interval, there is an imaginary fourth dimension, usually related to time as a dimension.

The Interval is the constant observation of an event by all observers. This can be represented by the constant radius, r, making a perfect circle.   
Alternatively, re-arrange the Interval in terms of light speed constant, c:

c² = I²/t² + u² = I²/t'² + u'²

c² = I²/t² - (iu)² = I²/t'² - (iu')².

The geometric mean (GM) is then the speed of light, with a dispersion between bounds, (I/t - iu) and (I/t + iu). Subtracting each bound from the GM, which is c, and adding the two differences gives: 2c - 2I/t = 2(c - I/t). That is for one of the observers. The same would apply to an other observer with the indexed velocities and times, t' and u'.

This consideration leads us to a statistical definition of different observers, in special relativity. Their respective observations measure different velocity dispersions about the same geometric mean velocity of light.

Looking at the total dispersion, 2(c - I/t), we have a theoretical upper bound of twice light speed and a lower bound of zero speed, as I/t varies from zero to c.

The speed of light is constant and this would be represented by the constant radius, r, making a perfect circle. One side of the dispersion is from the zero-valued origin of the circle to the circumference. The other side of the dispersion is from an outer circle, twice the radius of the geometric mean radius of the light speed up to the inner circle of radius, r. The shape of these two sides of the dispersion might look like a caldera.

In classical physics, a constant light speed is a fundamental postulate of special relativity. Statistics treats light speed as an average about differing velocity dispersions measured by local observers.

This formulation reduces light speed constancy from an axiom of the Special Relativity theoretical system to a derivation. Light speed constancy is explained as a statistical consequence rather than asserted as an essential condition of special relativity.

This is in keeping with quantum physics. To quote from Richard Feynman, in _QED_ :

"It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, _c_. The amplitudes for these possibilities are very small compared to the contribution from speed _c_ ; in fact, they cancel out when light travels over long distances. However, when the distances are short...these other posibilities become vitally important and must be considered."

A caldera is a volcano with increasingly steep inner and outer walls - However, in special relativity, a "caldera" is modeled so that the inner and outer walls never quite meet. A further difference, from the geographical caldera, is that, for the light speed model, the walls would be of infinite height that no body could ever reach. You have to imagine someone climbing the ever steepening inner side, finding it tougher and tougher going with every step up. And never reaching a summit. (The outer wall would be just as impassable, but not for catching up to the speed of light, but for slowing down to the speed of light.)

The inner wall represents the climb of objects towards the speed of light. The outer wall represents the climb of faster than light objects (tachyons) never quite going slow enough to reach light speed. Tachyons are moving inwards from the outer to the inner circle, on a graph of space and time co-ordinates, so they are moving backward in time.

Slower-than-light objects are sometimes called tardyons. Lisa Randall, in Warped Passages, says that recent findings reveal tachyons as instabilities, in equations, to be resolved. This suggests that tachyons are a mathematical figment rather than a reality.

One physicist said of the failed attempts to detect tachyons, that they are no longer thought of as possible real objects but as fancies, like unicorns. However, the caldera model implies that tachyons will not be observed in our tardyonic cosmos. So, there is no conflict there.

But the model does imply the necessary existence of a complementary tachyonic cosmos to our tardyonic cosmos. This is because a statistical interpretation of light speed as a geometric mean speed implies it is the mean of a range of values, one of whose two dispersions is tachyonic, the other tardyonic.

Tachyons are unobservable by standards of classical and relativistic mechanics. But it cannot be ruled out that tachyons may be implied by quantum mechanical observations. Feynman diagrams may offer an intimation of the possibility of tachyons. Sometimes the diagrams have an ambiguous interpretation, that allows a particle, in a sub-atomic interaction of particles, to be considered as briefly moving backwards in time, in effect faster than light.

One might fancifully imagine a tachyonic Feynman, on the outer side of a caldera cosmos, showing a diagram, in which an interaction involves a particle, that might be thought of as moving forward in time.

A wave pulse can be made to appear to move backwards in time but no single photon can be identified as tachyonic. Paul Davies reported an experiment that reversed the effect of a medium on light waves, which normally makes their group velocity less than their phase velocity. When a stone is thrown in a pond, one crest appears faster than its group of waves.

When a laser excited cold caesium, "the peak of the wave pulse appears to exit the gas before it enters." However, Heisenberg Uncertainty Principle prevents a foton being picked out from the pulse as traveling faster than light.

Observable quantum effects might convey a tachyonic message from the outside of our caldera cosmos. For instance, relativity theory allows no escape of energy from inside a black hole event horizon. That is the radius within which even light cannot escape the gravitational pull of this super-massive object.

But a quantum effect allows a black hole to give off incredibly minute quantities of energy, called Hawking radiation (or Beckenstein-Hawking radiation). The vacuum of space, while being empty of matter, is also prone to the spontaneous creation and annihilation of matter, within the limits imposed by the uncertainty principle.

A particle and its anti-particle might be created. Being of zero net energy, energy is conserved. Before they annihilate each other, one particle might be captured by a black hole, releasing the other as Hawking radiation.

After their own fashion, cross-over effects might be detected in the context of a A black hole allows only one-way traffic into it. A tardyonic-tachyonic caldera wall offers an impassable barrier either way, as far as we know, apart from possible quantum effects.

A sub-atomic phenomenon called "quantum tunneling" allows particles out of a potential well of energy that they could not possibly escape, according to classical laws of motion.

Other weird and wonderful effects are continually being discovered in quantum physics, so it would be unwise to assume that there may be no possible experimental confirmation of a tachyonic outside wall to the tardyonic wall of a caldera cosmos.

A caldera is the crater left by a volcanic explosion. This big bang makes one think of the Big Bang in cosmology. The "calderic" property of light might be explained as a left-over of the big bang, just as a caldera is the left-over of an eruption.

The calderic "wall" of light is infinitely high. Any massive object experiences an accelerating difficulty in approaching light speed. Tho subject to further revisions, current atronomical data has come round to the view that the universe is set on an infinite expansion, indeed that the expansion is accelerating.

I don't know whether there is any correspondence, in the previous paragraph, to the light caldera and the expanding universe, as such. I am not saying there has to be. But one can't help speculate about a possible correspondence between a light caldera that is not infinitely high and another universe that does not go on infinitely expanding.

Why might there be a correspondence?   
Well, the analogy is only the roughest of guides. In some cases, the height of a volcano caldera might be governed by the violence of the eruption. And indeed, other factors. The analogy isn't to be taken too literally. It just helps with the drift of ideas, about a "multiverse," without being applied too rigidly.

A later chapter extends the model of a caldera cosmos to an hour-glass universe, indeed a whole pack or multiverse of them.

### Foot-note on circular functions.

As mentioned above, all the complex geometric means and their ranges show that the two sides of the dispersion are by themselves asymmetric. But when you add them, they make a symmetric contribution to the over-all dispersion. Intuitively, this makes sense, in that the mathematics is that of circular functions. Circles are symmetric, yet they can be described by exponents, which relate to series, or ranges, which are asymmetric about their average, a geometric mean. At least, real number ranges, that can be graphed as an exponential curve, are asymmetric about their most typical average, a geometric mean.

Complex number, z, equals r.e^iQ, where r is the radius of a circle; "e" stands for the infinite number constant called the exponent, of about 2.718..., describing constancy in the rate of change of the rate of change, and so on. The exponent is progressively calculated by successive terms of a series, called the exponential series, the importance of whose successive terms, starting with 1, followed by another 1, then increasingly rapidly falls off. In other words, it is an asymmetric series, unlike the normal distribution, which is a symmetric series. The exponent is taken to the power of imaginary number, i, multiplied by an angle Q, made by the radius line above the positive x-axis, within the (upper right) quadrant, mapped by z.

Being an infinite series, the simple exponential series, that sums to 2.718..., lacks a lower bound, with which to calculate a geometric mean. But in practise the first few terms of the series may approximate a pattern in a collection of data, and the geometric mean is taken of those most important terms.

The normal distribution, which is a symmetricised kind of exponential series, lacks upper or lower bounds, but since they are symmetrical, they average out, with the arithmetic mean, at the middle peak of its "bell curve." This shape graphicly shows the magnitude of successive terms, approximated by the binomial series.

#### References and comments

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Paul Davies. Light goes backwards in time. The Guardian. Science section, 20 july 2000.

Gary, previously known, I believe as "hitlong", from the special relativity e-mail group, read this essay, and said he had no problem with it.   
Of course, he is not to blame for its undoubted short-comings. Also, I have made a few extra comments since.  
Gary was the one person, who did credit, as "your discovery", my finding that applying the geometric mean, to averaging reflected light beam journeys, instead of the arithmetic mean, gives the right prediction to the Michelson-Morley experiment.

He had a counter-argument, which I deal with seperately, because it leads to wider issues.

Nevertheless, I did thank him for taking the trouble to read and comment on this chapter, in its original form as a web-page.

I think Gary must have been a teacher. When someone joined the group to ask about a physics problem, he immediately rumbled him as a pupil seeking to get someone do his homework.

Gary was one ray of light in the, too often, benighted atmosphere of email groups.

Another such group was beset by larking gossips with their Blackberries or whatever, re-sending interminable threads, by adding comments like: LOL (which I didn't understand at the time). Another group member, like myself, only with a dial-up connection, at the time, complained that the data over-load was obstructing his connection. I agreed. The moderator gave only empty sympathy. That's when I was obliged to leave.

One member was grossly and gratuitously rude to others (not me). When the moderator decided to expel him, others flocked to his defense, as tho their lives would be blighted without the fragrance of his presence. And he was allowed to carry on, in his own sweet way.

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# An hour-glass universe.

Table of contents.

#### Sections:

The light barrier as universal event horizon?

An Event-horizon universe.

A holographic principle of an event-horizon universe.

Modally rotated normal curve as an hour-glass model of the cosmos.

Relative tachyons and tardyons.

Necklace multiverse.

### The light barrier as universal event horizon?

This chapter is a speculative after-thought.   
Physicists, like Stephen Hawking, suggested that the universe itself might be a black hole (that has evolved from a virtual singularity). A black hole is an object from which not even light can escape, once it has passed a certain critical distance, forming a border called the event horizon. Light has no rest mass but it has mass in its energy of motion, which is therefore subject to gravitational attraction.

In terms of a black hole analogy with the universe, light cannot escape the event horizon but gravity extends beyond the event horizon, pulling objects towards it, until the point of no return. The light speed barrier might be the analog of the event horizon of the universe as black hole. There may be testable consequences that can disprove that assumption. There is the possibility that matter might be drawn into the observable universe. That sounds like a feature of the steady state model of the universe. Fred Hoyle may have been right, after all. (Hoyle so modified his early ideas that they anticipated later more orthodox thinking.)

The idea, that gravity extends outside the universe, where not even light, an electromagnetic force, can pass, does seem to offer an explanation to the puzzle of why gravity is so much the weakest of the four known forces in the universe. The force of gravity may be dissipated by extending outside a black hole of a universe.   
String theorists, like Lisa Randall, already mentioned, or Brian Greene, investigate the possibilities of where gravity might be supposed to have leaked into other dimensions, on the extra cosmic scale or extra small Planck scale.

### An Event-horizon universe.

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Back in 1985, Heinz Pagels (Perfect Symmetry) already anticipates much of the sophistication of modern cosmology. What follows is only a simplistic model. That simplicity attracts me, tho it does little justice to the complexities of reality.   
In the context of the notion that the light barrier might be a universal event horizon, it was thought that nothing could escape a black hole. That is, in a manner of speaking, until the theory of Hawking radiation (sometimes called Bekenstein-Hawking radiation). So, according to my way of thinking, Hawking radiation also escapes the light barrier, as a postulated event-horizon of the black hole of a universe.

Hawking radiation is supposed to happen like this: on the extremely small sub-atomic scale, the vacuum of space seethes with energy, that is continuously erupting short-lived sub-atomic particles. These are only virtual particles, bound by the restrictions of the Heisenberg Uncertainty principle. Quantum theory modifies classical energy conservation law, which would not allow energy to appear out of nothing.

For example, an electron and its anti-particle, a positron, may erupt from the energy sea. Being positive and negative, there is conservation of energy. They shoot off in opposite directions, also conserving linear momentum. In other words, the books are balanced, despite the vacuum having a zero-sum energy.   
These virtual particles will almost immediately collide with others from the seething energy sea. And when matter collides with anti-matter, it annihilates. Again, energy must be conserved and so it radiates in the form of fotons.

Hawking radiation predicts that the event horizon might capture only one of a pair of virtual particles that happens to be created right on its border. The virtual anti-particle gains the extra energy from black hole gravity to become a real particle and escape into space. Fotons are their own anti-particles but are capable of creation in quantum-entangled pairs, for instance as to the compensating effect on each other of changes in their respective polarisations. On the event-horizon, one of these might be captured and the other escape as Hawking radiation.

Since an emitted particle has to have positive energy, the captured particle must have a corresponding negative energy, to an observer in the outside universe. Therefore, the black hole is held to lose mass and eventually (on a cosmic time scale) evaporate, as a result of Hawking radiation.

My unqualified if fevered imagination links this with Feynman sum-over-histories conception of Quantum Electro-Dynamics. In his popular lectures, QED, Richard Feynman explains that on the extremely small sub-atomic scale, light does not travel in a straight line at a constant speed, as it is familiarly observed to do.

I say to myself, even on the large scale, the classical picture is not true of the erratic paths of lightning. The leap from the sub-atomic scale of Feynman integral paths to the human scale of lightning is as great as the difference between lightning and, say, some imagined cosmic scaled-up foton, in the Creation of the universe.   
This helps me to believe Feynman sub-atomic description of a foton not merely moving in a straight line but by all other possible paths and at varying speeds greater and lesser than observed light speed.   
The sum, of all these histories of the foton, averages out to the classical path of light.

Consider that the constant speed of light is the speed imposed by a universal event-horizon of gravity on light, such that performs the averaging of light speeds, in the sum-over-histories calculation.

Given that gravity moves at light speed, in the present universe, fotons and gravitons may be deeply related, in the unfolding universe, according to conservation principles.

"The cosmic egg" (to use a Terry Pratchett sort of concept) in a zero-sum energy vacuum of practicly infinite extent (nothing that is potentially something) might consist of, say, a particle-pair that realises an inconceivably small probability of creating a universe within the permitted bounds of the Uncertainty principle. The more energy that goes into a virtual particle-pair creation, the less time it is likely to last.

I don't set much store by this not-so-simple paragraph but I imagined one Creation scenario as a proto-foton, which has no rest mass. It might be one of a pair of quantum-entangled particles, the other becoming massively concentrated to a virtual singularity of a black hole. (Light still has energy of motion, so the universal black hole must have negative kinetic energy, whatever that is.) That is to say they are two parts of a conservative system, where a change in one part must be offset by a balancing change in the other part.

It is assumed such quantum entanglement is possible, because particles and black holes are both basicly similar, being described by the same properties of mass, force, charge and spin.

As the proto-foton grows universal, the increasing gravity of the black hole might relate to the foton, as sum-over histories of different light speeds. The gravity might capture slow speed light paths and energise them into tardyons, real particles, moving slower than observed light. Likewise fast light paths might be energised into tachyons, real particles moving faster than light.  
(This is an extension of the idea of Hawking radiation, where a virtual particle pair on a black hole event-horizon results in one escaping as a real foton.)

Thus, observed light velocity just happens to be the velocity of light at the event-horizon to a universal black hole.

The speed of light happens to be what it is, possibly because it just happens to be the speed associated with the event-horizon produced by the particular strength of the universal black hole in this particular universe. There may be a random selection of other universes with black holes of differing gravitational pulls and thus different event-horizons with different corresponding light speeds. That is a multiverse of "black holes and baby universes" to name the Hawking title.

Normally, light is not much altered by gravity which is too weak to affect it. The standard model of physics accounts for three forces of nature, because the fourth, gravity is so weak that it can be left out of account. Only on astronomic scales, involving the gravity of a star, does a passing light ray even bend out of its normal straight path.

Suppose, on the universal scale, light moves in a straight line because it is held apparently straight by its huge event-horizon, presumably a three-dimensional horizon that is the volume of space. The observed universe would be three-dimensional event-horizon.

An event horizon is not normally straight, it curves round the black hole, but on a universal scale, the horizon would be so large as to look straight, just as the Earth looks flat, to those on it, tho it is a sphere.  
The observable universe has an almost flat spatial geometry, in accord with the large-scale homogenous distribution of mass, not causing any marked curvature of space.

Finding a curvature of light in inter-stellar space would perhaps measure the circumference of the universe, as an event-horizon.

The early universe was thought to hold just energetic radiation that gradually underwent a transition to particle masses, like water crystalising, as it cools.  
Taking into account the infered existence of dark matter and dark energy, the universe appears close to critical density, for its relative stability or equilibrium. So, after Creation from a Big Bang, it does not collapse or fall back, under gravity, nor does it so escape from gravity as to rapidly expand to the point of dissipation.   
What is the reason for this happy coincidence?   
Suggested answer: because the universe, at critical density, is the poise of an event-horizon (of a) universe.

This critical density may be like an orbital trajectory, rather than a falling trajectory or escape trajectory, from the Big Bang. This is analogous to a planet having orbital velocity round a sun. Likewise, perhaps, the universe as event-horizon to a black hole.  
The universe isn't static but then an event-horizon doesn't have to be static, either.

Perhaps tardyons would be comparable to centripetal force. Tachyons would compare to centrifugal force. And observed light, as marking a universal event-horizon, would be analogous to orbital velocity of a satellite to the universal black hole.

Ballistics comes from classical physics. Relativist physics would use gravitational mass to curve out spatial paths for attracted bodies. The black hole compares to the maelstrom in the Edgar Allen Poe story. The boat is caught in the curling waters drawn towards the giant whirlpool, which hurls the craft round the whirling rim, which is like the event-horizon.

In the story, the boat is spat out again by centrifugal force of the spinning waters that spares it, by great good fortune, from going down the plug-hole. In general relativistic terms, a drawn space-craft might be lucky enough to escape by the Poe path to the spatial curvatures near a spinning black hole.

Feynman toyed with the notion that there is only one electron in the universe. Feynman diagrams of sub-atomic interactions can be re-interpreted involving time-reversals. There is all the time in the world for one electron to be all electrons, which indeed appear to be the same.

(There is something called C-P-T [Charge-Parity-Time reversal] symmetry. Yang and Lee predicted that Parity or mirror reflection symmetry was not conserved. Madame Wu confirmed in a certain atomic decay experiment that you could define the difference between left and right. All physical reactions were not perfectly mirror-symmetrical in the universe: the so-called "left hand of God."   
Only a combination of all three properties was found enough to maintain conservation.)

In analogy to Feynman one-electron supposition, the universe of light might be thought of as a sum-over-histories of all possible light paths. These consists of paths both slower and faster than the observed light speed, which they average, over (observable) distance.

