

### Nuts & Bolts: Taking Apart Special Relativity

By Jim Spinosa

Published by Jim Spinosa at Smashwords

Copyright© 2010 Jim Spinosa

Smashwords Edition, License Notes: This e-book is licensed for your personal enjoyment only. This e-book may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each person. If you're reading this book and did not purchase it, or it was not purchased for your use only, then please return to Smashwords.com and purchase your own copy. Thank you for respecting the hard work of this author.

Ironic Dedication: To the unsung workhorses of Mathematics and Science—the subscript and the superscript.*

*Except for the overly complicated subscripts and superscripts, such as the letter _"T"_ appended with both the superscript _abc_ and the subscript _def,_ which are the sine qua non of optometrists, not to mention the sub-subscripts, such as the letter _"T"_ appended with both the superscript _abc_ and the subscript _def_ each of which is further appended by its own subscript, such as superscript _a_ appended by the number one etc. **

**At this juncture, it seems appropriate to mention that Smashwords does not accurately reproduce complicated subscripts or superscripts especially if they are floating images. Smashwords does reproduce endnote numbers, but these superscripts are lowered to midscripts.

Mene, mene, tekel, upharsin

CONTENTS

Introduction

1. The Definition of Simultaneity

Step 1: An Error Found or an Error Manufactured?

Step 2: The Michelson-Morley Experiment Is Flawed

Step 3: Inspired Insight or Inspired Ambiguity?

Step 4: The Method of Synchronizing the Clocks Is Missing

Step 5: The Relativity of Simultaneity

Step 6: Another Interpretation

Step 7: Einstein's Dilemma

2. The Twisted Path to the Transformation Equations and Beyond

Step 1: Metal Rods Turned into Cartesian Coordinates

Step 2: The Misapplication of Mathematical Rules

Step 3: A Change of Direction

Step 4: Return of the Misapplication of Mathematical Rules

Step 5: More Strange Equations

Step 6: The Transformation Equations

Step 7: Not So Simple Calculation

Step 8: Another Version of the Not So Simple Calculation

Step 9: Another Metal Rod Coordinate System

3. Components of a Radius and Confusing Clocks

4. The Limits of Special Cases and Running On a Train

5. Scalars or Vectors or Neither?

6. Conclusion: Sorting Through a Bag of Broken Parts

Note on Equations (3.18) and (3.19)

Endnotes for the Introduction and Chapters 1, 2, 3, 4, 5, 6 and the Note on Equations (3.18) and (3.19)

About the Author and Contact Information

Introduction

_Zur Electrodynamik bewegter Körper (On the Electrodynamics of Moving Bodies)_ was published in Vol. 17 of the _Annalen der Physik_ in 1905. Two other papers by Albert Einstein appeared in Vol. 17: _On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular–Kinetic Theory of Heat_ and _On a Heuristic Point of View Concerning the Production and Transformation of Light._ Arthur I. Miller writes, in his book _Albert Einstein's Special Theory of Relativity,_ "As far as we know the editorial policy of the _Annalen_ was that an author's initial contributions were scrutinized by either the editor or a member of the Curatorium; subsequent papers may have been published with no refereeing. Einstein's having appeared in print in the _Annalen_ five times by 1905, his relativity paper was probably accepted on receipt."1

Arthur I. Miller also includes this incident in his description of the initial reception of Albert Einstein's paper, "In the fall of 1907 the relativity paper was rejected by the University of Bern as his _Habilitationsschrift._ One experimentalist wrote, 'I cannot at all understand what you have written.'"2 As A. I. Miller notes, in the endnote that accompanies the previous quotation, this assessment was by a professor of experimental physics Aimé Forster.

One of the obstacles that may have hindered Aimé Forster's appreciation of the relativity paper is addressed by A. I. Miller, "the first part of the special relativity paper is, in fact, nothing less than an epistemological analysis of the nature of space and time."3 Epistemology is the branch of philosophy that concerns itself with theories about the nature, sources and limits of knowledge. With this in mind, it seems reasonable to try to gain a clear understanding of Einstein's philosophy of science.

It is difficult to address the significance of Einstein's many comments on the general principles that govern the field of knowledge known as physics. He certainly did not dismiss the notion that the empirical testability of a theory was a crucial criterion for judging a theory's validity. However, it was also crucial to Einstein that the premises of a theory have a naturalness and logical simplicity. This is what Miller refers to as Einstein's idea of the "inner perfection"4 of a theory. In the endnotes that accompany the "Introduction" to his book, Miller cites a well-known statement by Einstein, which Einstein wrote forty-five years after his completion of the special relativity paper, Miller writes, "in _Reply to Criticisms_ (1949) he [Einstein] described, 'concepts and theories as free inventions of the human spirit (not logically derivable from what is empirically given).'"5 Statements such as the one above, which seem to refer to the inner perfection of a theory are balanced by statements such as, "The first point is obvious: the theory must not contradict empirical facts. However evident this demand may in the first place appear, its application turns out to be quite delicate."6 Miller's conclusion is that the essence of Einstein's scientific method was inarticulable.

If we turn our attention to Einstein's book _Relativity: The Special and the General Theory,_ we can examine Einstein's "inarticulable" scientific method in operation. In chapter eight, which is entitled "On the Idea of Time in Physics," he gives us a definition for determining the simultaneity of distant events. Einstein's definition is written in the form of a dialogue between himself and the reader. In this dialogue, Einstein is investigating a meteorological phenomenon that — according to weather lore, familiar to us all — should be restricted to works of fiction and thought experiments. Namely, he is investigating lightning flashes that strike points _A_ and _B,_ repeatedly. Incidentally, the weather lore that lightning never strikes the same place twice is considered discredited by the multiple lightning strikes received by skyscrapers such as the Empire State Building. Einstein is not interested in exploring the conjunction of the atmospheric conditions and topographic features that conspire to produce such results. Instead, he is interested in determining whether these distant lightning flashes occur simultaneously. It should be further noted that a railroad line runs through points _A_ and _B._