Quantum entanglement seems to imply expansion of the universe from quantised units of space and time, around the Planck scale. Because, at that scale, space and time are thought to break down and the property of quantum-entanglement is that in a conservative system of, say, a pair of particles, a change in one automaticly produces a balancing effect on the other, over any distance and in no time.

The Big Bang might be the agent of transfering timeless and spaceless properties of sub-atomic particles from the quantum scale to the cosmic scale.

To sum-up this section:

The light speed barrier, that no massive object can surpass, suggests the universe has a black hole event-horizon, that no mass can surpass, not even light kinetic energy, the electro-magnetic force. But gravity extends from the black hole of the universe, explaining its relative weakness to the other three forces.  
Ultimately, that means that light only moves in straight lines, in the sense that the horizon of the sea appears straight.

Consider the Creation from a primitive particle pair, as a quantum-entangled conservative system, in sub-space and sub-time scales, possibly under (Heisenberg Uncertainty) random disturbances, driven into extreme opposites of gravitational singularity and universal light.

A cosmological Inflation of a proto-foton as a Feynman sum-over histories might be gravitationly realised analogously to the equilibrium of satellite ballistics.

If the observed constant light speed is analogous to a satellite orbit round a more massive star, then slow light or tardyons and fast light or tachyons respectively compare to perfectly balanced forces of centripetal and centrifugal force, that vector an event-horizon equilibrium of observed light velocity.  
An event-horizon universe might explain the "critical density" problem.  
Many of the other seemingly accidental physical parameters of the universe may be similarly explained as conservative equilibrium effects.

### Holographic principle of an event-horizon universe.

(Horizon, on BBC2. 18 january 2011).

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Leonard Susskind gave a simple explanation of his holographic principle. Usually books inform my viewing, rather than the other way around. I'd heard of the principle and of Susskind.

He was shocked by Stephen Hawking revelations about black holes. In particular, the evaporation of black holes seemed to imply that the information or precise properties of matter, drawn into them, was lost. This would break a basic law of the conservation of information.

Susskind got round this problem by suggesting that the two-dimensional surface of a black hole event-horizon might act like a hologram, storing the three-dimensional information of the captured objects passing thru that surface of no-return.

### Modally rotated normal curve as an hour-glass model of the cosmos.

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Cosmology, like mathematical physics in general, is a subject I am hopelessly unqualified to discuss. Astronomy has evolved hugely sophisticated observations, meeting its aspirations to become as effective as a laboratory science. Witness the 2009 observations of the Fermi telescope of extremely unequally energetic fotons moving at the same speed for over 7 billion light years.

My chapter, on Mach principle, or a self-representative universe, applied to mathematics and special relativity, derived a caldera model of the universe, treating the Interval in terms of a circular function but with an extra circle, twice the radius, from a common origin.   
By Mach principle, I meant a self-consistency principle applied to any self-contained discipline, mathematics, as much as physics. So, all numbers might be represented by other numbers, in terms of averages. This also applies to complex numbers, without which, the number system is incomplete.   
To this end, geometric mean calculations on complex numbers produced a second circle, twice the radius of the first.

The inner circle radius was not quite the conventional Interval measure (every observers common space-time measure) itself. Instead, the Interval equation was simply transformed so that light speed was the length of the inner circle radius (that is conventionally assigned to the Interval term, itself). Light speed was on the inner circle circumference. Every speed less than that radius was sub-light speed (or tardyonic speed). Every speed correspondingly above light speed was super-luminal (or tachyonic speed).

The caldera model meant that faster-than-light particles or tachyons were confined to an outer caldera slope of our tardyonic universe. However, the outer and inner slopes are of infinite height and never quite meet.  
Tho, "quantum tunneling" might display tachyonic symptoms of the near presence of the outer universe to our inner universe.

On the cosmic scale, my caldera model of the cosmos perhaps left out half the picture. Suppose the normal distribution is rotated about its norm, to look like an hour-glass. The caldera model may be refined into an "hour-glass." Why an hour-glass?

A volcanic explosion producing a caldera does not explain where the explosion came from. But a back-to-back caldera, like an hour-glass, can explode both ways into a universe and an anti-universe. This is a bit like two people pulling a party cracker. (One can just imagine George Gamov using such a picture, in his Mr Tompkins book.)

The point of such a both-ways explosion is that it is in keeping with the law of conservation of energy and conservation of momentum. For instance, even the vacuum of space is thought a seething sea of virtual energy. Virtual particles and their anti-particles can spontaneously emerge, from each other, shooting off in opposite directions, to balance the energy books, within the probabilities estimated by the Uncertainty principle.   
They are annihilated in collisions with other oppositely charged particles, likewise created and annihilated.

It has been suggested (for instance, in Black Holes And Baby Universes, by Stephen Hawking or Physics Of The Impossible, by Michio Kaku) that whole matter and anti-matter universes might be created on such lines. The hour-glass model follows this idea. One half of the hour-glass is a universe of matter. The other half is a universe made of anti-matter. When matter and anti-matter meet, they annihilate and their energy goes into the resulting explosion of radiation.

### Figure 1: Hour-glass universal expansion.

Figure 1 is not to scale, of course, and the hour-glass shape is not meant to be taken too literally. The normal curve in the hour-glass shape is itself just a mathematical illustration of some sort of exponential function to describe a process like the Big Bang.   
Current measurements show the universal space-time expansion to be accelerating. The figure for that would be a flaring, not a vertically-ended, hour-glass.

Figure 1 is a crude model of the Big Bang. The usual example, for popular reading, shows the universe expanding from virtually nothing, like a balloon. The surface is imagined as a two-dimensional simplification of a stretching "fabric" of the four dimensions of space-time.

The orthodox "Inflation" model has a super-fast space-time "balloon" expansion, to remove certain inconsistencies that would otherwise arise from certain astronomical observations of a homogenous universe, assuming that light was the same constant speed at the beginning of the universe, as it is now. The inflation would carry the light in the fabric of space-time to all corners of the universe, at a hyper-speed, that present light-speed could not have managed on its own.

Thus figure 1, the horizontal axis thru origin, 0, shows velocity. It is greatest at the origin, decreasing to either left or right (or indeed radially considered like a three-dimensional hour-glass). The shape of the hour-glass from its origin or middle, traveling verticly with time, governs the acceleration, or rate of change of velocity. Starting from the middle, it is at its narrowest stage, meaning that velocity remains very high over time; velocity then reduces rapidly over time, as the hour-glass billows out; until the walls become nearly vertical, meaning relatively less change in velocity over time.

### Relative tachyons and tardyons.

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A recent physical theory replaces an Inflationary preliminary to the Big Bang, with an explosion of light, at infinite speed, gradually slowing down, as the universe expands to its present size, by which time, light has assumed its familiar constant speed, at least within current limits of experimental accuracy. This Deformed (or Doubly) Special Relativity theory, or its second version, known as DSR2, is described by Lee Smolin, in "the trouble with physics."

Inflation is a rather ad hoc explanation, which just had to be brought-in to straighten-out the scenario, without naturally arising from it. Inflation may be dubious but that doesn't necessarily mean it is wrong. And most physicists seem to think that Inflation right, tho there are many competing models of it.

The declined light-speed model conforms to an implicit mutability principle of a temporal universe, where nothing lasts forever, without under-going transformation, possibly not even the universe itself. Even the proton is assumed to decay, tho the decay rate has not been determined, at the time of writing.

It is always possible to argue the point, that is why science requires evidence. However, it could be argued that, since time comes to a stop at light speed, it has the property of immortality. Another possibility is that light-speed varies with chance from universe to universe, while remaining constant within each universe. This assumption does away with the need for some sophisticated theory to derive the light speed constant, not to mention the other physical constants, that are just measured and plugged-in, without being logicly derivable from a theory like the Standard Model (of three of the four known forces of nature).

Suppose a Big Bang of light in equal and opposite directions of infinite speed, thus balancing the energy books, at any rate with regard to momentum. But suppose this two-ways imparting of light, at infinite speed, comes out of a sea of zero-speed light energy, located around the origin or middle of the hour-glass in figure 1.

The Creation is supposed an infinite light speed event, balanced in positive and negative directions. Then there might be an external residue of light at zero speed. Suppose, as the infinitely speeding light slowed down to its presently observed constant speed, the corresponding energy-sea light, outside the hour-glass, would be infinite speed minus current light speed. That is to say, external energy-sea light speed would have become greater than observed light speed, or tachyonic.

In special relativity, no masses can ever reach the speed of light. They are always observed to be slower than light speed. But the mathematics also allows for a sort of mirror-image process to normal reality, an Alice-in-Wonderland world of tachyons or faster-than-light particles that can never become slow enough to reach down to observed light speed.

A virtue of a varying external light speed, converse to a varying internal light speed, is that external light starts off from zero speed, so it is evident that external particles could not be less than zero light speed. (Quantum uncertainty might further explain why they could not be exactly zero speed, either.) As external light speed increased, this tachyonic property of particles, not being able to go under light speed, might be conserved, conversely to the observed tardyonic universe.

Even if the universe had no declining light speed, tachyons and tardyons could still be relative, if not within the universe, across universes, in a multiverse, consisting of a statistical array of different constant light speeds in each universe.

### Necklace multiverse.

At present, physicists measurements suggest an accelerating universe. According to Victor S Stenger, author of The Comprehensible Cosmos, this finding is not at all adverse to the possibility of a closing universe.  
In terms of an hour-glass model, the hour-glass would join to the base of other hour-glasses like a string of beads. To carry the analogy further, the string might even be in a loop, like a necklace. Given that the beads are "strung" in the time dimension, then time itself would be in a loop, being re-born every time the universe re-opened.

More-over, the row of hour-glass beads could be nested in multiple rows of beads. This would be a multiverse of parallel universes. In terms of figure 1, they could be imagined as side by side.

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# The Michelson-Morley experiment and the Principle of Least Action.

Table of contents.

#### Sections:

The legendary Michelson interferometer.

Potential and kinetic energy.

Least Action path of the original MMX (complex).

Geometric Mean anti-differentiation of its differential.

The normal curve as a probability distribution.

Unity-surpassing & Negative probabilities.

### The legendary Michelson interferometer.

The Michelson-Morley experiment was perhaps as revolutionary a chronometer as the salvaged "two thousand year old computer", the mechanical planetarium, that intensive research attributed to Archimedes. The Michelson interferometer used a synchronisation of light waves to tell the time. These are engineers of genius in their ingenuity. These are the people who make a difference.

The Michelson-Morley experiment found that perpendicularly split light beams relative to Earth motion took exactly the same time to be reflected back to source over the same distance.   
They calculated that an aligned beam, with respect to Earth motion, would take longer than a cross-ways beam. As far as I know, no one in physics has ever questioned the correctness of this calculation. At least, that is what I was told by a physics moderator, who refused to post my alternative suggestion.

Before the kind reader automatically assumes, that I must be wrong, recall the letters of Richard Feynman, Don't You Have Time To Think? edited by his daughter.   
A woman wrote to ask why she was marked wrong, when she was just following the Feynman lectures. Feynman replied: Because he was wrong on this point. We goofed. That will teach you not to follow authority.

The point about the Michelson-Morley calculation is that it is based on averaging the times taken by each beam over both the back and forth part of its journey. The average used for the earth-aligned return journey was the arithmetic mean. As I've said, more times than I can remember, the geometric mean is the average that gives the correct prediction.

In special relativity, massive objects significantly approaching light-speed, need disproportionately more energy, the closer they get to that limit. The path of this deceleration could be mapped out as a geometric series, averaged by the geometric mean. So there is nothing odd about using the geometric mean in the context of special relativity.   
Of course, special relativity hadn't been invented by the time of the Michelson-Morley experiment, which led to it.

The Minkowski Interval has a geometric mean form compatible with the Michelson-Morley experiment result. Whereas the original Michelson-Morley calculation, applied to the Lorentz transformations and the Interval, I found, don't compute.

### Potential and kinetic energy.

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Calculation in accord with the Principle of Least Action may compare with the way the physicists came to understand the Michelson-Morley experiment (MMX).   
In a conservative system, an object moving on a path, like a ball being thrown, always has a fixed total energy, tho in differing combinations of kinetic energy and potential energy. The actual path followed is determined by least difference between the kinetic energy and potential energy, integrated over time. In the case of the thrown ball, this least action path is a parabola.

Finding this path by calculation involves an approximation method. Likewise, experiment showed that the MMX calculation was the wrong answer. Straight away, this was treated as an approximation (by the Fitzgerald-Lorentz contraction) to the right answer.

When doing the Least Action calculation for MMX, I didn't appreciate that it was not just another demonstration of the need for using the geometric mean, as the right average in the MMX calculation. Rather, Least Action was an alternative that should be possible to apply. There may be a particular or analytic calculation, in terms of relative motion between particular objects. Alternatively, there is a holistic or systemic calculation, in terms of the varying (kinetic and potential) energies within a constant total energy of the system.

#### Diagram 1: Modified Michelson-Morley experiment.

Figure 1 is a schematic representation of a Modified Michelson-Morley experiment. The drawing is not to scale. The earth speed, u, has been greatly exaggerated in comparison with light speed, c, so that it shows at all. The light beam is split at source (on the lower left of the picture). The horizontal part of the beam, at light speed, c, goes against the earth motion, u. This gives the speed, c+u, because earth and beam are coming to meet each other, so their speed is combined. When reflected (on the lower right of the diagram) the speed with the direction of earth motion, is c-u.

Modified MMX also subjects the perpendicular beam to a back and forth change of magnitude, as if it also were moving relative to a second earth or simulacrum of earth, in a perpendicular direction to the original or real earth. The earth-perpendicular beam is still subject to drift from the perpendicular, with earth motion.

The mathematical meaning of this modified MMX is that the modified perpendicular beam changes from original MMX light speed value, c, to the same as the longitudinal value, I/t = (c² - u²)^1/2.   
Whereas, by theorem Pythagoras, the near-perpendicular drift line value changes to light speed, c.

The Minkowski Interval can summarise the difference between original and modified MMX. MMX has the same times for the two beams to return to source. Modified MMX has different times

Say the Interval for original MMX is I = t(cg), where observers, from earth-aligned or perpendicular view-points, relative to earth, measure the same time, t. The letter, g, stands for the Interval version of the gamma factor, which is more general than the Fitzgerald-Lorentz contraction factor. Then MMX, modified in the perpendicular direction can be summarised as I = (tg)c. In this case the perpendicular time is a multiple of the gamma factor, and so a shorter time taken than the earth-aligned beam.

You can visualise this from figure 1, where the drift-from-vertical beam now moves at speed, c, without any earth-relative motion drag to slow down its time. In original MMX, the drifted-perpendicular beam moves at the same speed as the earth-aligned beam and so takes the same time.

The Minkowski Interval, I, was not invented till some 21 years after the Michelson-Morley experiment. It is the agreed measure of an event by all physical observers, generalised from their local length and time measures:

I = t(c² - u²)^1/2 = t'(c² - u'²)^1/2.

The repetition of the Minkowski Interval, with the indices, merely signifies a second local observer with different local time and velocity, when he measures the same event. The equation shows, per observer, only one velocity, which may signify one dimension or a vector of two or three dimensions combined. The point is that both, and indeed all, observers measure the same Interval, I, for an event.

Here, the concern is simply with a velocity, I/t = (c² - u²)^1/2. It is shown as the vertical line, in line with the perpendicular mirror. Velocity, c, is shown on the diagram as the slightly slanting arrow, from perpendicular, under a drift effect from earth velocity, as this cross-ways beam goes forth and back from the perpendicular mirror.

This arrangement is confusing, because the positions of I/t and c are reversed, when the Minkowski Interval is graphed, on a complex plane, where one of the rectangular co-ordinates has an imaginary coefficient. But, for the modified MMX, in above diagram 1, this is not the case. Here the light speed, c, is the hypotenuse and the shorter vertical side, I/t = (c² - u²)^1/2. This is the "common sense" arrangement, because the figure 1 picture shows the longer side looking longer.

Diagram 2, below, of the complex plane of the Interval, I = t(c² - u²)^1/2, as hypotenuse, shows it looking longer, tho it has the lesser magnitude than ordinate, ct, light speed multiplied by a locally observed time, where either ordinate may have an imaginary coefficient, i. (Thus, either ict or iut.)

In diagram 1, above, the out-bound earth-aligned relative light speed, c+u, is co-terminous with the angle, J (in the top left corner. The vertical side, of the triangle formed, is I/t = (c² - u²)^1/2. Theorem Pythagoras can be used to calculate the hypotenuse velocity, F. Its square is the sum of the squares of the other two sides:

F² = (I/t)² + (c+u)² = (c² - u²) + (c+u)²

= (c+u)(c - u + c + u) = 2c(c+u).

Therefore: F = {2c(c+u)}^1/2.

The same procedure is followed with respect to the returning beam earth-relative velocity, c-u. The extent of its line, on the diagram, is co-terminous with angle, L. The right-angled triangle formed has a hypotenuse, H, also found by theorem Pythagoras:

H² = (I/t)² + (c-u)²

= (c² - u²) + (c²+ u² - 2uc)

= 2c² -2uc.

H = {2c(c-u)}^1/2.

The next step is to take the squared sines of the angles, J and L. (The sine is the ratio of the opposite side over the hypotenuse side, with respect to an angle in a right-angled triangle.)

Convention makes (sin J)² = sin²J.

sin²J = (c+u)²/F²

= (c+u)²/2c(c+u) = (c+u)/2c.

Likewise:

sin²L = (c-u)²/H²

= (c-u)²/2c(c-u) = (c-u)/2c.

Therefore:

sin²J + sin²L = (c+u)/2c + (c-u)/2c = 1.

Unity is the over-all ratio of earth-relative light velocity to earth-independent light velocity.  
I liken this to a total energy equation. Total energy adds kinetic energy, or energy of motion, to potential energy, which may be considered stored energy.

On a roller coaster, when the car goes up-hill, it is slowing down, losing kinetic energy, but gaining potential energy. The (c-u)/2c term, in the total energy equation, is equivalent to the up-hill motion, because it is losing energy of motion, in chasing earth motion, taking its objective away from it, but storing-up energy for the return journey, when its relative velocity increases, with the earth carrying its objective towards it, the condition given in the term, (c+u)/2c.

Therefore, (c+u)/2c corresponds to a kinetic energy term and (c-u)/2c corresponds to a potential energy term.

Another energy condition (called the Lagrangian) is the kinetic energy minus the potential energy. The Hamilton principle of least action established that, in a field of force, a freely moving body will graph a Lagrangian over time, that minimises the area under the curve.

This area is found by integration (or anti-differentiation) over time, from start to finish. (An angle, like Q, is a function of time, having a dimension of angular distance, being angular velocity multiplied by time.)

Thus, subtracting the kinetic and potential energy terms:

sin²J - sin²L = (c+u)/2c - (c-u)/2c = u/c.

However, the subtraction can be either way or in either order, so the more complete answer is: ±u/c.

This is the ratio of earth velocity to light velocity.

From diagram 1, u/c = sin Q.

And: ±u/c = ±sin Q.

This angle measures the amount the cross-ways light beam, in the Modified Michelson-Morley experiment, drifts with earth motion, on its way to, or from, the perpendicular reflector.

To integrate, which means anti-differentiate, sin Q, gives by calculus rule: -cos Q. This equals: -I/ct = -(1 - u²/c²)^1/2 = -g.

Conversely, -sin Q integrates to cos Q. This equals: I/ct = (1 - u²/c²)^1/2 = g.

But ±u/c can be considered as a range about unity (analogous to a total energy value). That is from one plus to one minus local observer velocity, u, divided by light speed: 1+ u/c and 1- u/c. The range can be averaged or represented by its geometric mean:

{(1+ u/c)(1- u/c)}^1/2 = ±(1 - u²/c²)^1/2 = ±I/ct.

This positive or negative geometric mean also constitutes its own range, about unity, which, in turn, can be averaged by the geometric mean:

{(1+ I/ct)(1 - I/ct)}^1/2 = {1- (1 - u²/c²)}^1/2

= ±u/c.

Thus, the energy sum and difference relation, in its cyclical trigonometric integration, is mimicked or replicated by the statistics of continuous range averaging. (This seemed a hopeful find, in the course of my attempt to put traditional calculus on a statistical basis, and extend it to a geometric mean differentiation.)  
There appears to be a trigonometric oscillation between observers local velocity measures, as range limits, and the (geometric mean) average of those velocities ranged about light speed. This averaged range takes place on one dimension, because this is solely a real-valued modification of MMX, not the original complex (real and imaginary) two-dimensional MMX.

In particular, the geometric mean interpretation of a modified MMX is justified in terms of the Interval form and the principle of Least Action.

The modified MMX, is modified to be without an imaginary coefficient, which is to say with only real variables. So, if we call this just MMX real, its Path of Least Action appears to be a geometric series, exhibited in high energy physics, by the path of bodies with exponentially increasing mass as they approach the speed of light. This series can be averaged by a geometric mean, such as the Minkowski Interval version, g, of the gamma factor.

This factor, g, expands by the binomial theorem into a distribution. It is the kinematic version of the similar dynamics form, the Poincare-Einstein equation, E=mc². This is appropriate, because the Least Action principle is in terms of energy, kinetic and potential energy. The value, E=mc², is just the first two terms of the distribution, the others being relatively insignificant.

To speculate (meaning I don't really know) modified MMX or MMX real, has a local velocities path of least action, given by integrating for its graphed area below the path, which is the g factor distribution, from which E=mc² is derived.

* * *

### Least Action path of the original MMX (complex).

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Finding an energy relation for the (unmodified) MMX is not so simple.

### Diagram 2: (Original) Michelson-Morley experiment.

To set-up the Least Action principle for the original MMX, diagram 2 is nearly identical with diagram 1. The essential difference is that the perpendicular line is now light speed, c, instead of I/t (Interval distance divided by time). And the transverse or cross-ways beam (under earth motion drift from complete perpendicular) is I/t instead of c.

In this case, we take the tangent of angle J, to obtain again the desired ratio, (c+u)/c. Likewise tan L = (c-u)/c. This original MMX treatment (with complex variables) looks simpler but turns out more difficult than the Modified MMX treatment (just with real variables), obtained an energy total, set at unity. (Or set at c, if multiplying the kinetic and potential energy terms thru by c.)  
The corresponding result, in this case, is 2, or 2c, multiplying thru by c:

(tan J + tan L) = [(c+u)/c + (c-u)/c] = 2.

c(tan J + tan L) = [(c+u) + (c-u)] = 2c.

Subtracting these two terms produces 2u/c, or 2u, which can be positive or negative, depending on which term is subtracted from the other.

By the way, this result might be generalised for All-angle Michelson-Morley experiments. That is to say not only for a beam split at a perpendicular angle or ninety degrees, and not only for the LISA experiment, which splits the beam by sixty degrees. Call the beam split angle, angle N, (for want of a better letter). Then the general rule is:

(tan J)(sin N) = (c+u)/c.

And

(tan L)(sin N) = (c-u)/c.

This rule works for MMX because sin 90° equals one.   
In general, where tangent equals opposite over adjacent, and sine equals opposite over hypotenuse, the tangent adjacent is a similar perpendicular to the sine opposite, and the two cancel.

Returning to the Least Action procedure for LISA, take the integral of ±2u/c. This, by rule, is: ±2(u/c)²/2 = ±(u/c)².   
By multiplying thru by c, or setting c = 1, this might be rendered: ±u². Integration is a summation (in the continuous terms characteristic of the calculus). In this sort of problem, the integral represents the sum area under a graph of the path of least action, which a moving object generally follows.

The ratio, u/c is tan Q. Alternatively, one might take the integral (symbolised by a dollar sign, $) of a tangent:

$ tan Q.dQ = log.sec Q.

The sign, dQ, is sometimes translated as the integral "with respect to Q". It is a reference to the differentials or bits of Q which are being summed or added together. (Please excuse my lack of mathematical proficiency.) The term, log, refers to the log, base e, or logarithm calculated with the exponent, e, as its base number. The term, sec, is short for the secant, which is the trigonometry ratio, hypotenuse over adjacent, or inverse of the cosine ratio.

The secant of the triangle containing angle Q is [(c²-u²)^1/2]/c. Thus:

$ tan Q.dQ = log.sec Q

= log.[(c²-u²)^1/2]/c.

But this is also possibly calculated as: ±(u/c)². Therefore:

±(u/c)² = log.[(1-u²/c²)^1/2].

Calculus texts give the rule that a variable, which equals the exponential logarithm of an other variable, is also the index or power to the exponent, e, which complete term equates to the other variable.   
In symbols, one version might look like the gamma factor, g, but is actually a more general version than the Lorentz factor, that relates instead to the Minkowski Interval, and is here equal to an exponential function:

g = {(1-u²/c²)^1/2} = [1 +{(-1)^n}(1/2)!]e^-{(u/c)²}.

The mysterious factor, in the square brackets, is unity plus factorial one half, with an alternating sign coefficient. I am not well versed in integral calculus, so I don't know whether this resulting auxiliary multiple is to be expected. But it makes the equation true.

The left side of the equation, g, is the geometric mean of light speed with respect to locally observed motion of a given event, considered as more or less relative motion to light speed. Taking the geometric mean implies that, on average, this locally measured motion can never exceed light speed. The right side shows an exponential decay function of local velocity, u, as a ratio of light speed, which never exceeds light speed.

Expanding from both sides of the equation:

(1-u²/c²)^1/2

= 1 - (1/2)(u/c)² - (1/2)(-1/2)[(u/c)²]²/2! - (1/2)(-1/2)(-3/2)[(u/c)²]^3/3! \- (1/2)(-1/2)(-3/2)(-5/2)[(u/c)²]^4/4! -....

Notice this gives alternating signs:

= 1 - (1/2)(u/c)² + (1/2)(1/2)[(u/c)²]²/2! - (1/2)(1/2)(3/2)[(u/c)²]^3/3! +...

Compare, term for term, an expansion from the right side of the equation:

e^-[(u/c)²] = 1 - (u/c)² + [(u/c)²]²/2! - [(u/c)²]^3/3! + [(u/c)²]^4/4! -....

The coefficient series that provides the equality between the two series is:

[1 +{(-1)^n}(1/2)!] = 1 + 1/2 - (1/2)(-1/2) + (1/2)(-1/2)(-3/2) - (1/2)(-1/2)(-3/2)(-5/2) + ..  
where n = 0, 1, 2, 3,...

There may be some systematic way this relationship could be deduced. However, it is true, I have derived this equation in an ad hoc manner or by trial and error, rather than relying wholly on deduction, or a step by step logic. This no doubt is unsatisfactory.

This is like the old problem in scientific philosophy of the contrast of induction to deduction. Induction is like an observation that always seems true, unless and until you find an exception, but cannot be logically justified \-- at least not until some logical justification can indeed be found.

However, traditional calculus texts consist of many pages of integrals found by trial and error methods. Only differentiation follows from first principles. Anti-differentiation or integration, of a given function, works backwards till it finds the anti-derivative or integral, that logically differentiates into the given function.

* * *

### Geometric Mean anti-differentiation of its differential.

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In calculus, a simple ordinary differentiation, say, of the equation, that graphs as the curve called a parabola, y = x², differentiates to dy/dx = 2x. This is a sloping straight-line, tangential to the curve. The derivative is the ratio of changes to the dependent and independent variables, y and x.

In differentiation from first principles, the change to the independent variable is taken to a limit approaching zero, having isolated these changes (I call them variations), to find their values, as distinct from the original y and x (variables) values.

I began a statistical way of looking at this differentiation process. I rather fancy that the limiting process is something like reducing a range to its average. You cannot make the range disappear altogether, because an average is, by definition, of a range.

It so happens that a straight sloping line is also the graphical representation of an arithmetic series, whose average is the arithmetic mean. I showed that conventional differentiation, from first principles, amounted to either arithmetic mean differentiation or harmonic mean differentiation.

The Einstein equations of curved space are an example of non-linear equations, not solvable by the classical differential equations of linear analysis.

My reasoning is that a new form of differentiation, geometric mean differentiation might be suitable for non-linear analysis, perhaps even of general relativity.

Traditional differentiation, I maintain, is essentially a business of taking an ultimate arithmetic mean or ultimate harmonic mean. I proposed a geometric mean differentiation, which like traditional calculus, would also have anti-derivatives or integrals. In particular, this integration process could be applied to exponential functions.

Thus, it is assumed that function, e^-{(u/c)²/2}, is the end result of a geometric mean differentiation from first principles:

(dy/dx)^dx = e^-{(u/c)²/2}.

Working by a mix of retro logic and trial and error, the first step backwards might look like this:

(&y/&x)^&x = (1 - {(u/c)²/2}/&x)^&x.

Where &x is dx, before it has been infinitely increased. The powers, &x, may now be removed from both sides of the equation. Also, multiply both equation sides by &x:

&y = &x - (u/c)²/2

where {(u/c)²/2} is equated to variable, x:

&y = &x - x.

Implicit to the logic of this equation is:

&y - y = f(&x - x), where y = 0 and f = 1, an over-simple way of feeling one's way into the problem.

Multiply thru by 2:

2&y = 2&x - (u/c)².

If 2&x = 1, &x = 1/2. Then:

2&y = 1 - (u/c)² = (I/ct)².

&y = (I/ct)²/2.

&y/&x = {(I/ct)²/2}2 = (I/ct)².

(&y/&x)^&x = {(I/ct)²}^1/2 = (I/ct) = g.

Thus, g, the Minkowski Interval version of gamma (the "contraction factor") is a geometric mean differential. In traditional calculus, the term, differential, is used to mean a change in a variable, that is still a finite amount, not reduced to approaching a limit of zero. A derivative (which used to be called the differential coefficient) sends the independent change variable, &x, (which I call a variation) towards the zero limit.

By contrast, a geometric mean derivative sends the variation to an unlimit of infinity. So, my equation of the gamma factor to an exponential function, considered as a GM derivative, was actually relating a geometric mean differential to a geometric mean derivative.   
This suggests that such a relationship has some justification.

Apparently, g is not merely a geometric mean but a geometric mean differential, of which a geometric mean derivative is a function.

And there is some reason for confidence that an exponential solution is suitable in the Minkowski Interval application to the MMX case (rather than the modified MMX case). This is because the Interval deals in complex variables, representing a circular function, which is an exponential function (tho the circular function index is imaginary).

The Least Action path is the locally measured velocities, u, u', u", etc, of any number of observers, forming a random distribution (the normal distribution) of view-points, on a given event. The Least Action path integral graphs as a minimal area underneath. This is given by the result, e^-[(u/c)²/2], which gives a normal distribution area. The index value, u/c, is a reduction of a fuller operation in the index: {(c±u) - (c±u')}.

With respect to that index form, in an unskewed normal distribution, velocity, u', is zero, and the arithmetic mean or norm of the normal distribution is light speed, c. Otherwise, u' measures the amount of skew to the norm or mode. The value, c±u, measures the range of possible sample local velocities, as measured by all possible local observers of an event. Whatever turns out to be the average of the sample total of these local velocities, is the measure, (c±u'). This sample can range from zero to 2c, because, ±u can vary from -c to +c.

A standard practise for the normal curve, is to reduce the mid-point of the horizontal axis to zero. So, light speed, c, is conventionally equated to zero, and , by convention, the left side of the mid-point becomes negative, and the right side positive. In this case, it is on a scale of plus or minus the velocity, u. This transformation brings us back to the original form of the normal distribution equation, as the exponent to the power of a function of velocity, u.

The normal distribution formula for the norm is the arithmetic mean, which equals sample size, n, multiplied by the probability, p. The sample size is the maximum possible size of the sample, which is 2c. For the perfectly symmetrical bell curve, the probability of being in one or other side of the sample is exactly one half. The average sample is, therefore, np = 2c.(1/2) = c. Taking the horizontal axis of a graph, measured from zero, to 2c, then position, c, marks off the mid-point. It is directly below the highest point of the bell curve distribution, the single most populous point in the bell shaped distribution. The vertical axis measures from zero occurence of the distribution to this high point at the center.

For the arithmetic mean, A.M. = (c±u') where u' = 0. Then A.M. = c. Given n = 2c. Probability = A.M./n = c/2c = 1/2.

If A.M. = (c±u'), probability = (c±u')/2c.

This reminds of the "energy" equation, considered as a probability equation:

(c+u)/2c + (c-u)/2c = 1.

Where either side can be the probability in question.

### The normal curve as a probability distribution.

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After treating the equation of the normal curve, in terms of a geometric mean derivative, there follows some explanation of this curve as a probability distribution.

The normal curve was first published by De Moivre in 1733, and independently discovered by Gauss and Laplace. This is the exponential version of the binomial distribution. In graphical terms, the normal curve is the famous bell-shaped curve, a smoothed-out version of the binomial distribution.

If you take the numbers in the rows of triangle Pascal and graph them, the further down the triangle you go, the more detailed the distribution, the less crude the graph, the closer to the normal curve. (That is as long as the scale or magnification is not increased to show the increased detail of bigger distributions.)

A simple form of the equation for the normal curve is:

w = exp^-z².

Here "e" or "exp" is short for the exponent, an infinite number starting 2.718... The sign,^, is Google notation to show that the exponent is raised to the power, -z².  
This is not quite the standard form of the equation, which includes a constant coefficient (the inverse of the square root of twice pi) whose purpose is merely to standardise to unity the area under the curve. (The index is also divided by 2.)

A standard form of the normal distribution is:

F = {1/(√2π)}exp^-z²/2.

The value, F, is the height at any graph point along the bell-shaped curve. For example it could measure the number of people by height in ordered steps. It is a relative measure of how many (the frequency) people are of a given height when ordered in a line from shortest to tallest.

Most people come more or less in a compact range in the middle of the bell-shape. There are a relatively few very tall and very short people, whose numbers fall off very rapidly. These are the big drop in numbers represented by the wings of the bell compared to the high frequencies of middling heights represented by the bell dome.

The variable, z, is a standardising term, so that all normal curves may be compared. z = (m - np)/s. This formula is most simply explained in terms of the binomial distribution (approximation to the normal curve), for the relative probability that an event will happen, zero, or one, or two, or three or more times.

Suppose we have a bag of sweets, with an equal number of black to white. Suppose we randomly pick out samples of four sweets: n = 4. Then we would expect, in terms of the binomial distribution, that in one case, none of the sweets is black; in four cases, one of the four sweets is black; in six cases, two of the sweets are black; in four cases, three of the sweets are black; and in one case, four of the sweets are black. (The value, m, is the range of possibilities, 1,2,3,... or however many possibilities of selection.)

These results are summed-up as follows:

Example of binomial distribution for sample size, n = 4.

1 | 4 | 6 | 4 | 1 : Frequency, F,

0 | 1 | 2 | 3 | 4 : out of four sweets black.

-2 -1 | 0 |+1 +2: Deviations from Arithmetic Mean number of black sweets randomly picked, with AM, of two, standardised to zero.

The same argument follows for white instead of black, as they are in equal numbers.

This random distribution is reached by an expansion of the binomial theorem, in this case (1 + 1)^4 = 1+4+6+4+1. = 2^4 = 16. This is a two-term factor - the bi in binomial - for example, one black plus one white sweet, raised to the power of n. In this case, n = 4, the size of random samples of four sweets. Two to the power of four derives the sixteen possibilities mentioned, which occur with the Pascal triangle (row four) probability distribution.

The variable, n, does not have to equal four. For instance, it could equal eight, in which case the black and white frequencies of the random possibilities would follow the binomial distribution, or Pascal triangle (row eight), for n = 8.

The larger number of items in a sample, the more gradual the changes between them for any given measure, until the change seems virtually continuous. Thus, the normal distribution curve implies large numbers in its distribution or range. It is solved for the frequencies, F, the vertical height of the curve, for any given value of the range, m, on the horizontal axis.

For the sweets example, the average probability is that the event happens two times. Formally, this average, the arithmetic mean (A.M.) can be calculated most simply here by A.M. = np. Here p means a probability of one half.   
That is: np = 4 x 1/2 = 2.   
For example, from all the random samples of four sweets each, on average, two of the sweets are black (or by the same reasoning, two are white).

The remaining probability is assigned the letter q. In this case q is one half, because total probability is unity. If p is probability of success, q is probability of failure. If you were only allowed to pick out randomly four sweets at a time, but wanted black liquorice sweets, then your probability of success with picking the black sweets would equal your probability of failure from picking the white sweets.

Unequal probabilities, or probabilities other than one half, say one-third probability of success, p, apply, say, if only one-third of three kinds of sweets are desired. This leaves two-thirds probability of failure, q. The greater the inequality of probabilities, the more skewed the binomial distribution.

The letter, s, in the normal curve formula, stands for the standard deviation. (Texts use the Greek letter sigma, σ.) Whereas an average represents the most typical item in a distribution, the standard deviation is a measure of the dispersion or spread of a distribution. Its formula is the square root of the multiple, npq, or:   
s = (npq)^1/2.

Note the close relation of the standard deviation to the geometric mean, which takes the square root of two values, in this case the probabilities of success or failure.

In the above example, s = {4.(1/2)(1/2)}^1/2 = ±1. Two standard deviations, either way (positively or negatively) about an average, are reckoned to cover about 95% of the area under a normal curve or its stepped version, the binomial distribution. Our simplified example is unrealisticly small for statistical purposes. And in its case, two standard deviations covers the whole spread of the distribution, which is only plus two or minus two about the average.