"After thinking the matter over for some time you then offer the following suggestion with which to test simultaneity. By measuring along the rails, the connecting line _AB_ should be measured up and an observer placed at the mid-point _M_ of the distance _AB._ This observer should be supplied with an arrangement ( _e.g.,_ two mirrors inclined at 90 degrees) which allows him visually to observe both places _A_ and _B_ at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous. I am very pleased with this suggestion, but for all that I cannot regard the matter as quite settled, because I feel constrained to raise the following objection: 'Your definition would certainly be right, if only I knew that the light by means of which the observer at _M_ perceives the lightning flashes travels along the length _A_ => _M_ with the same velocity as along the length _B_ => _M._ But an examination of this supposition would only be possible if we already had at our disposal the means of measuring time. It would thus appear as though we were moving here in a logical circle.'"7

A minor error in terminology occurs when Einstein expresses his desire to be certain that the velocity of light along the length _A_ => _M_ is the same as the velocity of light along the length _B_ => _M._ The velocity of light is a vector. A vector has both magnitude and direction. The light beams are traveling in opposite directions so they cannot have the same velocity. However, the light beams can have the same magnitude. If the light-beam vectors have the same magnitude, it would mean their speeds are equivalent, which is the point Einstein is trying to make.

There is another error that is more serious. The definition does not take into consideration that the earth is in motion and further that the motions of the light beams are independent of the earth's motion. In other words, the light beams are not carried along by the earth's motion. All the earthbound objects we commonly observe in motion such cars, planes and trains are carried along by the earth's motion. Also, all the objects we commonly observe as being at rest such as buildings, bridges and telephone poles are carried along by the earth's motion. Thus, the observer standing still at the midpoint _M_ of the length _AB_ is being carried along by the earth's motion. Since the endpoints of the length _AB_ are also being carried along by the earth's motion, the observer at the midpoint _M_ is at rest relative to the endpoints of the length _AB._ Thus, the observer at the midpoint _M_ maintains a constant distance from the endpoints. This is not the case with the light beams that originate from either endpoint. The observer at the midpoint _M_ is rushing toward one light beam and away from the other light beam, although he seems to be standing still. This is because the earth is in motion and the motions of the light beams are independent of the earth's motion. Of course, both light beams are traveling toward the observer at the midpoint _M_ at the speed of light.

Einstein's definition of a test to determine the simultaneity of distant events provides an example of the naturalness and the logical simplicity that he believed should characterize the premises of a theory. The naturalness of his definition resides in the fact that it agrees with our observations of the everyday world. For example, let an observer stand at the midpoint of a smooth and level stretch of two-lane highway 60 miles in length. Also, station an automobile at each end of this 60-mile length of highway, and let the automobiles start traveling at a given time with a constant speed of 30 mph. If the two automobiles pass by the observer standing at the midpoint at the same instant, the two automobiles began their journey at the same instant. This is precisely the same claim that Einstein makes for the flashes of lightning occurring at points _A_ and _B_. If the observer standing at the midpoint between points _A_ and _B_ sees the flashes of lightning at the same time, they were both produced at exactly the same instant.

The logical simplicity of his definition resides in minimal number of measurements required to assess the simultaneity of distant events and the straightforwardness of the observations required.

We can now return to Einstein's dialogue with the reader—picking up where we left off.

"After further consideration you cast a somewhat disdainful glance at me—and rightly so—and you declare: 'I maintain my previous definition nevertheless, because in reality it assumes nothing about light. There is only _one_ demand to be made of the definition of simultaneity, namely, that in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand is indisputable. That light requires the same time to traverse the path _A_ => _M_ as for the path _B_ => _M_ is in reality neither a _supposition nor a hypothesis_ about the physical nature of light, but a _stipulation_ which I can make of my own free will in order to arrive at a definition of simultaneity.'"8

Einstein overreaches with his statement, "There is only _one_ demand to be made of the definition of simultaneity, namely, that in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled." The following example shows that the _one_ demand Einstein makes on the definition of a test to determine simultaneity is not enough to obtain an accurate definition of a test to determine simultaneity. A definition of a test to determine simultaneity that everyone will agree is incorrect can still yield in every real case an empirical decision as to whether or not two distant events occur at the same instant. For example, if instead of stationing the observer at the midpoint _M_ between points _A_ and _B,_ let's say we station the observer at a point that is closer to point _A_ than it is to point _B_. Let's say the observer in our experiment is stationed one-third of the way from _A_ to _B_. This observer can provide us with an empirical decision in every real case as to whether or not the two distant events occurred at the same instant. Yet, no one would conclude that _this_ definition of distant simultaneous events fulfills the demands required of an accurate definition. If the flashes of lightning at points _A_ and _B_ occurred at the same instant, our observer stationed one-third of the way from point _A_ to point _B_ would observe that the flash of lightning at point _A_ occurred before the flash of lightning at point _B_ because he is closer to point _A_ than he is to point _B_. To reiterate, the lightning flash at point _A_ would have to travel a shorter distance to reach our observer than the lightning flash at point _B_. Since the lightning flash at point _A_ travels a shorter distance to reach our observer, it will reach our observer before the lightning flash from point _B_ even though the lightning flashes occurred at the same instant. Supplying an empirical decision in every real case is not a sufficient distinction for formulating an accurate definition of a test to determine the simultaneity of distant events. In the example above, for the sake of clarity, the effects of the motion of the earth on the observation of the distant flashes of lightning were ignored.