Going back to: z = (m - np)/2s, in its standardised form, np is re-set to zero, and m becomes, in the above example any of the values: -2, -1, 0, 1, 2. The minus signs don't affect the result because z is squared in the normal curve formula. This exponential formula yields counter-part values to the terms of the binomial distribution. The results don't coincide very closely for small distributions like this one.

(And you have to adjust with the above-mentioned area coefficient, the inverse of the square root of twice pi. This itself has to be multiplied by the inverse of the standard deviation, if the standard deviation has not been standardised out of existence, by reducing the values of the normal distribution, to units of 1, 2, 3, etc standard deviations.)

### Unity-surpassing & Negative probabilities.

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The foregoing rests under the traditional assumption of positive only probabilities, defined as: p+q = 1. Or, probability of success plus probability of failure equals the sum total probability of finding a sample of a distribution in one area of the distribution or another. For example, a perfectly symmetrical bell curve gives an equal probability of one half of finding the sample in one side of the distribution or another. The probability of it being in the side, you are looking for, is conventionally assigned probability of success. Otherwise is the probability of failure.

Total probability, p + q = 1 = p + 1-p, can be extended to consider an upper and lower range of probabilities, like so:

1+p + 1-p = 2.

And (1+p)/2 + (1-p)/2 = 1.

Compare this with:

(c+u)/2c + (c-u)/2c = 1

= (1+u/c)/2 + (1-u/c)/2 = 1.

This describes an arithmetic mean probability, suitable for averaging a constant change of motion, or velocity.

Therefore, it should be possible to consider a geometric mean probability measuring a change in a change of motion, or acceleration:

The arithmetic mean proceeds by addition of terms and dividing by the number of terms. The geometric mean is taken by multiplying terms and taking the root to the number of terms. For two terms the root is just the square root:

{(1-u/c)((1+u/c)}^1/2 = (1 - (u/c)²)^1/2.

This is the (Minkowski generalised version of the Fitzgerald-Lorentz) gamma factor, which is therefore an averaged measure of acceleration, given by the geometric mean. That is to say, the gamma factor is the representative motion of an acceleration (or deceleration) range of motions about an upper limit, (c+u) and a lower limit, (c-u).

The geometric mean probability can be stated as:

(u/c)² + 1-(u/c)² = (u/c)² + I/ct = 1,

where I/ct equals the Interval divided by light-speed times a time (varying with local observers).

These two GM probabilities were produced by following the Least Action principle for Modified MMX or MMX real. The least action path has the smallest area under its graphed curve. This suggests that the area is a probability distribution, with these two probability components comprising its total probability area of unity.

I have been struggling to make over-all sense of MMX real. Because the two probabilities oscillate between each other, as each others integrals, they measure distribution area. Conversely, they are least action paths, integrating each other.

Perhaps, the local velocities are least action paths, in the sense that they are observers natural local measures. Whereas the Interval version of the gamma factor is the space-time least action path common to all observers.

In election method (explained in the last chapter) the form of the definition of the keep value, k, for both surplus and deficit transfer value, t, candidates, carried over to probability, allows both p+q and p-q to equal one. In the latter, negative case, the probability, p, is, by definition, greater than one.

The transfer value is defined as one minus the keep value, which is defined as the quota (in votes per seat, V/S) divided by the total transferable vote, T:

k + t = 1.

Or,   
t = 1 - (V/S)/T.

This applies when a candidate is elected, that is when the total vote (all of which is transferable) is greater than or equal to the quota. In other words, when the keep value is less than or equal to one.

When a candidate has not reached the quota, their votes are in deficit of a quota, their keep value is greater than one, and their transfer value is therefore, negative. And k-t=1. It is perhaps more intuitive to re-state as: -t = 1-k. This form tells us that the keep value must be more than unity and the transfer value must be negative, which is to say a deficit of votes from the quota, not a surplus of votes to the quota.

FAB STV works by averaging counts in which a candidate may have a keep value greater than unity in one count but sufficiently less than unity in another count, to be elected on an over-all keep value, provided it equals or is less than unity.

#### Reference:

Brian Cox and Jeff Forshaw (2011): The Quantum Universe. Everything That Can Happen Does Happen.

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# LISA and the Michelson-Morley Minkowski (M3) clock of the universe model Extra Einstein Equivalence principle (E3).

Table of contents.

Calculating the Minkowski Interval for LISA.

MMX all-angles beam-splitting.

MMX kinematics of Einstein Equivalence principle.

M3 models E3.

### Calculating the Minkowski Interval for LISA.

The laser interferometer satellite antenna (LISA) has been prepared as a joint (if financially delayed) experiment between the National Aeronautics and Space Administration (NASA) and the European Space Agency (ESA).

A BBC Horizon televisation of a decade or two ago suggested that string theory might have to rely on mathematics to discover the theory of everything, because the prospects of it being tested were so remote.

Reading Brian Greene, it would appear that experiment is essential to narrow down the almost limitless variety of possible hyper-spatial (or, at any rate, geometrical) solutions to find those which apply to reality.   
Reading a book like Gravitys Engines, by the astronomer Caleb Scharf, it seems that revolution in physical theory is fast being caught, if not over-taken, by ever more accurate technology.

LISA is just such an experiment.   
Scientific qualifications, like general relativity, are needed to understand it. Enormous technical difficulties are being over-come to implement it. This member of the public feels that I have a nerve to discuss it at all.

The purpose here is a continuation of my amateur investigations into the Michelson-Morley experiment, because LISA is a descendant of that test.

A BBC Horizon science program reported an experiment showing the flatness of space-in-time to hold thru-out the entire observable universe, from here on earth to the cosmic backround microwave radiation, the explosive flash, left over from a few hundred thousand years after the Big Bang origin of creation. The experiment triangulated, from our earthly point to two prominences in the random early foton concentrations, and measured a Euclidean triangle indicating a flat space.

The main purpose of LISA is to detect incredibly small ripples in this flat fabric of space and time, such as would leave a measurable difference in the lengths of the sides of the satellite triangle. These ripples would be gravitational waves, predicted by general relativity to result from colossal disturbances, notably emanating from far-away but massive black hole collisions, and a host of other disruptive astrophysical phenomena, outlined on the NASA page: LISA Project Office.

Stemming from experimental efforts, as far back as the 1960s, ground-based gravitational wave detection, at last, was accomplished in 2016. This offers physicists a new way to view the universe, other than thru light and the entirety of the electro-magnetic spectrum.

Unlike the Michelson-Morley experiment, LISA could not use mirror reflections, which would be too faint to return over millions of kilometres. Nor are the beams split, for this purpose. Instead, when a beam travels from one satellite to the other, it triggers a return beam. However, for convenience, I continue to describe this trigger as a reflector, tho not a mirror reflector.

The LISA amplitude of the 60° transverse or cross-turned motion is less than the maximum amplitude of a 90° transverse motion found in the M-M experiment (MMX). Now I repeat my workings for a lesser amplitude experimental set-up, namely the LISA project.

The use of theorem Pythagoras assumes Euclid geometry of flat space, and, by extension, the Minkowski geometry, sometimes described as a four dimensional Euclidean space-time. On average, outer space geometry is that of a flat space-time, which is where LISA is to be launched.

In the M-M laboratory experiment, the transverse beam is at right angles to the earth-aligned beam.   
In the proposed cosmic scale LISA experiment, the laser beams are not at right angles to each other. They are mounted on three identical satellites at 60° to each other. Each side of the triangle is to be 2.5 million kilometres long. The three satellites are to be in Earth-like orbit about the sun.

For the sake of argument, I've assumed relative motion to the Earth as a reference point, so that one satellite moves in line with Earth motion and another moves 60° transversely to that line. This particular isn't essential to the following discussion (setting aside a deeper geometric interpretation, that Minkowski flat-space geometry guides motion in special relativity, as Riemann curved-space geometry guides motion in general relativity.)

The calculation for the earth-aligned relative velocity of a LISA beam, is easy. It is the same as for the original Michelson-Morley experiment. Or it would have been easy, if they had used the geometric mean rather than the arithmetic mean in their calculation of the average two-way time.

Consistent with the Minkowski Interval formula, I use the geometric mean to average the forth and back earth-aligned relative velocities.   
This contradicts well over a century of following the authority of the Michelson-Morley calculation, which uses the arithmetic mean. I know I have laid-on this simple point, pretty thick, but it is necessary to make plain that it has not been the conventional wisdom in physics.

The LISA triangle of satellites is a fixed formation. So, two satellites in line with earth will move at the same relative motion to and from earth motion. And a laser, firing to and from the two satellites, will have its earth-relative light speed, c, added or subtracted from earth speed, u, depending, on whether moving, respectively, against or with earth direction.

As in the LISA diagram, below, say the earth-aligned beam starts off in line but in the opposite direction, to Earth, moving the reflector towards it, at earth velocity, u. Then it will move relatively faster, that is, speed: (c+u), and take less time, to reach its reflection point.   
On reflection, returning to its source, the beam will be moving in the same direction to reference point, Earth, moving the source away from it, which means it loses relative velocity, by its own light speed, c, minus the reference bodys speed, u. That is: (c-u).

Now, a reflection implies a change in velocity, relative to the reference body motion. That means relative acceleration, for which the suitable average is the geometric mean.

For example, the suitable average to measure a range of simple interest over time, is the arithmetic mean: the addition to capital is a constant increase. But to measure the average interest, for a given growth of capital by compound interest, requires the geometric mean. Compound interest accelerates the increase in capital, with the interest taken on a capital, that successively adds previous interest.

The geometric mean of the earth-aligned beam journey is the square root of the multiple of the out-going and return journeys:   
{(c+u)(c-u)}^1/2 = (c² - u²)^1/2.

This equals the velocity measurement for a 90° transverse beam, and so gives the correct prediction for the Michelson-Morley experiment. The same velocities over the same distances imply the same time for the light beams to return to source, as is what happens.

This is also the form of the velocity part of the Minkowski Interval for any local observer of a given event, significantly approaching light speed dimensions.

The Minkowski Interval, I, is:

I² = t²(c² - u²) = t'²(c² - u'²).

The Interval is a constant common to all local observers of some light-speed proximate event. It has dimensions of distance, being equal to a multiple of time, t or t', by velocity, u or u'. The indexed letters merely mean that a different local observer measures different local times and velocities. But they all agree on a common so-called space-time measurement, of the event, called the Minkowski Interval.

The Minkowski Interval can be satisfied for the Michelson-Morley experiment by making the times of the two observations, aligned and oblique or cross-ways, the same. Whilst converting the cross-ways beam journey, into imaginary velocities, signifies ninety degrees turn of the reference body to the cross-ways beam.

In algebraic terms, the Minkowski version of Michelson-Morley might be rendered:

I² = t²(c² - u²) = t²(c² + [±iu]²)

There are two opposite directions in which the (two-dimensional) cross-ways or transverse beam could be transverse, for this equation, expressed by two terms with opposite signs, [iu]² to [-iu]².   
(This is also true of the (one-dimensional) aligned case, with [u]² to [-u]². This allows for a distinction between either of two ways round, on that one dimension.)

As the earth-aligned motion is the same for LISA as for M-M, all that needs to be done is find the cross-ways form for LISA. Because the earth-aligned case is in one dimension, it was straight-forward to take an average of that dimension. But the transverse case is in two dimensions.

Explanation will be guided by my LISA diagram below. It compares to the MMX diagram, in a previous chapter. The LISA diagram shows three successive triangle positions. Assuming the normal condition of space-time flatness, I calculate the Minkowski Interval for a LISA type triangle, shown in the diagram, with two laser beams, subject to earth-relative motion, one bouncing forth and back, from the source, marked, A, in the earth-aligned direction to the corner, B, and the other beam, earth-crossways, from corner A to the corner, marked C, and back again.

### The LISA laser beam journeys.

In the chapter on vector symmetry of MMX, the MMX vector analysis diagram shows the three successive perpendicular positions of the transverse mirror. The experiment aims the transverse beam at right angles, to the earth-aligned beam. But its light speed, c, is shifted, relative to earth velocity, from the perpendicular to a slightly sloping direction, by the same angle before and after its reflection, with respect to the constant Earth velocity, u.   
This forms two symmetrical triangles about the vertical mid-position. Averaging two velocities, of the same value, necessarily leaves the velocity the same for the whole journey. This is the velocity for the transverse journey of the MMX.

The LISA calculation is like MMX, tho appearances are somewhat different, because the transverse beam no longer forms a symmetrical reflection, when aimed at 60°, instead of 90°, to the earth-aligned beam. The red lines show this asymmetry, with the reflected line being shorter than the out-going line and creating a lop-sided appearance, unlike the tent-shaped symmetry of an MMX diagram.

In the LISA cross-ways journey of light beams, the slope-shifted or relativistic values, we seek, are given in the diagram. The outward journey is A,C2. In calculation, let A,C2 = f, for short. The return journey is C2,A3 (Let C2,A3 = g). The different lengths represent different velocities, so that their average should differ from both, by taking their geometric mean.

The asymmetrical red lines, of the out-going transverse beam, f, and its reflection, g, share the same magnitude vertical vector component, C2,D, to their otherwise different sized triangles.

C2,D can be found from knowing the other sides of right-angled triangle whose vertices are A2-C2-D.   
A2,C2 = c. That is the 60° transverse beam, as it reaches the reflector, at C2.   
A2,D = c/2, because C2,D drops a vertical line that halves the base of the middle equilateral triangle, A2-C2-B2.

By theorem Pythagoras, the triangle perpendicular is found as:

(C2,D)² = (A2,C2)² - (A2,D)²

= c² - (c/2)² = 3c²/4.

When it comes to measuring the transverse or cross-ways beam, this error-prone amateur and loner found making the transition from MMX to LISA really tricky, and made many mistakes. (The book may contain other errors.)   
With MMX, the light beam, fired in a perpendicular direction, is taken to travel at light speed, c. Likewise the light beam fired at 60°, to the eath-aligned beam, is taken to travel at light speed, c. Both MMX and LISA beams, however, under-go a drift with respect to the moving earth, which removes the transverse reflector from its original position, while the light beam is traveling there. In both experiments, the shifted paths have slightly different velocities from c, with respect to earth motion.

I also forgot that the two experiments, MMX and LISA, both are calculated on the basis of the Minkowski Interval, which graphs as a two-dimensional co-ordinate system with an i-for-imaginary axis, signifying turn thru 90° for the second dimension, as well as the one dimension of a real axis. In other words, one of the co-ordinates is multiplied by the operator, i, which calculates as the square root of minus one.

The earth-aligned beam calculation, being simply back and forth, is only in one dimension, and so does not need an imaginary variable, which serves as an operator, to signify turn thru 90°, into another dimension. This makes sense when one considers that a minus sign means in the opposite direction of (or at 180° to) a positive scale.  
A complex variable, or combination of a real and an imaginary variable, means measuring magnitude in the context of two dimensions, instead of only one.

In measuring transverse motion, which is partly in one dimension, irrespective of earth motion, and partly in another, relative to earth motion, we need to make one of the dimensions, or co-ordinates, imaginary, to calculate relative motion in two dimensions.

To find the earth-motion drift (red line f, on the diagram) of the LISA 60° transverse beam, by theorem Pythagoras, requires knowing C2,D and AD. We know A,A2, which equals earth velocity, u. That leaves A2,D, which equals half light speed, c. Therefore AD = u + c/2. This is on the single earth-aligned dimension. This is distinguished from its perpendicular dimension by an imaginary coefficient

Therefore:

f² = (C2,D)² + [i(AD)]²

= 3c²/4 + i²(c/2 + u)²

= 3c²/4 - (c²/4 + u² + 2uc/2)

= c²/2 - u² - uc.

Similar working applies to find, return transverse beam, g. In this case the light beam is chasing, instead of riding earth velocity, u, which is subtracted instead of added, giving the relevant base-line value A3.D as: c/2 - u. This accounts for the difference of g from f. Namely:

g² = 3c²/4 - {c/2 - u}².

= c²/2 - u² + uc.

The average velocity of the transverse beam over its whole journey is found by taking the geometric mean of f and g. (Note again that this is not the traditional physics procedure with MMX of averaging which used the arithmetic mean.) Values, f and g are the upper and lower limits of a range, whose average or representative value is being found. The geometric mean of two terms is the square root of their multiple.

(f.g)² = (c²/2 - u² - uc)(c²/2 - u² + uc)

= (c^4)/4 - u²c²/2 + (u.c^3)/2 - u²c²/2 + u^4 - cu^3 - (u.c^3)/2 + cu^3 - (uc)²

= (c^4)/4 + u^4 - 2u²c²

= c^4 + u^4 - 2u²c² - (c^4)3/4

= (c² - u²)² - (c^4)3/4

(f.g)² is the square of the geometric mean squared. The Minkowski Interval is usually left in its squared form and would have to be squared again to be equated to (f.g)².

The square of the geometric mean, fg, gives the velocity values which may be inserted in the Minkowski Interval, which has dimensions of distance, equated to velocity multiplied by time. The point is that different local observers of an event have their different local velocity and time measurements, but the Interval equation, gives every observer a common joint velocity and time measurement. This common measure is often conceived as a unitary space-time.

Given the Minkowski Interval, in its conventional squared form:

I² = t²(c² - u²) = t'²(c² - u'²).

Let left and right sides of the Interval stand respectively for the earth-aligned and cross-ways beams of a Michelson-Morley type experiment, The beams of MMX are perpendicular. Experiments, like LISA, with beams set at less than 90°, have a comparable earth-aligned beam reflection but lesser angled cross-ways journeys.

The LISA earth-aligned beam is the same as for MMX, say:

t²(c² - u²) = t²(c + u)(c - u).

The two factors on the right side are upper and lower velocity range limits, which the geometric mean averages by multiplying and taking their square root.   
For the LISA transverse value, we substitute in the other side of its Minkowski Interval: fg. The velocity, for the cross-ways beam journey, u', refers to the same magnitude, but different direction, in relation to earth velocity, u, as in the earth-aligned beam journey.

I^4 = t^4(c² - u²)² = {t'²(c² - u'²)}²

= t'^4{(c² - u²)² - (c^4)3/4}.

This equation of the Interval tells that time t', the transverse light beam time must be greater than the earth-aligned beam time, t, because the coefficient of a smaller total velocity.   
This disagrees with the traditional (arithmetic mean) calculation, carried out on MMX, where the earth-aligned journey would take more time than a (more or less) transverse journey.

### MMX all-angles beam-splitting.

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Consider applying, to the Minkowski Interval, the general case of MMX to splitting the beam at all angles.

In all cases there is an aligned or base-line direction, given for example, by the earth-aligned beam, and representable by the recognised Interval equation form:

I² = t²(c² - u²).

The transverse beam is varied thru all angles to the earth-aligned or base-line beam, not just 90° with MMX and sixty degrees with LISA. With reference to the LISA diagram, the Interval formula for angles of Michelson-Morley type experiments with beam splits at varying angles, Q, requires more general formulas for the out-going and returning transverse beams, f and g:

f² = (C2,D)² + i²(A,D)².

Therefore:

f² = (c.sinQ)² - [{c² - (c.sinQ)²}^1/2 + u]²

= (c.sinQ)² - c² + (c.sinQ)² - u² - 2u{c² - (c.sinQ)²}^1/2

= 2(c.sinQ)² - c² - u² - 2u{c² - (c.sinQ)²}^1/2.

The reasoning for return transverse beam, g, is similar:

g² = (C2,D)² + i²(A3,D)²

= (c.sinQ)² - [{c² - (c.sinQ)²}^1/2 - u]²

= (c.sinQ)² - c² + (c.sinQ)² - u² + 2u{c² - (c.sinQ)²}^1/2

= 2(c.sinQ)² - c² - u² + 2u{c² - (c.sinQ)²}^1/2.

Therefore:

(fg)² = {2(c.sinQ)² - c² - u²}² - 4u²{c² - (c.sinQ)²}

= 4(c.sinQ)^4 - 2(c.sinQ)²c² - 2(c.sinQ)²u² - 2(c.sinQ)²c² + c^4 + c²u² - 2(c.sinQ)²u² + c²u² + u^4 - 4c²u² + 4(c.sinQ)²u²

= 4(c.sinQ)^4 - 4(c.sinQ)²c² + c^4 -2c²u² +u^4

= 4c^4(sinQ)²{(sinQ)² - 1} + (c² - u²)².

This is the formula for all reflected-to-source beams, split at any angle. To check that this works, substitute in sin Q, the MMX case of sin 90° = 1. This correctly reduces the formula to:

(c² - u²)² = (c² + [iu]²)².

This equals the Interval form, (I/t)^4, where the two factors represent the MMX longitudinal and perpendicular beam journeys, taking the same time, t.

To check this general formula works for the LISA experiment, substitute sin 60° = (3^1/2)/2, in sin Q. Then:

4c^4(sinQ)²{(sinQ)² - 1} + (c² - u²)²

= 4c^4(3/4)(3/4 -1) + (c² - u²)²

= -3/4(c^4) + c^4 + u^4 - 2u²c²

= (c² - u²)² - (c^4)3/4.

This result, before putting in Interval times, t and t', of local observers, is the same as that directly obtained for the LISA experiment:

I^4 = t^4(c² - u²)² = {t'²(c² - u'²)}²= t'^4{(c² - u²)² - (c^4)3/4}.

Where there is no angle, or zero degrees, between the two beams, sin Q equals zero, the equation for (fg)² reduces to the form of the square of the squared Interval:

(fg)² = [c² - u²]².

Therefore.

fg = ±(c² - u²).

Multiply this by a local observers measured time, t, and you convert velocities, fg, into the distance dimension of the Interval:

fgt² = I² = ±t²(c² - u²).

Physics convention allows either signature of the Interval:

t²(c² - u²) or t²(u² - c²).

This generalisation of the Interval for all angles of MMX (including LISA) explains the existence of both conventions in a generalised Interval, that also has an extra term, a sine term. Accounting for the convention, of allowing both signatures, is a small confirmation of this reasoning.

In the previous chapter on Least Action, a Modified (real numbers only) MMX is investigated, because more amenable to treatment than the actual version. This left out the transverse beam as two-dimensional or with an imaginary ordinate, so that both co-ordinates are real. In effect, the earth-perpendicular beam also under-goes the earth-aligned type motion.

The only difference to the above calculations for f and g is that you leave out their respective imaginary coefficients. This simplifies the working, so that the two terms, (c.sinQ)², cancel, instead of add.

You end with the simpler result:

(fg)² = [c² - u²]² + 4u²c².sin²Q.

It has an interesting special case of the angle changing from 0° to 90°, so sin Q goes from = 0 to 1. This alternates the equation from [c² - u²]² to [c² + u²]².

Or alternates term, fg, from ±[c² - u²] to ±[c² + u²].

This term, fg, equates to the conventional Interval term, (I/t)² = (c² - u²). It also gives the conventional Interval a complementary positive factor, of straight-forward four-dimensional Euclidean space, with four positive terms, and not the imaginary fourth dimension. That is not so apparent from the summary two-term factors used here. But the velocity, u, can be considered as a three-vector, or one resultant velocity vector out of the velocity measured in a three dimensional framework.

Also, it might be worth comparing the equation for modified (real) MMX:

(fg)² = [c² - u²]² + 4u²c².sin²Q.

with the equation for MMX.

(fg)² = (c² - u²)² + 4c²(sinQ)².c²{(sinQ)² - 1}.

This comparison tells that MMX and Modified MMX are equal if:

u² = c²{(sinQ)² - 1}.

That is at perpendicular beams, or sin 90° = 1, if u = 0.

Or at 60° beams if:

u² = c²{3/4 - 1} = -c²/4.

Or: u = ic/2.

Or at 0° beam split if u² = -c², or: u = ic.

The equation for all-angles MMX gives an obvious interpretation of the classic Michelson-Morley experiment, when applied to it.

However, the "obvious" interpretation is that the positive or negative versions of the classic Interval form for velocities, ±[c² - u²], stand for which velocity, light velocity, c, or locally measured velocity, u, is being subtracted from which. Likewise, its perpendicular factor, ±[c² + u²] = ±[c² - (iu)²], can be perpendicular to the north or to the south, so to speak. That is to say like a rectangular or Cartesian co-ordinate system.

### MMX kinematics of Einstein Equivalence principle.

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Albert Einstein once asked a man, who survived a great fall off a building, if he felt a force on him. And he said he felt nothing of the sort.

An object in a lift, like some vertical lift-off rocket in the middle of empty space has an object, such as a man, floating in its interior, because there is no massive presence in the vicinity to gravitationally attract it. The lift man floats free, in a state of inertia, without apparently moving, while he does not push against floor or walls.

The Einstein thought experiment has a lift, in outer space, suddenly accelerate. He imagines the man, in the lift, then feel his feet hit floor, as if rooted there, under a force of gravity. His inertial mass apparently is transformed to gravitational mass. The two kinds of mass are shown to be the same, only exhibiting apparently different properties under different frames of reference: an inertial frame and an accelerating frame.

The new appearance of gravity could be due to no more than a chain, fastened to the roof, accelerating him away, ever faster.   
(Einstein thought the experiment, before rocket propulsion in space flight had got much beyond the minds of Ziolkowsky and Goddard.)

A beam of light passes thru a high lift window, and out another window, on the other side. To an outside observer, the beam is going in a straight line. The inside observer sees the light bend, because the lift, which is his reference frame, has lifted somewhat, before the beam reaches the other side of the lift. The man in the lift is imagined to see the light beam slightly dip from its entrance to its exit. He assumes it, too, feels the force of gravity, bending it down. To the out-sider, it's just a case of the lift accelerating so fast, that it is leaving the entering light beam behind. Likewise, the man in the lift would be left behind, if he were not caught by the floor.

Hence, Einstein derived his Principle of Equivalence of acceleration and gravity and predicted that light would bend under strong gravitational attraction.   
As early as 1919, the Eddington expedition confirmed that light, while passing by a star, should have its path bent slightly towards it. (Not without subsequent but indecisive controversy.)  
Light, trying to climb out of a stars gravitational well, will have its wavelength lengthened or red-shifted. Light velocity is a constant, that equals wave-length times frequency. Longer wave-length means lower frequency and less energy of the light, its mass of motion is reduced. Light diving straight into a gravitational well gains a more energetic frequency and is blue-shifted to shorter wave-lengths.

Tho not on such a small scale as traditional MMX, of course, in principle, "gravitational" effect comes from a return journey of a beam. The super-position of these wave trains, with and against Earth motion, should result in wave decoherence, as a result of this interference from moving in and out of Earths gravitational well.

It must be admitted that a contraction factor, of sorts, has sneaked its way back into the argument. Equally, there is also an expansion factor. And these factors do not affect the constant speed of light, and therefore the time it takes to travel. Light velocity equals frequency times wavelength. Light velocity is a constant equality to a counter-balancing frequency and wavelength.

Suppose, when the bending light beam reaches the other side of the Einstein lift, that it is instantly reversed, as if the lift was pulled by an oscillating spring, so that the light beam repeats its path in the other direction, forming a loop.

Two kinds of waves are transverse and longitudinal waves. Take a string tied to a post or wall and hold the other end. Pulling the string back and forth produces longitudinal waves. Jerking the string end, up and down (not to be confused with so-called up and down stream), will produce up and down waves along the string. These are called transverse waves. The more energy put into the vertical shaking, the more wave vibrations formed along the string. Since the string is fixed at both ends, the vibrations tend to form a definite number of up and down vibrating loops (refered to as harmonics). Just one loop is called the fundamental. This is like a reversed lift scenario, where the observer under-goes a single up and down oscillation with regard to the light beam. The more energeticly oscillatory the situation, the more oscillations.

More complicated oscillations presumably could result in standing waves in the harmonic series. To the observer in the lift, these light beam oscillations suggest gravitational waves, rather than reverse accelerations. The lift man (or rocket man) might assume he had been bobbed up and down, on a ripple in the fabric of space-time, caused by a super-nova explosion.   
To the outside observer, tho, this is not a gravitational ripple but just the lift being yanked up and down, with the light passing back and forth in a straight line.

The reference frame of the man inside the lift is perpendicular to the light beam. This compares to the perpendicular journey of the split MMX light beam, relative to earth motion. The split light beam, under "cross-stream drift", in the Michelson-Morley experiment, compares to the Einstein lift thought experiment bend of crossing light.

The classical analogy, of the Michelson-Morley experiment, to a boat both ways across a stream, is misleading, because it assumes two distinct linear drifts, implying two separate velocities. Rather, the journey is a curved one, tho an acutely angular one, involving change in direction of velocity. Hence, the MMX earth-perpendicular beam, like the Einstein lift beam is a curved beam of light under relative acceleration.

Whereas the MMX beam in a straight line with Earth motion corresponds to the light, passing thru the Einstein lift windows, seen as going in a straight line from an external observers view-point.

It appears that the Minkowski formulation of MMX shows an equivalence of times, corresponding to the Einstein equivalence of masses, inertial and gravitational. The Einstein Equivalence principle is a dynamics version of the kinematics of MMX, calculated by the Minkowski Interval.

The Equivalence principle of acceleration to gravity is like a mass-dynamics version of MMX time-kinematics, in that it involves an equivalence of masses. This equivalence is of inertial mass and gravitational mass, when the inertial mass is subject to an accelerating frame of reference.

### M3 models E3

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On a clock, the pointer tells the time. On the Michelson-Morley Minkowski (M3) clock of the universe, direction also tells the time, as signified by the sine-generalised Interval.

Every ninety degrees sweep of the M3 clock, local times are synchronised. Like a familiar analog clock dial, time is space-directional. Not for telling some absolute time or convention of "the time." But for relating locally observed times.

This is the case of a flat space universe, which outer-space appears to be, to its observable limits. Of course, present science may be as unaware of a transcendant curved space universe, as early man presumably was, of more than a locally flat earth.

The gist of the matter is that the Minkowski Interval describes the Michelson-Morley experiment, when the times are set equal, with one observer in a longitudinal reference frame (aligned relative motion, in one dimension) and the other observer in a transverse reference frame (perpendicular two dimensions). This is to say there is no difference between the magnitudes of their locally observed times, but there is directional difference, between observers, given by either a negative sign or an imaginary sign, signifying, respectively, the operations of turning thru 180 degrees or 90 degrees.

Taking into account non-perpendicular versions of MMX, like LISA, involves an extension of the Minkowski Interval for all-angles beam splitting:

(I/t)^4 = 4c^4(sinQ)²{(sinQ)² - 1} + (c² - u²)².

This is the sine-generalised Interval for one observer. It equates to another observer with their own local time, t' and velocity, u' measures. Angle Q is the angle of beam splitting: 90 degrees for MMX, 60 degrees for LISA, etc.

The Einstein Equivalence principle, derived from the lift thought experiment, is analogous to the Michelson-Morley experiment. Both experiments are in terms of frames of reference, which see a light beam, either straight or bent. In the MMX case, two similar split light beams, essentially one beam, are straight or bent, from their differing reference frames, with respect to earth motion. In the Einstein lift case, one light beam is either straight or bent, with respect to reference frames outside or inside the lift.

The Einstein lift beam compares to half of the MMX two-way journey. To complete the analogy would require the lift beam to be reflected.   
The outside observer would just see the light beam reflected back in a straight line. To the observer inside the accelerating lift, it would appear that the reflected light was also bent, in the other direction.

This could compare the walls, of the Einstein lift or rocket, to an MMX scenario, which imagined banks to an alleged ether stream, on which a crossing boat (the beam) drifts with the force of the stream.

(Alternatively, the light might look like the fundamental harmonic vibration of a string, as if from a gravitational ripple of space-time, when really caused by an oscillating motion of the lift.)

This completes the imaginative comparison of the Einstein lift to MMX. The Minkowski Interval derives the equality of times found in MMX. The Interval has the same form for the equality of masses, as it has for equal times. (In the next chapter, an electoral analogy accounts for the isomorphism of time and mass in special relativity kinematics and dynamics.) Thus, the Einstein Equivalence principle of gravitational mass to inertial mass is described by the equation of the Interval where local observers measure the same mass of an observed object, in the same formal way that the Minkowski Interval describes MMX equality of times.

That also means there is an Extra Einstein Equivalence principle (E3) that follows the same form as the generalised Interval for all-angles reference frames. Just as observers local times vary, so will their local masses, which is tantamount to saying there be (still calculable) departures from the equality of gravitational to inertial mass.

The Einstein Equivalence principle is right enough for the particular thought experiment, which inspired it. But E3 qualifies and extends it, by way of an all-angles MMX applied to the Interval.

An all-angles Interval, the sine-generalised Interval from rectilinear reference frames, to all-angles, allows angular approximations to a geometry of varying curvature. Moreover, masses are built into the geometrical equation, rather than having to be brought in extraineously.

(FAB STV 2D might be modifiable after the model of the sine-generalised Interval. Its electoral version might give electors a choice of frames of reference other than straight representation and straight arbitration. The second dimension of arbitration might be modified, by degrees, allowing other reference frames (other elections) of arbitration by less strictly neutral bargainers.)

#### Historical reference:

In my twenties I read, JWN Sullivan, on The Bases of Modern Science, which gives a lucid account, super up-to-date, for when it came out, in about 1928.

# Relative Motion Models Relative Choice and FAB STV 2D.

Table of contents.

FAB STV: Four Averages Binomial STV.

FAB STV 2D: Two dimensional Four Averages Binomial Single Transferable Vote.

Relative Motion Models Relative Choice.

Mach principle as statistics.

General Relativity and FAB STV.

### FAB STV: Four Averages Binomial STV.

FAB STV is a system of proportional representation, using a quota count to equitably elect personal representatives, in the voters order of individual choice.

True to the tradition of Thomas Hare, from John Stuart Mill to Enid Lakeman (via H.G. Wells) FAB STV prohibits any divisions into classes of voters, or candidates, as owes to collectivist bias in the ballot, like voting, above the line, for the Australian Senate, or an indirect vote for a party. Likewise shunned is any corporate count for privileged groups, or party lists, or other infringements on freely transferable voting.

Over the past half-century, the likes of Social Choice theory has concentrated academic minds on hypothetical rogue results from voting methods. Their imperfection is not a conclusion, that democracy is doomed as a rational pursuit, but a starting point for statistics, whose approximations increase, not decrease, accuracy.

FAB STV is a systematic and stable system of averaging-out perverse results from misrepresenting the vote.

STV originally used the Hare quota, the maximum threshold for PR, but later shifted to the Droop quota, the minimum threshold for PR.

FAB STV introduces my invention of the Harmonic Mean quota, which averages the maximum and minimum quotas to give a more representative quota. This is the first of the four averages, to FAB STV, which all serve the purpose of arriving at a more representative over-all result.

Meek method is a computer counted STV. But it departs from tradition, which took short-cuts, to make a manual count practical, without unduly distorting the results. Meek method does not miss out the transfer of later preferences to candidates, who have already received an elective quota. Meek method recognises the full popularity of elected candidates, thru the concept of the keep value. The bigger the surplus of votes, that an elected candidate receives, the smaller the proportion of votes that he keeps. (The most popular candidates have the smallest keep values.)

From 2000 to 2002, Meek method was programmed and prepared for electing 7 out of the 11 members to each of 21 (now 20) District Health Boards and some local councils, in New Zealand, a world first for its official use, as partly described by S W Todd, in Voting Matters, the specialist journal of the Arthur McDougall Trust, Issue 16, pp. 8-10, February 2003.

FAB STV extends the Meek method keep value to candidates, in deficit, as well as in surplus, of a quota. This gives a quantitative standard of comparison for the relative preference of all the candidates, in the election count.

FAB STV uses keep values also to measure the relative unpreference of the candidates. The preferences are counted in reverse, in what amounts to an exclusion count, instead of an election count. The exclusion count does away with a fault, of all the worlds electoral systems, in official use, called "premature exclusion". Inverting the exclusion keep values of each candidate amounts to a second opinion, compared to their election keep values. The respective election keep values and inverted exclusion keep values, of each candidate, are then averaged (with the geometric mean) to give a statistically more representative result. This is a second of the four averages to FAB STV.

Dr James Gilmour told me (in the STV voting email group) that the board of the Electoral Reform Society dissented from the Meek method practise of reducing the quota, as voters preferences are exhausted. Happily, FAB STV does not have this feature. FAB STV uses all the preference information, including abstentions, which would leave a seat empty, if they reach a quota. The level of abstentions is a measure of the over-all popularity of the candidates, as their keep values are a measure of their relative popularity. (The counting of abstentions is necessary, because it takes away some of the weight that would otherwise illegitimately go to unpreferences, compared to preferences.)

The most significant change to STV, from the voters point of view, is that Binomial STV gives Bidirectional preference (two-way choice): the voter can order candidates in order of preference or liking and can also order candidates in order of unpreference or disliking. Indifference towards candidates is shown by abstentions.

The combination of an election count with an exclusion count may be called first order Binomial STV. The second and higher orders are qualifications, of the basic first-order count, governed by higher order (non-commutative) expansions of the binomial theorem. Binomial literally means two names or terms, in this case, Preference, P, and Unpreference, U. Thus, first order binomial STV is represented symbolically by: (P+U)^1 or, for short: (P+U). This factor stands for two counts: a Preference election count and an Unpreference exclusion count.

Second order binomial STV is represented by:

(P+U)^2 = PP + UP + PU + UU.