We can safely ignore the effects of the motion of the earth in the above example because we are constructing a thought experiment to demonstrate that supplying an empirical decision in every real case does not validate Einstein's definition. If we attempt to construct a scenario where the motions of the earth have a negligible effect on the results of our observations, we will see that it is more difficult than it may seem. Let's imagine that points _A_ and _B_ are separated by ten or more miles of very flat terrain. The speed of light is approximately 186,000 miles per second while the orbital speed of the earth is approximately 18.5 miles per second. Since the orbital speed of the earth is only approximately 1/10,000 the speed of light, its effects on the example above would be negligible because our observer would be both several miles closer to point _A_ and several miles farther from point _B_ than an observer stationed at the midpoint between points _A_ and _B_. For example, let our observer was stationed three miles from point _A_ and seven miles from point _B,_ if we ignore the orbital speed of the earth and its other motions, as well, we can calculate that it would take approximately 16,000 nanoseconds for the lightning flash from point _A_ to reach our observer, and it would take approximately 37,000 nanoseconds for the lightning flash from point _B_ to reach our observer. It would take more than twice as long for the lightning flash from point _B_ to reach our observer as it would for the lightning flash from point _A._ So without taking into consideration the motion of the earth, the observed difference between the arrivals of the lightning flashes would be approximately 19,000 nanoseconds. If we take the orbital speed of the earth into account in our calculations, it would change this value by less than 6 nanoseconds. A nanosecond is one billionth of a second.

To be complete, we should note that the motion of the earth includes motions other than its orbital velocity. We have not taken into account the velocity of our solar system as it rotates around the center of the Milky Way Galaxy along with the other arms of our spiral galaxy. And, we have not taken into account the velocity of the Milky Way Galaxy as it races outward along with our neighboring galaxies in the expanding universe. It is possible that these velocities could affect the outcome of our thought experiment.

Einstein again overreaches with his statement, "That light requires the same time to traverse the path _A_ => _M_ as for the path _B_ => _M_ is in reality neither a _supposition nor a hypothesis_ about the physical nature of light, but a _stipulation_ which I can make of my own free will in order to arrive at a definition of simultaneity." The dilemma is that Einstein's definition of simultaneity requires that distant events that do not occur at the same moment must be regarded as simultaneous. Einstein's definition of distant simultaneous events is an example of "concepts and theories as free inventions of the human spirit (not logically derivable from what is empirically given)."9

In his paper _On the Electrodynamics of Moving Bodies,_ Einstein employs the notion that on a moving body the time it takes a light beam to travel from _A_ => _M_ is different from the time it takes a light beam to travel from _B_ => _M._ The reason Einstein gives for this phenomenon is the earth's motion and the independence of light beams from the earth's motion. He uses this phenomenon to demonstrate the relative simultaneity of distant events. For observers in a system at rest (the earth is not a system at rest), the time it takes a light beam to travel from _A_ => _M_ is equal to the time it takes a light beam to travel from _B_ => _M._ For observers in a moving system, the time it takes a light beam to travel from _A_ => _M_ is not equal to the time it takes a light beam to travel from _B_ => _M_.

In his paper, Einstein deals with the question of whether a light beam travels with the same speed regardless of the direction in which it is traveling by assuming, as a postulate of his theory, that the speed of light is constant.

Once Einstein has used the fact that on a moving body the time it takes a light beam to travel from _A_ => _M_ is not equal to the time it takes a light beam to travel from _B_ => _M,_ in order to demonstrate the relativity of the simultaneity of distant events, he alters or altogether rejects this notion. He uses a complex mathematical equation in an attempt to demonstrate that on a moving object (or in a moving system) the time it takes a light beam to travel from _A_ => _M_ is equal to the time it takes a light beam to travel from _B_ => _M._

If the first part of his special relativity paper is an epistemological examination of the nature of space and time, it is surely a glaring omission that Einstein never addresses why distant clocks must by synchronized solely by beams of light. Light beams have many attributes that are useful in the synchronization distant clocks, but they also have attributes that would not be useful. Perhaps, a discussion of the various other ways of synchronizing distant clocks such as mechanical devices, devices using sound waves, or electronic devices would have clarified Einstein's position. If each method of synchronizing distant clocks has its advantages and disadvantages, perhaps, a method of synchronization that employed both sound waves and light beams, for instance, could overcome—by combination—any disadvantage possessed by a system dependent on a single method of synchronization.

Christopher Jon Bjerknes begins his book _Anticipations of Einstein: In the General Theory of Relativity_ with a citation from Anthony Berkeley Cox that first appeared in _Scribner's Magazine,_ Volume 88, (July—December 1930). The citation appeared in Charles L. Poor's article for _Scribner's Magazine_ entitled "What Einstein Really Did," he gives us this statement by A. B. Cox, "Artistic proof is, like artistic anything else, simply a matter of selection. If you know what to put in and what to leave out you can prove anything you like, quite conclusively."10 With that said, there is only one remaining point to cover in this introduction.

Einstein's theory of special relativity is presented in the first part of his paper _On the Electrodynamics of Moving Bodies._ The notion that the first part of Einstein's paper—the "Kinematic Part"— is incorrect is opposed to the established views on the subject. Therefore, any counter claim must provide a detailed and concise examination of every equation and every significant statement that appears in the first part of Einstein's paper. Also, any counter claim must include an examination of all the intermediate steps between the various equations. Although there are many mathematical equations in each of the following six chapters of this book, they are not difficult to comprehend because they are explained in detail.

A Note on Quotations

A small number of the quotations in this book are not followed by an endnote number. For some, this may cause confusion. Such quotations are partial reiterations of previous quotations, and these quotations have their appropriate endnotes.

Chapter One: The Definition of Simultaneity

Chapter Summary: An Analysis of "Section 1. Definition of Simultaneity" and "Section 2. On the Relativity of Lengths and Times" from _On the Electrodynamics of Moving Bodies_

Step1: An Error Found or an Error Manufactured?

If we examine a number of passages from Albert Einstein's seminal paper, _On the Electrodynamics of Moving Bodies,_ the paper's inconsistencies and ambiguities will become apparent. These ambiguities and inconsistencies amount to errors that invalidate his claims regarding the concept referred to as the simultaneity of distant events. Einstein claims the simultaneity of distant events cannot have an absolute meaning and that instead its meaning must be relative to a particular coordinate system.

_On the Electrodynamics of Moving Bodies_ is a remarkable accomplishment that has had an enormous impact on the development of modern physics. John Stachel, the editor of _Einstein'_ s _Miraculous Year: Five Papers That Changed the Face of Physics,_ provides a concise description of Einstein's paper and its significance.