Each successive higher-order of binomial STV doubles the number of counts. So, the second order has four distinct counts. Two of them are qualified preference election counts and the other two are qualified unpreference exclusion counts.

All candidates, who have received a quota, are up for having their votes redistributed, to next prefered candidates, in an election count, or next unprefered candidates in an exclusion count. (Their own keep values are unchanged, when their votes are redistributed, to affect the keep values of other candidates.)

PP stands for a preference qualified preference count. With reference to the first order preference count, a quota-prefered candidate has all their votes redistributed to next prefered candidates. If there is more than one such candidate, resulting in more than one distinct set of changed keep values for all the candidates, then all the sets of modified keep values are averaged, using the arithmetic mean.

This is the third of the four averages binomial STV.

The same averaging procedure is followed for all four qualified second-order counts. (It would be followed for all eight qualified third order counts.)

Moving on, from the first qualified preference count, PP, to the second qualified preference count, UP, any candidate, who received an exclusion quota of unpreference (in the first order exclusion count), has all their votes redistributed to next preferences. If there is more than one such redistribution count, the respective keep values are averaged with the arithmetic mean.

Then, the two qualified preference counts, PP and UP, are themselves averaged (using the geometric mean) for an over-all qualified preference count.

Exactly the same process is repeated in reverse, for the remaining two counts, PU and UU, which are qualified unpreference counts, until an over-all qualified unpreference count is achieved. This last is inverted and averaged (using the geometric mean) with the over-all qualified preference count, to give the final keep values of a second order binomial STV.

The fourth average of FAB STV averages different orders of binomial STV, most simply to average first order and second order STV keep value results.

So far, the three averages used are the arithmetic mean, the harmonic mean and the geometric mean. The geometric mean is actually a power arithmetic mean. The fourth average in the set is a power harmonic mean. When the latter is used to average orders of STV, it gives a slightly greater weight to the higher orders of keep values. This is as one would expect, as the higher orders are, on balance, more complete and therefore more accurate expansions.

A fuller discussion of all these matters is given in my book, FAB STV: Four Averages Binomial Single Transferable Vote.

FAB STV is the only electoral system to give systematic in-depth representation, which is likely to give it a considerable advantage over all the other electoral systems used by the data mining community for information retrieval. As such, FAB STV could serve as a scientific signpost, that transferable voting is the way to go.

More grandiosely, the claim is that, as all the worlds electoral systems are uninomial (generally preference-only) counts, they are rendered obsolete by binomial STV of combined election and exclusion counts.

FAB STV is a difficult but necessary research. An election is a matter of both inclusion and exclusion of candidates. Meek method is the system that comes closest to a wholly rational inclusion count. FAB STV goes the extra distance. And then you automatically have also a wholly rational exclusion count, by counting the preferences in reverse order.

The essential point is that, in the past half-century, an academic industry has grown around proving the rational Impossibility (with a capital I) of democracy. This is largely, if not wholly, based on the irrationalities of elections without a rational exclusion count. Even Meek method resorts to a feeble Last Past The Post exclusion count, to facilitate a final result.   
FAB STV releases from this dead-end to democracy.

If Binomial STV did not go further than the combination of an inclusion count with an exclusion count, probably it would be simpler than Meek method. But it's this binary count that makes applicable the binomial theorem, making possible systematic refinements to the result. And it makes no sense to stop the mathematics at first order binomial STV.

These refinements are unlikely to be necessary in routine democratic elections (at least not beyond the second order, I would guess). But as a search engine application, FAB STV might provide just the systematic analysis required for information retrieval from the exponential growth of human knowledge. Of course, there is no way of knowing that, without actually implementing and testing the system.

As to FAB STV being feasible, I am ignorant of computer programming, it is true. And FAB STV is complicated. But it is like this. The basic first order STV is relatively simple, probably considerably simpler than Meek method. (Needless to say, I don't mean to demean that ground-breaking method.) FAB STV throws out Meek Methods two most makeshift algorithms, which are subject most to chance. The quota reduction is makeshift, because it is governed by the chance matter of when voters cease expressing a preference. And the exclusion of candidates is makeshift, because it is governed by the happenstance of which candidate has the least votes, when the surplus votes run-out.

First order STV consists of an exclusion count which is the mirror image of the inclusion count, the only difference being that the exclusion count is of the reverse order of preferences (or of unreferences). And the keep values of the exclusion count, of unpreferences, are inverted, so that they can be averaged (using the geometric mean) with the keep values of the inclusion count of preferences.

All this is made possible by the Meek calculation of keep values, which, however, has to be extended to all candidates, meaning those still in deficit of a quota, as well as candidates with election surpluses.

The complication increases exponentially with higher orders of STV. More or less simple arithmetic operations are involved, tho an unmanagable number, except for a computer. However, these calculations, being the taking of averages, guarantees the stability of their results.

These complications are not makeshifts but follow the logic of the binomial theorem, with respect to Preferences and Unpreferences. In my opnion, it is not the complication of FAB STV itself, making an insurmountable task of programming it, that is at issue, but its potential as a tool for systematic probing or searching the exponential growth of human knowledge. That potential of FAB STV can only be known by actual testing, when this mathematical system is translated into machine language.

(I don't anticipate that more than first or second order STV would be needed for the general run of democratic elections. Admittedly, that would involve higher democratic standards than the chronic erosion from the high points set by Hare and Mill and HG Wells.)

* * *

### FAB STV 2D: Two dimensional Four Averages Binomial Single Transferable Vote.

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As far as I know, FAB STV is the first election system to completely cover one dimension of voters choice. That means that voters can move or choose freely in either direction of the dimension, either from Preference to Unpreference or Unpreference to Preference. Other voting systems are generally only one way. They choose the candidate they like best, or choose the candidate most likely to stop the most disliked. With FAB STV, voters can prefer the most liked candidates and unprefer the least liked candidates, knowing that both directions of choice will count (probably).

One-dimensional or 1D FAB STV may not seem like much but, as just explained, it gives voters much more power than traditional elections. Even Meek method, with its more consistent transfer of surplus votes, only counts preferences, which is to say, it moves in one direction of choice, and only intermittently, having to be helped out, with rough and ready exclusions of candidates, to supply more preferential vote transfers.

Two-dimensional FAB STV (FAB STV 2D) works like two FAB STV elections of the same candidates, judged on two independent criteria or rules or dimensions. One dimension of the election would be the familiar choice by personal order of preference or unpreference. The other dimension might be an impersonal order of choice. For instance, voters might be asked which of these candidates would make the best (down to the worst) arbiters in a dispute.

In other words, one dimension would be of personal representation and the other dimension would be of impersonal representation or arbitration. Consider these two independent dimensions, as representation and arbitration ordinates to a co-ordinate system, geometrically represented on a plane, like a flat piece of paper.

The neutrality of the arbitration dimension, to the representation dimension, is geometrically represented by their being rectilinear co-ordinates, or at right angles to each other.

These ordinates bisect each other in half, at the so-called origin, which is a zero point, meaning where the voters neither have positive or negative attitudes towards candidates, whether considered as representatives or arbiters.

In a FAB STV 2D election, voters have two ballots for the same set of candidates. One ballot is to rank them in order of preference (and/or unpreference). The other ballot is to rank them as best arbiters (and/or worst).

If there is an even number of candidates, the first half are treated as on the positive axis and the last half are treated as on the negative axis. If there is an odd number of candidates, then the mid-ranked candidate is treated as at the origin.

Voters can wholly or partly abstain from stating preferences, so that a whole or part of their vote goes towards an abstention quota, which leaves a seat empty.

In my book on FAB STV, I thought of abstentions, mainly as being complete, a blank ballot, the equivalent of None Of The Above, or consisting mainly of abstentions from indifference to later preferences.

I suppose, a voter might return a ballot paper, much more chequered with abstentions. Short of NOTA, the voter might think these candidates are not very good but not totally useless. So, the voter might abstain from giving any of them a first or second preference, but hand out a third preference to some candidate or other. The voter might recognise an oasis of half decent candidates to be given middling preferences, while abstaining from ranking other candidates with higher or lower preferences.

The minimum requirement for formality or an unspoiled ballot would be in so far as the voter gives a correct number order of choice.

Up to a point, the counting of 2-D FAB STV would be the same as (1D) FAB STV. 1D has an inclusion count of preferences for representatives and an exclusion count of unpreferences for unrepresentatives. 2-D also has an inclusion count of preferences for arbiters and an exclusion count of unpreferences for bad arbiters.

1D inverts the candidates exclusion keep values, so they act like a second opinion election count, then takes the averages of the two sets of keep values, for an overall result.

2-D does not do this, but follows the rules of complex numbers, which measure all the possible positions on a plane, with respect to two dimensions. In this case, candidates positions are measured with respect to voters opinions of them as both representatives and arbiters.

A complex number consists of a real number and an imaginary number, given coefficient, i. These are historical names which have stuck. The leading 19th-century mathematician Gauss was one of the first to realise that the so-called imaginary number is just as real as the real number, merely applying to a second dimension.

The dimensions of representation, R, and arbitration, A, can be expressed as a complex variable: R + iA. It is convenient to treat the second dimension, as the so-called imaginary variable, iA. This complex variable consists of two positive variables, which stand for the candidates which the voters regard positively both as representatives and as arbiters.

But there are three other possible combinations of regard for the candidates.

R - iA symbolises a positive regard for candidates as representatives combined with a negative regard for them as arbiters.

The latter complex variable is called the complex conjugate of the former. Multiply the two factors:

(R + iA)(R – iA) = R² + A² = H².

The imaginary coefficient, i, equals the square root of -1, so the negatives cancel. The resulting form is familiar as theorem Pythagoras, where the symbol, H stands for hypotenuse.

There are two other possibilities: -R – iA and –R + iA, which are also complex conjugates, and derived the same result:

(-R – iA)(–R + iA) = R² + A² = H².

By convention, the real variable is on the horizontal axis, and the vertical axis is for the imaginary variable. Circumscribe these co-ordinates by a circle, and the four complex variables each represent a quadrant of the circle.

H for hypotenuse is the constant radius of the circle.

Guided by this understanding of complex variables, the computation of 2-D FAB STV proceeds as follows: Take the first complex variable and its conjugate, which both have positive real variables. The complex variable is of the keep values for positively prefered representatives plus the keep values for positively prefered arbiters. Its conjugate is the keep values for positively prefered representatives minus (because negatively prefered) arbiters keep values.

The effect of multiplying the complex variable and its conjugate is equivalent to theorem Pythagoras. So the over-all or two-dimensional keep value is found by squaring the positively prefered candidates representation keep values and adding them to the square of (positive multiplied by negative) arbitration keep values (where, as we saw, negatives cancel).

This sum of the squares of the real and imaginary numbers gives the square of the hypotenuse or radius. Taking its square root, gives the hypotenuse, considered as the two-dimensional representation and arbitration candidates keep values. But that is only the estimate for two quadrants of the circle, where the real variables of both quadrants are positive.

Exactly the same procedure follows for the other half of the circle, where the real variables of both quadrants are negative. The end result is another estimate for the hypotenuse or radius. The two radius estimates can be averaged for a final two-dimensional FAB STV set of candidates keep values for a representation and arbitration combine.

* * *

### Relative Motion Models Relative Choice.

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As far back as my twenties, nearly half a century before, I recognised a considerable analogy between relative motion and the relativity of choice. Many (failed or unsatisfactory) attempts were made, over the years, to arrive at a systematic comparison. But my ideas about both systems needed considerable improvement.

The following model is much my most comprehensive comparison.  
Complex variable treatment of elections, in two dimensions of voting (FAB STV 2D), compares to the complex variables of the Minkowski Interval.

In the scheme of the complex variable co-ordinates inscribed in a circle, the Interval term itself is represented by the radius. Don't expect to tumble, all at once, to my electoral model (or "electics") of relative choice, for the mechanics of relative motion. Remember, it has taken me a long working lifetime, just to get this far, and my understanding and presentation may not be all that felicitous, for those coming after.

But here goes. The equation of the Interval, for one local observer, is:

I² = t²(c² – u²) = t²(c + u)(c – u).

with observers local time, t, and local velocity, u. For any given event, other observers measure the same light-speed, c, and share a common space-time Interval, I. Their times and velocities differ with different locations. These differences are usually noted by indexing the time, t' and velocity, u'. Consequently, ut = x, a locally observed distance; and u't' = x' is another locally observed distance.

In the Michelson-Morley experiment (MMX), the Interval factors, shown above, (c+u) and (c-u) have a specific interpretation: (c+u) is interpreted as a light beam speed, c, plus Earth velocity, u, when they are going in opposite directions; (c-u) is the relative decrease in speed from the light beam having to catch up with the Earth going in the same direction.

In MMX, one light beam is reflected to go both ways with respect to earth motion. Another light beam is split off in a perpendicular direction to earth motion. This beam is also reflected back to source. Earth motion drifts it slightly, both forth and back, by the same amount.

Measuring this triangular MMX set-up amounts to trigonometry of Earth-relative light speeds, and an alternative treatment, previously shown.

MMX showed that the two light beams took the same time to return to source. So, at least in that context, just consider (c+u) and (c-u). Light-speed moving oppositely to earth motion is given an access of relative speed. This implies greater energy of motion or kinetic energy. Colliding objects impact with greater destructive energy.

In theory, a light beam has greater energy, from greater frequency, when its wave-length is shortened, from hitting a mirror, in oncoming earth motion.

The light beam, chasing earth motion, loses kinetic energy but gains potential energy, from the fact that when it is reflected, it will be on collision course with Earth motion, gaining kinetic energy.

Generally, the total energy (t.e.) of a closed system is a constant, because the changes in kinetic energy (k.e.) and potential energy (p.e.) always balance each other out: k.e. + p.e. = t.e.

Likewise, (c+u) + (c-u) = 2c. The constant can be changed, for instace, to: c or 1. Light speed usually is set at one. This means that locally observed velocities, u, are some fraction of unity.

The interest in setting the total energy at unity is that it compares to probability theory, where total probability is also set at unity. This is another extremely simple formula, namely that probability of success plus probability of failure equals unity: p + q = 1.

The single transferable vote, especially from Meek method to FAB STV, employs a similar elementary equation, namely that the keep value plus the transfer value equals unity: k + f = 1.

The keep value is the quota, Q, or elective proportion, which is a function of votes per seat, V/S (not necessarily the Hare quota), divided by the total transferable vote, T. Or: k = (V/S)/T = Q/T.

The returning officer to an STV election refers to the total transferable vote, as all the votes an elected candidate receives, not merely their surplus over the quota. All of these votes equally help determine the already elected candidates next preferences but only in proportion to the size of that candidates surplus vote over from the quota. A small surplus means that the next preferences are only valued at a small fraction of one vote (which each voter equally has), which is to say, a small transfer value.

Such a narrowly elected candidate with small transfer value of votes, means that their keep value, the fraction of votes kept is correspondingly not much less than one. A keep value of unity means the candidate has only a quota of votes just sufficient to elect, with no surplus to give to next preferences, or a zero transfer value.

Meek method only calculates keep values for candidates, who have reached, or are in surplus of, a quota. That is where T equals or is greater than Q. In those cases, the transfer value is some positive fraction, or at least never less than zero.

FAB STV also calculates the keep values for the remaining candidates, who do not reach a quota and have keep values greater than unity. This means they must have negative transfer values, which is to say, votes in deficit of a quota, rather than surplus votes to transfer to the voters next prefered candidates.

Allowing for the possibility, that k is greater than one, implies a minus transfer value, or –f, So, for quota deficit candidates: k-f = 1.

This generalises the probability formula from just p+q = 1. In classical probabilty either p or q, as complementary fraction or ratio, may arbitrarily stand in for some probability in question. It's just a question of more or less difference between the two options.

Introducing a quota to one of the probabilities increases the mensural power of the probability theory from a majority measure to a proportional measure. Just as proportional representation (as originally and properly conceived by the mathematician Carl Andrae, and by Thomas Hare) is a more powerful measure of the voters wishes than plurality counting (First Past The Post).

Classical probability knows only positive probabilities. Quota probability enlarges the dimensions of measurement by including the negative number scale, in probability theory. (I believe a Richard Feynman paper speculated on negative probabilities.) There is a similar such measurement enlargement by FAB STV also allowing negative transfer values to unelected candidates with keep values above unity.

Moreover, there is symmetry between the Interval, with its (c+u) and (c-u) factors, and between FAB STV, with its (k+f) and (k-f) factors.

Statistically speaking, (c+u) is the upper limit and (c-u) is the lower limit of a range of velocities above and below the light speed, c. The Interval effectively finds the average of the range by taking its geometric mean, which is the square root of the mutiple of the two factors. Tho, usually, the formula is left as the Interval squared:

(I/t)² = (c + u)(c – u).

A basic rationale of FAB STV is the averaging of counts by the geometric mean. In one count, a candidate may have a keep value over unity, because in deficit of a quota. In another count, the same candidate may have a keep value below unity with a surplus transfer value. If the average keep value is still below unity or equal to it, then that over-all result constitutes an elective proportion of the votes.

In Special Relativity, the Interval can take either kinematic or dynamic form. The form already given is kinematic. The conversion to a dynamic Interval, Y, is simply accomplished by substituting the variable, time, for variable, mass, m. (In classical physics, mass is constant. In Relativity, mass is variable.)

Y² = m²(c² – u²).

It turns out that there is a simple way to mimick this physics dualism, with FAB STV, because the equation relating keep value to transfer value can be expressed in two ways.

Starting from:

k± f = 1, where k = (V/S)T

f = 1± (V/S)/T.

So:

f = (T± V/S)/T.

Or:

f = (S± V/T)/S.

The plus or minus possibilities can be averaged by the geometric mean. Hence, the two forms, respectively become:

f² = {(T+ V/S)(T- V/S)}/T²

= (T² – (V/S)²)/T²

= 1² – {(V/S)/T}².

The value, 1² or one squared equals one, but this is in relation to term, V/S as a squared fraction. Tho squaring unity leaves it the same, squaring V/S much reduces it.

The unity squared may be compared to the square of light speed, c, which is also conveniently set at unity, to simplify the computation.

The keep value term, (V/S)/T, may compare to some value in the dynamic version of the Interval.

Moving on to the alternative verson of the transfer value:

f² = {(S + V/T)(S - V/T)}/S²

= (S² – (V/T)²)/S²

= 1² – {(V/T)/S}².

This compares to the kinematic Interval, because the keep value is now not in terms of a total transferable vote, T, but in terms of a total of transferable seats. What we have here is a different sort of election altogether, tho similar in form.

The transfer of seats might take place, if some local communities had more than others, and you wanted to create a uniform member system, such as in Tasmania or Ulster. (Whether it was a 5, 6, or 7 member system has changed, over the years, regretably, down to a proportionally less accurate five-member system in both cases.)