"Einstein was the first physicist to formulate clearly the new kinematical foundation for all of physics inherent in Lorentz's electron theory. This kinematics emerged in 1905 from his critical examination of the physical significance of the concepts of spatial and temporal intervals. The examination, based on a careful definition of the simultaneity of distant events, showed that the concept of a universal or absolute time, on which Newtonian kinematics is based, has to be abandoned; and that the Galilean transformations between the coordinates of two inertial frames of reference has to be replaced by a set of spatial and temporal transformations that agree formally with a set that Lorentz had introduced earlier with a quite different interpretation. Through the interpretation of these transformations as elements of a space-time symmetry group corresponding to the new kinematics, the special theory of relativity (as it later came to be called) provided physicists with a powerful guide in the search for new dynamical theories of fields and particles and gradually led to a deeper appreciation of the role of symmetry criteria in physics."11

The amount of care (to borrow John Stachel's terminology) Einstein employed in his formulation of the definition of distant simultaneous events is unclear. Certainly, in his thought experiment(s) Einstein did not describe the method by which the clocks in a moving system are synchronized with the clocks in a system at rest. This oversight is significant because testing the synchronization of distant clocks, which are located in a moving system where they are at rest relative to each other, is central to Einstein's demonstration of the relativity of simultaneity. In Einstein's crucial thought experiment two different sets of observers must test distant clocks in a moving system for synchronization. One set of observers is co-moving with the moving system, and the other set of observers is at rest in the rest system. In Einstein's thought experiments, a moving rod represents the moving system. He merely informs us that the clocks at either end of the rod are synchronized with the clocks in the rest system.

"Further, we imagine the two ends (A and B) of the rod equipped with clocks that are synchronous with the clocks of the rest system, i.e., whose readings always correspond to the 'time of the system at rest' at the locations the clocks happen to occupy; hence, these clocks are 'synchronous in the rest system.'"12

Einstein does not tell us the method by which the clocks in the moving system are synchronized with the clocks in the rest system. Furthermore, he never tells us if the rest system is in a state of absolute rest or if it is merely labeled the rest system for convenience or for some deeper reason. If Einstein were to tell us the method by which the clocks in the moving system were synchronized with the clocks in the rest system, this would spell out whether the rest system was in motion or in a state of absolute rest.

It is interesting that John Stachel refers to Einstein's definition of the simultaneity of distant events as "careful." In Einstein's book _Relativity: The Special and the General Theory,_ the definition of simultaneity is referred to as "most natural."13 The definition of the simultaneity of distant events is not referred to as scientifically rigorous in either instance, which is unexpected for the central tenet of a theory that formed "the new kinematical foundation for all of physics."14

Step 2: The Michelson-Morley Experiment Is Flawed

In order to clarify Einstein's ideas, in the remaining "Steps" of Chapter One, we will analyze many passages from his paper. The following passages are taken either from the introductory paragraphs of Einstein's paper or from the first part of his paper, which is entitled "A. Kinematic Part." In the first part of his paper, he formulates the thought experiments that overturn the concept of absolute time. The second part of his paper is entitled "B. Electrodynamic Part." In this part, John Stachel writes, "he applied his kinematical results to the solution of a number of problems in the optics and electrodynamics of moving bodies."15

The passage below is taken from Einstein's introductory paragraphs. There are two points that should be noted. First, near the beginning of the passage, there is an oblique reference to the Michelson-Morley experiment in the phrase, "unsuccessful attempts to detect a motion of the earth relative to the 'light medium.'"16 Secondly, the last sentence of the passage is consequential, "The introduction of a'light ether' will prove superfluous, inasmuch as the view developed here will not require a 'space at absolute rest' endowed with special properties where electro-magnetic processes are taking place."17 Does Einstein mean that his analysis does not require a 'space at absolute rest' or that his analysis does not require a 'space at absolute rest' endowed with special properties? Only a few paragraphs later Einstein introduces the "rest system."18 The "rest system" would seem to be a system at absolute rest except for the fact that Einstein is compelled to establish the definition of a common time for points _A_ and _B_ of the rest system _"by definition,"_ 19 as opposed to by utilizing the physical properties of the system. These two points will be expanded upon in the analysis following the quotation below.

"It is well known that Maxwell's electrodynamics—as usually understood at present—when applied to moving bodies, leads to asymmetries that do not seem to be inherent in the phenomena. . . . Examples of this sort, together with the unsuccessful attempts to detect a motion of the earth relative to the 'light medium,' lead to the conjecture that not only the phenomena of mechanics but also those of electrodynamics have no properties that correspond to the concept of absolute rest. Rather, the same laws of electrodynamics and optics will be valid for all coordinate systems in which the equations of mechanics hold, as has already been shown for quantities of the first order. We shall raise this conjecture (whose content will hereafter be called "the principle of relativity") to the status of a postulate and shall also introduce another postulate, which is only seemingly incompatible with it, namely that light always propagates in empty space with a definite velocity _V_ that is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent electrodynamics of moving bodies based on Maxwell's theory for bodies at rest. The introduction of a 'light ether' will prove to be superfluous, inasmuch as the view developed here will not require a 'space at absolute rest' endowed with special properties where electromagnetic processes are taking place."20

It should be noted that Einstein employs the term _V_ for the velocity of light instead of the more familiar term _c._ In Einstein's book _Relativity: The Special and the General Theory_ the term _c_ is employed to represent the velocity of light in empty space, and also there are translations of his 1905 paper that employ the more familiar term _c_ instead of _V_ , such as Arthur I. Miller's translation, which appears in the Appendix of his book, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905—1911)._

Einstein's oblique reference to the Michelson-Morley experiment has caught the attention of many commentators. Though most of them have failed to indicate how ill-suited the Michelson-Morley experiment is to make a determination of the absolute motion of the earth with regard to the light ether. The light ether can be conceived of in two ways. First, the light ether can be conceived of as a medium for the transmission of light. In this instance, the light ether is in a state of absolute rest. When the earth moves through this medium, an apparent ether wind is produced that can affect a beam of light. The apparent ether wind is similar to the wind you feel when you ride a bicycle on a windless day. You feel and see the effects of this localized wind, but it is only an apparent wind because when the bicycle stops the wind ceases. Secondly, the light ether can be conceived of as empty space, which can include an observer in a state of absolute rest. In this conception of the light ether, the ether has no influence on a beam of light. Whether the first or second conception of the light ether is employed, the Michelson-Morley experiment is ill-equipped to detect either its influence on a beam of light in the first instance or the influence of the earth's motion on a beam of light in the second instance.