Nearly half a century since, I could not help but notice that Special Relativity concerned uniform motion in a straight line, whereas elections tended to be uniform member systems of seats per constituency.

Now, just match the two pairs of physics and elective equations. Just multiply thru the denominator, S, for the electoral version of the kinematic Interval, I; and multiply thru the denominator, T, for the electoral version of the dynamic Interval, Y.

Thus:

S²f² = S² – (V/T)²

compares to:

I² = (ct)² - x².

And:

T²f² = T² – (V/S)²

compares to:

Y² = (E/c)² - p².

One has to remember that the electoral versions have the analog of c² implicit in their equations as 1², meaning that the terms, V/T and V/S are a fraction of unity.

In comparing special relativity to binomial STV, I looked for the formers use of the binomial theorem. In fact, the Poincare-Einstein equation, E = MC², is derived from an expansion of the binomial theorem, in form, looking like the Fitzgerald-Lorentz contraction factor, (1 - v²/c²)^1/2, also called the "gamma factor".

(By the way, this famous equation is an approximation, based on only the first two terms in the expansion into a series. Probably, general elections could be conducted satisfactorily with an expansion of Binomial STV to no more than second order STV.)

This electoral model of the kinematics and dynamics of special relativity can be used to explain why the two forms of the Interval are so alike. The velocities involved are the same. They just have different coefficients. The kinematic coefficient is time and the dynamic coefficient is mass. Yet these two terms appear to be radically different observables. Mass can be felt and takes physical effort to move. Time cannot be physically felt and cannot be moved. Time is more like a mental state of awareness.

So it is not at all obvious why mass and time should be so ready substitutes in the Interval formula. The question then is can they be mathematically derived from each other.

Richard Feynman was asked this question and came back the next day, to admit he couldn't do it! I was impressed as much by his honesty (which he inculcated in his students) as by his legendary reputation for fundamental originality and surpassing computational skills, that made him widely regarded as the greatest in his profession, during the mid-twentieth century.

Actually, you don't need to be a prodigy to explain the time-mass symmetry of the kinematic and dynamic Intervals. But it helps to have the electoral model to hand. Like the sharp contrast in physics between the concepts of time and mass, the electoral situation also offers a sharp contrast in kinds of elections. One counts votes per candidate. The other counts seats per constituency or district.   
The number of seats per district may be governed by the shifting proportions of people with population migrations across district boundaries. This is the kind of election known as voting with ones feet.

The electoral model offers two alternative keep values, from the same basic formula. This is analogous to the time-mass situation in special relativity. In order to make this clear, it is necessary to show an isomorphism or one-to-one relation of terms between special relativity physics and election method or "electics".

A start was made, in identifying light speed with a unity term. But we need to transform the election count formula till its form coincides with the Interval. We multiply the two electics formulas by their denominators, in either, S, the seats (representatives), or T, the total transferable vote (which term shortens to "transferable vote"). For purposes of comparison with physics, these may be called the kinematic and dynamic elections, respectively.

Indeed, I assumed that the electics term, S, for seats (reps.), corresponds to the physics term for time, t. And that term, T, corresponds to physics term, mass. On this basis, I managed to work thru to a consistent comparison between Special Relativity (SR) physics and FAB STV electics. It is convenient to put the whole comparitive scheme into a table.

Physics and Electics SR dynamics | STV dynamics | SR kinematics | STV kinematics  
---|---|---|---  
m, mass | T, transferable   
vote | t, time | S, seats  
(reps.)  
c=1,   
light speed | k=1, quota   
keep value | c=1 | k=1

u=(mx/t)/m =   
(mut/t)/m =   
(mu)/m = p/m  
= x/t; velocity   
equals distance   
over time.

| k=(V/S)/T,   
keep value =  
(votes per   
seat) quota,   
over  
transferable   
vote. | u=(mx/m)/t  
= x/t  | k=(V/T)/S,   
keep value  
= (votes per   
transferable   
vote) quota,   
over seats.  
p=mu,   
momentum =  
mass times   
velocity. | T.(V/S)/T = V/S   
(votes per seat) quota. | x = ut, distance   
equals velocity   
times time. | V/T, votes over transferable votes.  
pt=mx,   
momentum   
times time =   
mass times   
distance. | (V/S).S = V,   
votes | tx, time times   
distance | S.V/T = V/(T/S)   
votes per   
transferable   
vote per seat.  
E/c = mc,   
Energy over   
light equals   
mass times   
light (speed). | T.1,   
transferable   
vote times   
unity keep   
value. | tc, light speed   
multiplied by   
time. | S.1, seats   
times unity   
keep value.  
mcg= Y =  
mc{1-(u/c)²}^1/2  
mass times light   
times Interval   
gamma factor. | Tf,   
transferable   
vote times   
transfer value | tcg=tc{1-(u/c)²}^1/2  
= I. Time times light   
speed times Interval   
gamma factor. | Sf, seats times   
transfer value.  
Y² = (E/c)² - p², dynamic   
Interval squared. | T²f² =   
T²(1²)–(V/S)² | I² = (ct)² - x²,   
kinematic   
Interval squared. | S²f² =   
S²(1²) – (V/T)²

In the table, term, g, stands for the Interval generalisation of the gamma factor, the Fitzgerald-Lorentz contraction factor. The Interval version, g, applies to any local velocities of different observers of an event. The contraction factor only equals the Interval version, when one of the local velocities is zero, or at rest relative to any other observer.

In this table, the key row is for velocity. The kinematics and the dynamics pair of boxes for electics and physics transform the same formula to give two different forms. In physics, the result is kinematics and dynamics. But it is the electoral model, which hints at what is going on here.

In proportional elections, the concept of a quota of votes per seat is familiar. Another possibility is a quota of votes per transferable votes. This would be a quota to determine the re-apportionment of votes from one constituency to another, because population shifts made it more equitable to give one multi-member constituency an extra seat, for extra population, taken from a community constituency with declining population.

I suggest here that physics of dynamics and kinematics operates analogously to two different versions of the keep value, with different quotas in the numerator. In the kinematics case, the quota is mx/m, such that the mass terms cancel, so that the numerator as quota appears simply as x, distance. But in the dynamics case, the quota in the numerator, mx/t, cannot cancel the mass term. But it cancels the time term, t, because:

mx/t = mut/t = mu = p, for momentum. Thus, a difference in quotas, as keep value numerators, canceling either the mass or time term, explains how the same basic physics formula transforms into either dynamic (mass) or kinematic (time) form.  
At least, in a formal sense, physics is, in some sense, electoral. It is tempting to go a guess further and suggest that as electics deals in the quota, physics deals in quanta. That is to say, discrete proportions, are the basis of both sciences.

The electoral model highlights the quota nature of physics. Proportional counts have a maximum and a minimum quota. Likewise, physics has its maximum and minimum quotas. The maximum velocity ratio is light speed. The minimum energy transfer is the quantum.

It is interesting to note that the Heisenberg Uncertainty Principle, of subatomic physics, is usually framed either in terms of the trade-off between measuring momentum and position, or in measuring energy as against time. In special relativity, compared to its electoral model, momentum and distance are the dynamic quota and kinematic quota. Whereas, mass, which is equivalent to energy, compares to a total transferable vote, and time compares to total transferability, in terms of seats per district, for unequal populations in local communities to be transformed into a uniform multi-member system.

An electoral model might involve a normal distribution of seats per constituency in proportion to the normal inequalities of populations in local communities.

There is not a uniform increase in population, moving from village to town to city. Rather, there is a great mass of communities, the norm, close to the average population, and therefore with an average number of seats per district. And there is a rapid falling off in the number of the most sparsely populated and also the most densely populated communities.

Graphically, this shows as the famous bell curve of the normal distribution, a non-uniform distribution. Whereas a uniform distribution would graph as a straight line, from rural to urban population distributions, tho it might tilt one way or the other.

Like the kinematic version of Electics, in terms of seats per constituency or district, d, we are looking for a normal distribution to a dynamics of Electics. This also involves a new variable to the electoral model, namely candidates, c. The number of votes that candidates receive is well known to form a normal distribution.

(To satisfy the principle of democratic equality, the quota, or elective proportion of the votes, the votes per seat or representative, V/S, remains constant.)

Non-uniform and uniform seats per district are possible (posssibly analogous to general and special relativity cases). Then non-uniform and uniform votes per candidate should also apply. The special case of equal votes per candidate amounts to a quota. This would seem to imply some sort of primary election of candidates, as second-order representatives.

In principle, higher orders of representation would also be possible. The distribution of primary, secondary, tertiary, etc candidates might be mathematically represented, for example, by the formula for exponential growth. That is to say, by the exponent with a positive index, instead of the opposite case, above considered, with a negative index (known as exponential decay). It may be noted that above derivation allowed for both possibilities.

Mankind is inescapably territorial. He does not exist in a vacuum. Tho, it is possible to make districts implicit, by considering only one district, or by treating man as belonging to one world.

Likewise, it could be argued that just as territories are ecologically necessary, as an enduring base for a species, so is the tribalism that defends them, which takes political form as parties, made up of candidates, contesting their positions.

The complementary nature of territories and tribes or districts and parties might be expressed in simple mathematics. If there are two basic kinds of district, rural and urban, their distribution can be expressed as an expansion of the binomial theorem, taken to the power of the number of seats or representatives per district, s/d. The binomial distribution expresses how rural, r, and urban, u, districts have numbers of seats per district in proportion to rural and urban population densities.

To put this in symbols: d = (r+u)^s/d. If the population is a normal distribution, unskewed in either the rural or urban direction, then r and u can be given equal unit values. In effect, d = 2^(s/d).

### Mach principle as statistics.

To top.

The Mach principle related the motions of bodies to the motions of all other bodies in the universe, rather than put them in some arbitrary context of any imposable space and time frame-work. Barbour and Bertotti derived Newtonian laws from Mach principle, which they also found was implicit in General Relativity.

Their process of "best matching" of bodies from one configuration to another may be likened to a statistical idea of finding the ranges of bodies motions by putting successive motions of bodies in a series. There are "goodness of fit" tests in statistics, such as the chi-squared test, that show how well actual ranges of values match theoreticly expected values. In principle, this class of tests might be used to best match configuration changes. At any rate, Barbour says: "it is very convenient to measure how far each body moves by making a comparison with a certain average of all the bodies in the universe."

As the Barbour title, The End of Time, suggests, he is trying to replace time as a basic concept in classical as well as quantum mechanics. A note suggests what I take to be a use of topology to remove distance measurements, that define the traditional concept of space, in physical theory. This would be consistent with a statistical treatment that also does not need to assume space and time.

In so far as Barbour and Bertotti worked a statistical treatment of General Relativity, it has the merit of showing the necessity of that treatment, without having pre-supposed that need with a program for reformulating GR in terms of statistics.

Mach principle appears to be a statistical program under another name. And it shows the kinship of science and democracy. It is scientific because it insists on reference only to observables. It does not impose any outside reference, which anyone can impose, resulting in a dead-lock of prejudices. Progress in knowledge depends on shedding unfounded assumptions for which there is no basis in agreement from common experience.

This is democratic progress because it depends on all points of view agreeing on the rules or laws, rather than an arbitrary rule, essentially a privileged anarchy, in imposing the conventions. Relativity is a democracy of observers all equally free to observe physics laws. Mach principle is a democracy of the observed physical phenomena, representatively measured, rather than externally measured.

* * *

### General relativity and FAB STV.

To top

Minkowski four-dimensional space-time geometry inspired Einstein to the curved geometry of General relativity.

In his book, The End of Time, Julian Barbour says:

"Alternatively, and much closer to what happens in general relativity, I could compare the curvatures at matching points. The essential point is that some intrinsic property is compared at each pair of matched points, and an average of all the resulting differences is then determined. This average, one number, is the provisional difference. I leave out the mathematical details, which are intricate even though the underlying idea is simple....

"We shall know that we have found the best matching pairing and corresponding intrinsic difference when the provisional difference remains unchanged if we go from the given pairing to any other pairing that differs from it ever so slightly. (In mathematics, the fulfilment of this condition indicates that one has found a maximum, a minimum or a so-called stationary point of the quantity being considered..)...

"... It soon became clear in the discussions with Karel [Kuchar] that the idea of best matching and the whole way of thinking about duration as a measure of difference were already both contained within the mathematics of general relativity, though not in a transparent form.... General relativity was discovered as a theory of four-dimensional space-time, and that is still essentially the way it is presented. The fact that it is simultaneously a dynamical theory describing the changes of three-dimensional things is given much less weight. That is why so few people are aware that there is such a deep issue and crisis about the nature of time at the heart of general relativity."

The procedure of best matching has some striking similarities to FAB STV. It too is a succession of provisional approximations to an ideal average, the over-all keep value. And this value also can be maximal, minimal or stationary. The stationary value corresponds to the quota, which is a stationary or constant value, or a required keep value of one, just sufficient to elect a candidate. Maximal values, which is to say those keep values higher than unity, actually apply to candidates, whose votes are in deficit of a quota. The minimal keep values, of less than unity, belong to the candidates who already have a quota, with (a surplus of) votes to spare.

The size of the surplus is the transfer value, to next prefered candidates. Thus, a minimal keep value goes with a surplus transfer value, which sums to unity, or the rationing of the electorate to one person one vote. Conversely a maximal keep value of more than one, sums to unity, with a corresponding negative transfer value in deficit of a quota.

One could stretch the comparison of electics with physics, by comparing positive and negative transfer values with geometries of positive and negative curvature, introduced into general relativity.

An election consists of two corresponding frames of reference, the vote and the count. The votes expresses inequality of choice. The count gives proportion to its representation. The more efficient the electoral system, like FAB STV, the clearer this is. Usually, the sum of votes form an exponential distribution. It may be the normal distribution. And the value of the transfer of votes from more to less prefered candidates falls off exponentially.

Compared to general relativity, this is an "acceleration" (or deceleration) reference frame in elections.

This acceleration is transformed into a "velocity," which is the quota count of transferable voting. Because, the quota is a uniform count of votes per seat, which is a constant ratio, like velocity is a constant ratio.

FAB STV is particularly suitable for comparison with natural science, because it offers one complete dimension of choice, that the voter can move along forward or backward, in preference or unpreference, also counting abstentions, so as to leave no gaps in the preferential measurement scale.

And once you've got one dimension, potentially you have got them all. I have given the rules for two dimensions (FAB STV 2-D).

Relative choice is strikingly similar to relative motion. I would imagine possible a more systematic comparison than I have given, even an isomorphism. Yet I would not attempt a slavish imitation of physics by electics. They are, after all, very different subjects and one would expect considerable divergences. It is enough, perhaps, that they sufficiently share a common mensural framework, as to draw attention to itself.

Physics has been trying to remove the special status of some arbitrary frame of reference, like absolute space and time and velocity, and demote all possible arbitrary frames of reference to equally insignificant status, as points of observation. This is about the fundamental importance of symmetry in science.

Is not logic of measurement, a frame of reference we cannot put by, whatever scientific picture we look at? (This was the subject of my book, Science is Ethics as Electics.)

# Disclaimer.

Table of contents

This book has not been independently checked, tho I guiltily corrected such mistakes as I found. To my horror, one of my fumbled web-pages, that I had long since removed, must have been cached somewhere, because it was viewable still on the internet!

My grammar school failure, to qualify for the natural sciences, gave me a phobia, and left me without the faintest idea of scientific discovery. My further education amended that clueless state, but concluded I was not right for the social sciences, either.

These pages began as a teach-myself exercise, at the turn of the century (and millenium), one winter when I was confined, with an infection, to a room with a gas-fire (since removed), and didn't go to bed for a month or three.

John Gribbin, in one of his many books on popular science, derided special relativity as "ancient history." So great was the growth of natural science, in the twentieth century compared to the rest of history, a theory from the beginning of that century must seem out-moded. His book on time still had to deal with it, under whatever caption. It remains a starting point for modern physics.

Long since, thousands of books were written about relativity. Often, a good idea is given of the mathematics of special relativity. This can be done without going much, if at all, beyond high school standard. (As if I could.)

I checked in the local library (June 2011), to pick up a new physics text book. As expected, it described the Michelson-Morley calculation of different times for two perpendicularly reflected split light beams to return to source.  
This text happened to be published in 2008, four years after I first showed, on my web-pages, that the calculation uses the wrong average.

Also in 2011, I e-mailed a physics forum moderator, who refused to post my message, with this remark:

'Your aside that "a common mistake still found in modern physics texts" occurs "in the Feynman Lectures on Physics, vol. 1" is not relevant because that so-called "mistake" is so common it can be found in all physics textbooks, and all books and papers that discuss the M-M experiment that I am aware of, except your web pages.'

I pointed out that the unquestioning following of the historic Michelson-Morley calculation is not a rational argument for it. It's a simple issue: use the arithmetic mean or the geometric mean. Nothing to do with my "promoting personal beliefs," as his e-mail also said, but an objective scientific question of right and wrong. That appears to be how the professional closed shop distinguishes the ideas of non-professionals: stigmatises them as personal beliefs to be ignored. The amateur seems to be regarded as the wrong person to be right.

Somewhat bitterly, I rejoined: What little I may have learned, imperfect tho I fully admitted it to be, I studied for it all my life, without anybodys help, since I left college, and - I would hope - without having to put up with wilful obstruction.

I did find a physics forum that admits non-professionals. This was PhysForum, Science, Physics and Technology Discussion Forums. So far as I know, nobody discussed my post titled: Michelson-Morley Experiment, Geometric mean (not AM) predicts truly.

My post said:

The Michelson-Morley experiment would have been correctly predicted by using not the arithmetic mean but the geometric mean to average the return journeys of the reflected light beam in line with, and split at right angles to, earth motion.

That is a change in velocity of light relative to earth motion or relative acceleration, for which a geometric mean is the appropriate average.

The Minkowski Interval is in effect a geometric mean form, which correctly predicts the Michelson-Morley experiment, as I showed on one web page. (Links not allowed.)

Another web page disproves the experiment's use of the arithmetic mean by how it gives non-solutions of the Lorentz transformations.

The titles of the respective web pages [revised in this book] are:

The Minkowski Interval predicts the Michelson-Morley experiment.

Lorentz transformation non-solutions of the Michelson-Morley calculation.

Like everyone else, I make mistakes and no doubt my web pages on popular physics are riddled with errors, for which I can only apologise and give fair warning. I did try to take care but I know I am prone to confusion.

And my education is in social science not natural science.

[End of forum post]

In a 2017 John Gribbin book on Fifty Experiments, recall of the original sensation appears to have been lost. A now old popularisation, by Sir James Jeans, The New Background of Science (1933) catches the original consternation of the physics community.

The maths of general relativity is much too advanced for me. And I don't know of any good popular account of it. Roger Penrose, in The Emperor's New Mind, gives a thumb-nail sketch of the maths involved. He also prefaces Six Not So Easy Pieces, by Richard Feynman, which gives the best conceptual explanation of General Relativity, that I came across. The "pieces" are, in fact, taken from his under-graduate text-book on physics. Don't let that put you off. That treatment gives much insight.   
If you are a beginner in the subject, it would be better to read first several introductory works, to fully appreciate Feynman on special and general relativity. I am glad I didn't come across Six Not So Easy Pieces early in my reading. In fact, my explanation is not influenced by that book.

Round about 1960, the master mathematician, Roger Penrose made general relativity more accessible to the profession, sparking a renascence in its research. Penrose, eventually, brought out a text book, The Road To Reality. However, this opus was so ambitious in scope that it could not be contained in one volume. Explanations were farmed out to other texts. I did not live near a good library. And I also knew from experience, that such delegations of responsibility generally don't afford a seamless explanation of a subject.   
I did try to trace the meaning of all the strange mathemetical concepts indexed. Wikipedia informed me (but left me none the wiser) that much of this forbidding jargon belonged to General Relativity. Perhaps not surprising from the physicists mathematician of GR.

By the way, the Penrose magnum opus does not mention the Michelson-Morley experiment. It's a bit like missing out Archimedes. I did post his secretary that the geometric mean, not the arithmetic mean, should be used in the traditional Michelson-Morley calculation, and could not help but wonder if that advice eclipsed the great MMX from his great volume.

When I was tired of compiling and revising the draft of my book, I decided to take a rest (most recently and seriously disrupted by an unhelpful social services of my attempts to continue being a carer). I thought I would indulge a little recreational booklet and pass the time with some preparatory work for another book or three, not to mention a few not so brief -- and quite draining \-- projects. The booklet turned into an arduous full-length biography, of my father, and social history. Another lengthy and exhausting work on alphabet reform somehow got itself published.

And then there was FAB STV: Four Averages Binomial Single Transferable Vote. That arithmetic marathon did however turn up something useful for this book, when I found out what the fourth average is.

The first three averages were familiar: the arithmetic mean, the harmonic mean, and the geometric mean. The geometric mean is actually a power arithmetic mean and, to cut a long story short, the fourth average is a power harmonic mean.

Then I knew that there could be not only three forms of average to differentiate but also a fourth, namely power harmonic mean differentiation. Being another power series, like the geometric series, its differentiation should be somewhat similar yet distinct from geometric mean differentiation.

## Moral

At the end of a work that tries to promote an alliance between science and democracy, it might be worth mentioning how damaging or fatal science can be, in undemocratic hands.   
The twentieth century was called the nuclear age, for its Pandora story of the release of nuclear energy on the world, whether in the form of nuclear arsenal over-kill potential, or for "peacable" purposes, which, however, permanently pollute the planet with radioactive waste products from uranium fission energy.

Vested interests keep pressing their poisonous fission plants on the world, before they become obsolete, and their investments worthless, with advances in renewable energy and its storage. (Fusion energy remains out of the picture.)

Dismaying was the determination of the British government to press on with nuclear power regardless, even in 2019, when Hitachi pulled-out. Science-educated Angela Merkel took the heavy hint of Fukushima, that nuclear is not a good idea. Wind and solar power continue their long evident improvement. Nuclear problems proliferate and costs escalate, and pose an unsolved threat to future generations of life on earth for hundreds of thousands of years.

"The Nuclear Barons" subsidise politicians to subsidise nuclear. This corruption goes mostly unchallenged by an obsequious media. The European Union allowed the British government exemption from its fair competition rules.   
What use the EU, if it cannot obey its own laws, when it really matters, like on a question of life or death for the future Earth? (This questionable ruling was challenged by governments of Austria and Luxembourg, as well as Bavarian solar power firms.)

It is disturbing that the sway of the British political class has been so biased in favor of an inefficient and needlessly dangerous Soviet-style command economy answer to energy supply.   
British rulers (or politicians the world over) remain predominantly feudal in attitude, not fit to govern a democracy, or indeed a biocracy, when you consider the feet-dragging on containing global warming.

* * *

## Other Works by this author.

Table of contents

* * *

### Single-stroke English (Summary edition)

This is the booklet for learning Single-stroke English. The essential English letters of the alfabet at a stroke -- so far as practical. More memorable than year-zero shorthands. By-passes the hopeless tangle of English spellings.

* * *

## Guide to five volume collected verse by Richard Lung

Table of contents

The following descriptions give information on my other books. My up-to-date books list, with links, can be found on my profile page: here.

* * *

### The Valesman.

The first volume is mainly traditional nature poetry.   
(160 poems, including longer narrative verse in section three.)  
The nature poet Dorothy Cowlin reconnected me with my rural origins. Many of the poems, about animals and birds and the environs, could never have been written without her companionship.

The unity of themes, especially across the first two sections, as well as within the third section, makes this volume my most strongly constructed collection. I guess most people would think it my best. Moreover, there is something for all ages here.

1. How we lived for thousands of years.  
Dorothy thought my best poems were those of the farming grand-father, the Valesman.

2. Flash-backs from the early train.  
More memories of early childhood on the farm and first year at the village school.

3. Trickster.  
Narrative verse about boyish pranks and prat-falls.

4. Oyh! Old Yorkshire Holidays.Features playtime aspects of old rural and sea-side Yorkshire.

* * *

### Dates and Dorothy

Book two begins with eight-chapter review of works, plus list of publications & prizes by Dorothy Cowlin.

This second volume continues with the second instalment of my own poems, classed as life and love poetry.  
The Dates are historical and romantic plus the friendship of Dorothy and the romance of religion. 169 poems plus two short essays.

Prelude: review of Dorothy Cowlin.

Dates, historical and romantic, and Dorothy:

1. dates.  
2. the Dorothy poems.  
3. loves loneliness loves company.  
4. the romance of religion.

The hidden influence of Dorothy, in the first volume, shows in this second volume. The first two sections were written mostly after she died. Thus, the first section, Dates, reads like a count-down before meeting her, in the second section, as prentice poet.

She was warmly responsive to the romantic lyrics of the third section. This was reassuring because some originated in my twenties. (I gave-up writing formal poetry during my thirties, to all practical purposes. There were scarcely three exceptions.) These surviving early poems, like most of my out-put, under-went intensive revision.

The fourth section probably stems from the importance attached to religion at primary school. Here humanitarian Dorothys influence made itself felt, somewhat, by her liking to visit churches.

The fotos, I took of Dorothy, are published for the first time. I welcome this opportunity to publish my literary review of her work, as an extra to volume 2 of my verse.

* * *

### He's a good dog. (He just doesnt like you to laf.)

The third volume is a miscellaneous collection of 163 poems/pieces, making-up sections, one, three and four, with the arts and politics the strongest themes, as well as themes found in other volumes. There is also a story, and a final short essay.

1. with children  
2. or animals  
3. never act  
4. the political malaise  
5. the lost  
6. short essay:

Proportional Representation for peace-making power-sharing.

One section includes a sort of verse novela and dramatic poem with an eye on the centenary of the First World War. The idea stemmed from an incident related by Dorothy Cowlin (yet again). Her uncle was stopped flying a kite on the beach, because he might be signaling to the enemy battle fleet.   
In this miscellany, previous themes appear, such as children, animals and birds. Verse on the arts comes in. I organised these poems on the WC Fields principle: Never act with children or animals.  
The fourth section collects political satires from over the years.

The fifth section reflects on loneliness.

This volume is classed as of "presentatives" because largely about politics and the arts, with politicians acting like performing artists or representatives degenerating into presentatives on behalf of the few rather than the many.

However, the title poem, He's a good dog..., hints how eccentric and resistent to classification is this third volume. (There are six dog poems in the volume.) This title poem is based on a true war-time air incident. The good dog is also derived from a true dog, whose own story is told in the poem, the bleat dog (in volume 1).

* * *

### In the meadow of night

The fourth volume is of 160 poems and three short stories on the theme of progress or lack of it.

part one: allure.  
The allure of astronomy and the glamor of the stars.

part two: endeavor.  
The romance and the terror of the onset of the space age and the cold war.

part three: fate.  
An uncertain future of technologies and possible dystopias. Ultimate questions of reality.

This fourth volume is of SF poetry. SF stands for science fiction, or, more recently, speculative fiction. The verse ranges from hard science to fantasy.  
This literary tradition of HG Wells and other futurists exert a strong influence.  
Otherwise, I have followed my own star, neither of my nature poet friends, Dorothy and Nikki, having a regard for SF poetry.   
Yet science fiction poetry is a continuation of nature poetry by other means.  
This may be my most imaginative collection. Its very diversity discourages summary.

* * *

### Radical!

Volume 5 opens with a play (since published in a separate book) about the most radical of us all, Mother Teresa: If the poor are on the moon...  
This is freely available, for the time being, on my website: Poetry and novels of Dorothy Cowlin. (Performers are asked to give author royalties to the Mother Teresa Mission of Charity.)

The previously unpublished content consists largely of fairly long verse monologs, starting with artistic radicals, in "The dream flights of Berlioz and Sibelius," which is a sequence of The Impresario Berlioz, and The Senses of Sibelius.

Next, the intellectual radical, Sigmund Freud, followed by short poems on a sprinkling of more great names, who no doubt deserved longer. (Art is long, life is short.)

The title sequence, Radical! is made-up of verse about John Stuart Mill, Arthur Conan Doyle, George Bernard Shaw, HG Wells, George Orwell and JB Priestley.

_If you read and enjoy any of these books, please post on-line a review of why you liked the work.  
_

* * *
Seperately from "Radical!" the play about Mother Teresa, also appears as a book, on its own, here.

## Guide to two more book series.

Table of contents.

## The Commentaries series

* * *
Commentaries book one:

### Literary Liberties

Literary Liberties with reality allow us to do the impossible of being other people, from all over the world. Our imagined other lives make the many worlds theory a fact thru fiction.

This book of books or illustrated reviews span fiction, faction and non-fiction.  
It goes some way to substantiate the belief of Benedetto Croce that history is the history of liberty.  
I only wrote of books that I appreciated, so that I could pass on that appreciation to others. It must be admitted that I went with novels that looked over horizons confined to family values. (Family is, of course, a basic trial of liberty, compromised by obligations to partner and children.)

Likewise, these reviews themselves need not be bounded by the horizons of literary criticism but reach out to solutions for the problem novel or the non-fiction book with a cause.

In promoting others writings, I hoped to promote my own, any-way, the liberal values that inform my writings. It took a lot more preparation than I had anticipated. This is usually the case with my books.

Literary Liberties is the first of a series of Commentaries. This author also has a Democracy Science series. The series of Collected Verse was the first to be completed.

* * *
Commentaries book two:

### Science and Democracy reviews

As they separately pursue their shared ethic of progress, scientific research and democratic reform conduct themselves as two different journeys, both here followed, as the evidence mounts that they depend on each other to meet the stresses that survival poses.

Works reviewed and studied here include the following.

The physicist, John Davidson under-took an epic investigation into the mystic meaning of Jesuses teachings, as for our other-worldly salvation, supplemented by a revelation in non-canonic texts of the gnostics.

The Life and Struggles of William Lovett, 1876 autobiography of the "moral force" Chartist and author of the famous six points for equal representation.   
Organiser who anticipated the peace and cultural initiatives of the UN, such as UNESCO.

Jill Liddington: Rebel Girls. Largely new historical evidence for the role especially of working women in Yorkshire campaigning for the suffrage.

"How the banks robbed the world" is an abridged description of the BBC2 program explanation of the fraud in corporate finance, that destroys public investments.

David Craig and Matthew Elliott: Fleeced!  
How we've been betrayed by the politicians, bureaucrats and bankers and how much they've cost us.

The political system fails the eco-system.  
Green warnings, over the years, by campaigners and the media, and the hope for grass roots reforms.   
From Paul Harrison, how expensively professionalised services deprive the poor of even their most essential needs. And the developed countries are over-strained, on this account, drawing-in trained people from deprived countries.  
Why society should deprofessionalise basic skills important for peoples most essential needs, whether in the third world or the "over-developed" countries.

The sixth extinction  
Richard Leakey and other experts on how mankind is the agent of destruction for countless life forms including possibly itself, in the sixth mass extinction, that planet earth has endured in its history. Why world politicians must work together to counter the effects of global warming.

On a topic where science and democracy have not harmonised, a few essays from 2006 to 2010, after "nuclear croneyism" infested New Labour and before Japans tsunami-induced chronic nuclear pollution. There's a 2015 after-word.

Some women scientists who _should_ have won nobel prizes.   
Lise Meitner, Madame Wu, Rosalind Franklin and Jocelyn Bell, Alice Stewart, to name some.

Reading of their work in popular science accounts led me, by chance, to think they deserved nobel prizes; no feminist program at work here.

Julian Barbour: _The End Of Time._   
Applying the Mach principle, to an external frame-work of Newtonian absolute space and time, both in classical physics and to Schrödinger wave equation of quantum mechanics, by which the universe is made properly self-referential, as a timeless "relative configuration space" or Platonia.

Murray Gell-Mann: _The Quark and the Jaguar._   
Themes, including complex systems analysis, which the reviewer illustrates by voting methods.

Brian Greene: The Elegant Universe.   
Beyond point particle physics to a theory of "strings" that may under-lie the four known forces of nature, and its material constituents, thru super-symmetry, given that the "super-strings," as such, are allowed to vibrate, their characteristic particle patterns, in extra hidden dimensions of space.

Brian Greene: The Hidden Reality.   
A survey of the more extravagant physics theories that have invoked many worlds or a multiverse.

Lee Smolin: Three roads to quantum gravity.   
Reviewing the other two roads (besides string theory) namely black hole cosmology and loop quantum gravity. All three approaches are converging on a discrete view of space and time, in basic units, on the Planck scale. General relativitys space-time continuum is being quantised, rather as nineteenth century thermo-dynamics of continuous radiation was quantised.

Lee Smolin: the trouble with physics.   
Impatience with the remoteness of string theory and hope for progress from theories with more experimental predictions. How to make research more effective. Smolin on a scientific ethic. Reviewer criticises the artificial divide academics make between science and ethics.

* * *
Commentaries book three.

### Echoes Of A Friend: Letters from Dorothy Cowlin.  
Comment by Richard Lung.

Dates And Dorothy started with a literary appreciation of the professional writer, traveler, nature walker, and poet, combined with my second book of verse, that includes the story of our friendship.

My second book, about Dorothy, is a memorial, she graces. by speaking thru letters to me, as well as assessments of this writer, she made into a maker and aided as a reformer.  
In widowhood, she yet became companionable and widely liked. Her quiet and sunny disposition held in reserve a deeply serious nature.

* * *
Commentaries book four:

### War from War.

Biography of the authors father, in his faraway origins, over-taken by war, on two fronts, and how to confront it!

## The Democracy Science series.

Table of Contents.

The Democracy Science series of books, by Richard Lung, Some material is edited and renovated from this authors pages on the Democracy Science web-site, largely superseded.

* * *
Book 0:

### Single-stroke English (long edition).

This is the long edition of the English short-hand alfabet, with extra information on making English easier to learn and use.

* * *
Book 1:

### Peace-making Power-sharing.

The first book on voting method, has more to do with electoral reform. (The second is more about electoral research.)  
"Peace-making Power-sharing" features new approaches to electoral reform, like the Canadian Citizens Assemblies and referendums.   
I followed and took part in the Canadian debate from before the assemblies were set-up, right thru the referendums.

This was a democratic tragedy and an epic in the dashing of idealistic hopes.  
Some developments in America are reviewed.

The anarchy of voting methods, from the power struggle in Britain, is investigated over a century of ruling class resistance to electoral reform.

A penultimate chapter gives the simplest way to explain transferable voting, on to the more formal treatment of a small club election.

The last chapter is the earliest extant version of my work on scientific measurement of elections (in French).

* * *
Book 2:

### Scientific Method of Elections.

The previous book had a last chapter in French, which is the earliest surviving version of the foundation of this sequel, Scientific Method of Elections. I base voting method on a widely accepted logic of measurement, to be found in the sciences. This is supported by reflections on the philosophy of science.

The more familiar approach, of judging voting methods by (questionable) selections of basic rules or criteria, is critically examined.  
This author is a researcher, as well as a reformer, and my innovations of Binomial STV and the Harmonic Mean quota are explained.  
This second book has more emphasis on electoral research, to progress freedom thru knowledge.

Two great pioneers of electoral reform are represented here, in speeches (also letters) of John Stuart Mill on parliamentary reform (obtained from Hansard on-line).   
And there is commentary and bibliography of HG Wells on proportional representation (mainly).

Official reports of British commissions on election systems are assessed. These reports are of Plant, Jenkins, Kerley, Sunderland, Arbuthnott, Richard, and (Helena Kennedy) Power report.

The work begins with a short history on the sheer difficulty of genuine electoral reform. The defeat of democracy is also a defeat for science. Freedom and knowledge depend on each other.   
Therein is the remedy.

* * *
Book 3:

### Science is Ethics as Electics.

Political elections, that absorbed the first two books in this series, are only the tip of the iceberg, where choice is concerned. Book three takes an electoral perspective on the social sciences and natural sciences, from physics to metaphysics of a free universe within limits of determinism and chance.

* * *
Book 4:

### FAB STV: Four Averages Binomial Single Transferable Vote.

Rambling discussions about and on voting method, for the general reader, in the first part. The second part is for specialists, a technical account of FAB STV.

Book 5:

### Statistical Relativity Elections.

* * *
In French/En Francais:

### Modele Scientifique du Proces Electoral.

On pouvrait considérer le problème de la représentation comme va problème scientifique de mesure. Pour cela, il y a à notre disposition quatre échelles possibles pour mesurer la représentation. L'échelle classifiée ou nominale, l'échelle ordinale, l'échelle à intervalles, et l'échelle à raison.

Le scrutin transférable (ST, ou STV, en anglais) est un système co-ordonné du vote au dépouillement, dans un ordre de préférence empirique 1, 2, 3,.. à l'ordre rational de 1, 2, 3,.. membres majoritaires.

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### The Angels Weep: H.G. Wells on Electoral Reform

The never bettered reality show of the role of domestic power politics and human frailty in the misrepresentation of the people. This collection, which amounts to a substantial volume, may establish Wells as a worthy successor to JS Mill, champion of liberty, equality and fraternity in electoral method.

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### John Stuart Mill: Proportional Representation is Personal Representation.

Essential writings of Mill on proportional representation, which he called "Mr Hare's system" of "Personal Representation." A most passionate defender of electoral liberty of the individual. This collection is augmented by characterisations and commentaries by contempararies, including the French historian, Taine.

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### 1,2,3, Representation by Enid Lakeman

Collected writings of the world authority on voting methods, Enid Lakeman, author of How Democracies Vote. Former director of the Electoral Reform Society.

Table of contents.