If we view the light ether as a medium for light transmission, the motion of the earth through the ether will influence the path of a light beam in the same way the current of a river will affect a swimmer. The mathematical underpinnings of the Michelson-Morley experiment require that one beam of light that has been split in half by the experimental apparatus must form two rays of light that are perpendicular to each other. The experimental apparatus does produce two rays of light, but the flaw is that the two rays of light are never perpendicular to one another, and no amount of alteration of the experimental apparatus can change this.

If we view the light ether as empty space in which an observer in a state of absolute rest can be located, the empty space will have no influence on a beam of light, but the motion of the earth will have an influence. Under these conditions, the experimental apparatus can produce two light rays that are perpendicular to each other. The flaw in this instance is that the light ray returning along the arm of the experimental apparatus, which is perpendicular to the motion of the earth, will fail to coincide with the ray of light returning along the arm of the apparatus parallel to the motion of the earth. The experimental apparatus will move forward with velocity _v_ , which is the velocity of the earth. The light ray perpendicular to the motion of the earth will not move forward with velocity _v_ so it will lag behind the experimental apparatus. The light ray will, of course, travel with the velocity _c_ in the direction perpendicular to the motion of the earth.

Step 3: Inspired Insight Or Inspired Ambiguity?

Now, we should return our attention to Einstein's thought experiments in which he will analyze distant simultaneous events. The following five passages are taken from the first section of "A. Kinematic Part," which is entitled, "1. Definition of Simultaneity." The five passages discussed below appear in the same order of occurrence as they appear in the original text.

"Consider a coordinate system in which Newton's mechanical equations are valid. To distinguish this system verbally from those to be introduced later, and to make our presentation more precise, we will call it the 'rest system.'"21

There are several reasons why Einstein could have enclosed the phrase "rest system" in quotation marks. The quotation marks may have been used to emphasize the term, to denote the term as a label, or to suggest doubt or skepticism. Einstein could have used the quotation marks to accomplish all three of the tasks. Einstein must be using the term "rest system" as a label since he states, "To distinguish this system verbally from those to be introduced later and to make our presentation more precise, we will call it the "'rest system.'"22 However, as the following analysis will show, he is also using the quotation marks to suggest doubt. He wants to cast doubt on the notion that the concept of a rest system and the concept of a moving system are distinct from each another.

Is Einstein's "rest system" at rest relative to some other system that is in motion? For example, trees, buildings and telephone poles are all systems that are at rest relative to the motion of the earth. Is the "rest system" in a state of absolute rest? Two statements from the second introductory paragraph suggest otherwise, since Einstein tells us, "the phenomena . . . of electrodynamics have no properties that correspond to the concept of absolute rest . . . . And "the view developed here will not require a 'space at absolute rest' endowed with special properties where electromagnetic processes are taking place."23

However, Einstein is inconsistent. As he continues his description of the "rest system," the system he describes must be a system in a state of absolute rest. This is so because his mathematical description of a light beam's round-trip journey on a moving body is different from his mathematical description of a light beam's round-trip journey on a body at rest in the "rest system." The mathematical descriptions of a light beam's round-trip journey from point _A_ to point _B_ and back to point _A_ on a moving body are _t_ B – _t_ A = _r_ AB _/(V –_ _v)_ and _t'_ A _– t_ B = _r_ AB _/(V_ \+ _v)_ , where _r_ AB denotes the distance from point _A_ to point _B._ The duration of the outbound leg of the light beam's round-trip journey is denoted by _t_ B – _t_ A. The term _t_ B denotes the time at which the light beam arrives at point _B._ The term _t_ A denotes the time at which the light beam began its journey by leaving point _A._ The duration of the return leg of the light beam's round-trip journey is denoted by _t'_ A _– t_ B. The term _t'_ A denotes the time at which the light beam returns to point _A_. The term _t_ B denotes the time at which the light beam leaves point _B._ Since the light beam is reflected from point _B_ at the very instant it arrives at point _B,_ the time _t_ B denotes both the time the light beam arrives at point _B_ and the time it leaves point _B. V_ represents the speed of light, and _v_ represents the speed of the moving body. Also, the moving body is moving in the direction that makes point _B_ the lead point of the system. If these two formulas represent a light beam's round-trip journey on a moving body, it is reasonable that the formula that represents a light beam's round-trip journey in the "rest system," _t_ B – _t_ A = _t'_ A _– t_ B, should represent a system in a state of absolute rest. Since points _A_ and _B_ are not in motion in the "rest system" the distance a light beam must travel to go from point _A_ to point _B_ is the same distance the light beam must travel to go from point _B_ to point _A,_ and, hence, the duration of each leg of the round-trip journey is the same. Thus, the "rest system" is a system at absolute rest. The duration of each leg of a round-trip journey of a light beam is not the same on a moving body.

We should recall that a light beam is not carried along by the motion of the moving body on which it travels. We are familiar with the motions of many objects such as cars, trains, ships and planes. When each of these objects is in motion, its motion is carried along by the motion of the earth. A light beam in motion on the earth will not be carried along by the motion of the earth. In our example, the earth is moving such that point _B_ is in the lead position and point _A_ is in the rear position. The result is the following. When the light beam emanates from point _A_ , point _B_ is rushing away from the oncoming light beam. When the reflected light beam—emanating from point _B—_ is returning to point _A_ , point _A_ is rushing toward the oncoming light beam. Thus, the journey from point _A_ to point _B_ is of greater duration than the return journey from point _B_ to point _A._ During the first leg of the journey—from point _A_ to point _B—_ point _B_ is rushing away from the oncoming light beam so the light beam has to travel an extra distance to reach point _B_. This extra distance equals the velocity of the earth multiplied by the duration of the first leg of the journey. During the return leg of the journey—from point _B_ to point _A_ —point _A_ is rushing toward the returning light beam so the returning light beam has to travel a shortened distance. This shortened distance equals the velocity of the earth multiplied by the duration of the return leg of the journey subtracted from _r_ AB, which denotes the distance from point _B_ to point _A_. The distance from point _A_ to point _B_ is equal to the distance from point _B_ to point _A._

Before we delve any deeper into the nature of the "rest system," we should explore Einstein's development of the relationship between "time" and _simultaneous events._

"If we want to describe the _motion_ of a particle, we give the values of its coordinates as functions of time. However, we must keep in mind that a mathematical description of this kind only has physical meaning if we are already clear as to what we understand here by 'time.' We have to bear in mind that all our judgments involving time are always judgments about _simultaneous events_. If, for example, I say that 'the train arrives here at 7 o'clock,' that means, more or less, 'the pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.'"24

Einstein's use of quotation marks in this excerpt is more straightforward. The word "time" is enclosed in quotation marks to call special attention to it. The phrase "the train arrives here at 7 o'clock," is a half-direct and half-indirect quotation and the quotation marks are used to emphasize the phrase. (Notice there is no comma, and the word, _the,_ is not capitalized.) In this passage, the final use of quotation marks is for the definition "the pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events," which defines the passage's previous quotation, "the train arrives here at 7 o'clock."

Einstein is linking our understanding of time to our understanding of simultaneous events. In this passage the simultaneous events occur at the same location. In the next passage Einstein explains how to evaluate events that do not occur at the same location. This explanation will have a very significant bearing on the meaning of the "rest system."

"If there is a clock at point _A_ in space, then an observer located at _A_ can evaluate the time of events in the immediate vicinity of _A_ by finding the positions of the hands of the clock that are simultaneous with these events. If there is another clock at point _B_ that in all respects resembles the one at _A,_ then the time of events in the immediate vicinity of _B_ can be evaluated by an observer at _B_. But it is not possible to compare the time of an event at _A_ with one at _B_ without further stipulation. So far we have defined only an ' _A_ -time' and a ' _B_ -time,' but not a common 'time' for _A_ and _B_. The latter can now be determined by establishing _by definition_ that the 'time' required for light to travel from _A_ to _B_ is equal to the 'time' it requires to travel from _B_ to _A_. For, suppose a ray of light leaves from _A_ for _B_ at ' _A_ -time' _t_ A, is reflected from _B_ toward _A_ at ' _B_ -time' _t_ B, and arrives back at _A_ at ' _A_ -time' _t'_ A _._ The two clocks are synchronous by definition if _t_ B – _t_ A = _t'_ A _– t_ B _._ We assume that it is possible for this definition of synchronism to be free of contradictions and to be so for arbitrarily many points. . . ."25

When Einstein describes the "rest system" as a system where, "the 'time' required for light to travel from _A_ to _B_ is equal to the 'time' it requires to travel from _B_ to _A_ ," he must be describing a system at absolute rest. Yet, there is a caveat that must be noted. Einstein prefaces the previous quotation with the italicized phrase, _by definition._ The quotation can be reformulated as, "[The common 'time' for _A_ and _B_ is] determined by establishing _by definition_ that the 'time' required for light to travel from _A_ to _B_ is equal to the 'time' it requires to travel from _B_ to _A_."

If the "rest system" was at absolute rest, the common time for points _A_ and _B_ would not need to be established _by definition_. Einstein writes, "We assume that it is possible for this definition of synchronism to be free of contradictions. . . ." Does he mean free from contradictions in the "rest system" only, or does he mean free from contradictions for both a system at rest and a system in motion? Twelve paragraphs later, Einstein tells us that, for a system in motion, a light beam making a round-trip journey from point _A_ to point _B_ and back to point _A_ does not adhere to his requirements for the definition of a common time for points _A_ and _B_ in the "rest system." The ambiguity continues in the next passage.

"By means of certain (imagined) physical experiments, we have established what is to be understood by synchronous clocks at rest relative to each other and located at different places, and thereby obviously arrived at definitions of 'synchronous' and 'time.'"26

"Clocks at rest relative to each other," may either refer to clocks in motion or clocks at absolute rest. Passengers on a plane who are sitting in their seats are at relative rest to each other, but of course they are in motion. Clocks at absolute rest would be at rest relative to each other, as well. Einstein states, "By means of certain (imagined) physical experiments, we . . . arrived at definitions of "synchronous" and "time." He does not restrict these definitions to the "rest system" in this passage, which is in marked contrast to the following passage.

"It is essential that we have defined time by means of clocks at rest in the rest system; because the time just defined is related to the system at rest, we call it 'the time of the rest system.'"27

In this passage, Einstein concludes that his definition of time is more appropriately understood as "the time of the rest system." Einstein suggests in this passage that his definition of time is limited to the rest system. When Einstein writes, "we call it 'the time of the rest system,'" is he implying that a system in motion will have another definition of time? If so, that notion would contradict a prior statement he has made about the definition of the time of the rest system, "We assume that it is possible for this definition of synchronism to be free of contradictions. . . ."28

Step 4: The Method of Synchronizing the Clocks Is Missing

We will now turn our attention to the second section of "A. Kinematic Part," which is entitled, "2. On the Relativity of Lengths and Times." The following three passages appear in the same order of occurrence as they do in the text of Einstein's paper. We begin with an excerpt from the opening paragraph of the second section.

"The following considerations are based on the principle of relativity and the principle of the constancy of the velocity of light."29

Were the notions examined in the first section also based on the principle of relativity and the principle of the constancy of the velocity of light? Why did Einstein wait until the opening paragraph of the second section to explicitly state that his two postulates formed the foundation of his argument? The constancy of the velocity of light is a necessary postulate for the formation of his definition of a common time for point _A_ and point _B_ , which he puts forth in the first section. Perhaps, Einstein's two postulates were not in play in the first section. This would give us another explanation for the fact that his definition of time could only be established _by definition_. Einstein's definition of a common time should be based on the principle of the constancy of the velocity of light. If the "rest system" was not at absolute rest and in fact was in motion, the principle of the constancy of the velocity of light would pose an obstacle for Einstein's definition of a common time. Without the introduction of the principle of the constancy of the velocity of light, the "rest system" can be in motion, and this motion will be less of an obstacle to Einstein's definition of a common time, as given in the first section, because a light beam with a non constant velocity is a loophole of sorts.

The next passage lays the groundwork for the thought experiment in which Einstein claims to demonstrate the relativity of simultaneity.

"Take a rigid rod at rest; let its length, measured by a measuring rod that is also at rest, be _l_. Now imagine the axis of the rod placed along the _X_ -axis of the rest coordinate system, and the rod then set into uniform parallel translational motion (with velocity _v_ ) along the _X_ -axis in the direction of increasing _x_."30

The purpose of the phrase, "along the _X_ -axis in the direction of increasing _x,_ " is to give a direction to the rod's velocity. It is similar to stating the rod is traveling due east. The rigid rod's motion is similar to a train traveling on an endless length of straight track at a constant velocity. The next passage elaborates on this rigid rod "set into uniform parallel translational motion."

"Further, we imagine the two ends ( _A_ and _B_ ) of the rod equipped with clocks that are synchronous with the clocks of the rest system, i.e., whose readings always correspond to the 'time of the system at rest' at the locations the clocks happen to occupy; hence, these clocks are 'synchronous in the rest system.'"31

Notice that Einstein does not describe the method by which the clocks in the moving system of the rigid rod are to be synchronized with the clocks in the "rest system." He states only that the clocks in the moving system are synchronized with the clocks in the "rest system." We know that according to Einstein's definition of common time, clocks are synchronized when the time it takes a light beam to travel from point _A_ to point B is equal to the time it takes a light beam to travel from point _B_ to point _A._

Let us suppose point _A_ is located in the "rest system" and point _B_ is located at one end of the rigid rod. The rigid rod is designated as the moving system by Einstein. Along with the clock that Einstein indicates is at point _B_ , let us also suppose there is a mirror. The mirror is connected to the clock so that when a powerful light beam strikes the mirror the clock will instantly note the time of this event. At point _A_ there is a clock and a device that can project a powerful light beam to the mirror on the moving rigid rod. The clock at point _A_ is connected to the light-beam projecting device so that it will instantly note the time when the light beam begins its journey to the mirror at point _B._ The clock at point _A_ is also connected to another device that can detect the returning light beam from the mirror at point _B._ The instant this device detects the light beam returning from the mirror, the clock at point _A_ notes the time of this event, as well. According to Einstein the clocks at points _A_ and _B_ are synchronized if the following equation is valid: _t_ B – _t_ A = _t'_ A _– t_ B _._

This equation is only valid if the "rest system" is at absolute rest and the light beam begins its journey from the "rest system." It is only under these conditions that the distances the light beam travels in each leg of its journey are equal. The moving system of the rigid rod is traveling away from point _A_ with a constant velocity. When the light beam from point _A_ strikes the mirror at point _B_ , it has traveled a distance _d._ Since the light beam is reflected instantly from the mirror, when it returns to point _A_ , the return distance it has traveled is the same as its outbound distance, i.e., distance _d._

Next, let us change the experimental set up. Now, the light beam is projected from point _B_ of the moving system of the rigid rod and the mirror has been relocated to point _A_ of the "rest system." When the light beam from point _B_ strikes the mirror relocated to point _A_ , it has traveled a distance _d_. When the light beam returning from the mirror strikes the detection device relocated to point _B_ , it has traveled the distance _d_ \+ _v (t'_ B _)._ The distance _d_ is the distance that separated point _B_ from point _A_ at the instant the light beam began its journey to point _A._ Since that instant, the moving system of the rigid rod has been moving further away from point _A_ with a constant velocity _v._ While the light beam was traveling from point _B_ to point _A_ , the rigid rod was traveling further from point _A._ Also, while the light beam was returning from point _A_ to point _B_ , the rigid rod was traveling further from point _A._ The total increase in the distance of point _B_ from point _A_ is the velocity of the rigid rod times the duration of the light beam's round-trip journey or _v( t'_ B _)._ We are assuming the clock at point _B_ read 00:00:00 at the beginning of the light beam's journey.

The time it takes the light beam to travel from point _B_ to point _A_ is the following: _t_ A – _t_ B = _d / V._ The time it takes the light beam to return from point _A_ to point _B_ is the following: _t'_ B _– t_ A = _(d_ \+ _v t'_ B _)/ V_. When the light beam travels from point _B_ on the moving system to point _A_ on the rest system and back again to point _B_ , Einstein's definition for synchronized clocks, _t_ A – _t_ B = _t'_ B _– t_ A, is not met. Instead, an inequality is produced that is represented by the following: _t_ A – _t_ B < _t'_ B _– t_ A _._ This inequality is produced because _d /V < (d_ \+ _vt'_ B _) / V._

The conclusion of this analysis is that the clocks in the moving system of the rigid rod can be synchronized with the clocks in the "rest system." However, there are two important stipulations: first, the "rest system" must be at absolute rest, and, secondly the light ray must always originate from the rest system. The fact that the clocks in the moving system of the rigid rod can only be synchronized with the clocks in the "rest system" by modifying Einstein's definition of time is enough to undermine his claim of the relativity of simultaneity. As Einstein stated, "The two clocks are synchronous by definition if _t_ B – _t_ A = _t'_ A _– t_ B _._ We assume that it is possible for this definition of synchronism to be free of contradictions, and to be so for arbitrarily many points. . . . "32 Since the clocks in the moving system of the rigid rod can only be synchronized with the clocks in the "rest system" by modifying Einstein's definition of time, therefore, if we strictly adhere to his definition, the clocks in the two systems cannot be synchronized. But, according to Einstein the clocks in the "rest system" are synchronized with the clocks in the moving system.

Step 5: The Relativity of Simultaneity

Although Einstein never describes the method by which the clocks in the "rest system" are synchronized with the clocks in the moving system, this synchronization sets the stage for his demonstration of the relativity of simultaneity. The essence of Einstein's argument is that although the clocks in the moving system are synchronized with the clocks in the "rest system" the observers co-moving with the moving system will determine that their two clocks at either end of their rod are not synchronized.

"We further imagine that each clock has an observer co-moving with it, and that these observers apply to the two clocks the criterion for the synchronous rate of two clocks formulated in section 1. Let a ray of light start out from _A_ at time _t_ A; it is reflected from _B_ at time _t_ B, and arrives back at _A_ at time _t'_ A. Taking into account the principle of the constancy of the velocity of light, we find that _t_ B – _t_ A = _r_ AB _/(V –_ _v)_ and _t'_ A _– t_ B = _r_ AB _/(V_ \+ _v)_ , where _r_ AB denotes the length of the moving rod, measured in the rest system. Observers co-moving with the rod would thus find that the two clocks do not run synchronously, while observers in the system at rest would declare them to be running synchronously. Thus we see that we cannot ascribe _absolute_ meaning to the concept of simultaneity; instead, two events that are simultaneous when observed from some particular coordinate system can no longer be considered simultaneous when observed from a system that is moving relative to that system."33

What compels the observers co-moving with the rod to use the following definition of two synchronous clocks: _t_ A – _t_ B = _t'_ B _– t_ A, to determine if the clock at one end of their rod is synchronized with the clock at the other end of their rod? Einstein's closing comment in the first section on his definition of time is, "It is essential that we have defined time by means of clocks at rest in the rest system; because the time just defined is related to the system at rest, we call it 'the time of the rest system.'"34 Observers co-moving with the rod would be justified in questioning whether Einstein's definition of synchronous clocks was appropriate for their situation. Einstein writes, "We have defined time by means of clocks at rest in the rest system. . . ."35 Clocks at rest in the rest system are essential to Einstein's definition. The clock at each end of the rod is not at rest in the rest system. Instead, they are at rest in the moving system.

Einstein's penultimate comment in the first section on his definition of time is, "By means of certain (imagined) physical experiments, we have established what is to be understood by synchronous clocks at rest relative to each other and located at different places, and thereby obviously arrived at definitions of 'synchronous' and 'time.'"36 What should the observers co-moving with the rod make of the above statement? Their rod has a clock at each end. Their clocks are at rest relative to each other and they are located in different places. The observers co-moving with the rod might conclude that Einstein's definition of time does apply to their situation. However, after reflection, the observers co-moving with the rod might conclude that it was ambiguous whether or not Einstein meant his definition of time to apply to their circumstance.

What conclusion would the observers co-moving with the rod arrive at if they applied Einstein's definition of time in the following thought experiment? The observers assume that their two clocks are synchronized. Next, they follow Einstein's criterion for determining if two clocks are synchronized. Einstein gives an encapsulated version of their procedure and results when he writes, "Let a ray of light start out from _A_ at time _t_ A; it is reflected from _B_ at time _t_ B, and arrives back at _A_ at time _t'_ A. Taking into account, the principle of the constancy of the velocity of light, we find that _t_ B – _t_ A = _r_ AB _/(V_ – _v)_ and _t'_ A _– t_ B = _r_ AB _/(V_ \+ _v)_ , where _r_ AB denotes the length of the moving rod, measured in the rest system."37 The observers co-moving with the rod could conclude from their thought experiment that Einstein's definition of time does not apply to their situation, if the round-trip journey of their light beam matched the two equations just presented.

We should expand on Einstein's encapsulated version to explain how the observers would reach their conclusion. In Einstein's thought experiment, the observers are using his definition of synchronized clocks, _t_ A – _t_ B = _t'_ B _– t_ A. That is they are using his definition of time to determine if their clocks are synchronized. When the result of their test determines that their clocks are not synchronized, the observers unquestioningly accept the results. In our thought experiment the observers question whether Einstein's definition of time applies to their circumstances. Our thought experiment is the same as Einstein's except that in our thought experiment the observers assume their clocks are synchronized. Since the observers are doing a thought experiment, this assumption is legitimate.

First, the observers will analyze the journey of the light beam from point _A_ to point _B._ Their rod is moving forward, i.e., along the _X_ -axis in the direction of increasing _x_. Point _B_ is at the front of their rod, and point _A_ is at the rear of their rod. When the light beam emanating from point _A_ is projected toward the mirror at point _B_ , point _B_ is racing away from the oncoming light beam with the velocity _v._ The total distance the light beam must travel is the length of the rod, _r_ AB, plus the distance forward point _B_ has traveled in the time it takes the light beam to travel from point _A_ to the mirror at point _B_ , which is _v(t_ B – _t_ A _)_. The total distance the light beam travels is _[r_ AB \+ _v_ ( _t_ B – _t_ A _)]_ , the velocity of the rod is _v,_ and the time it takes the light beam to travel from point _A_ to point _B_ is _(t_ B – _t_ A _)_. The duration of the light beam's journey equals the distance traveled by the light beam divided by the speed of the light beam, _(t_ B – _t_ A _)_ = _[r_ AB + _v(t_ B – _t_ A _)]/ V._ If we multiply each side of the equation by _V_ we obtain the following: _V(t_ B _−t_ A _)_ = _r_ AB \+ _v(t_ B _−t_ A _)._ Next, we subtract _v(t_ B _−t_ A _)_ from each side of the equation, and we obtain _V(t_ B _−t_ A _)_ − _v(t_ B _−t_ A _)= r_ AB _._ On the left side of the equation, we observe that _(t_ B _− t_ A _)_ is a common factor. We can rewrite the equation as _(t_ B _−t_ A _)(V − v)_ = _r_ AB _._ If we divide each side of the equation by _(V_ – _v)_ , we have _(t_ B – _t_ A _)_ = _r_ AB _/(V –_ _v)_. The observer's thought experiment has produced the same equation for _(t_ B – _t_ A _)_ as Einstein's did.

In the second portion of our thought experiment the observers will analyze the return journey of the light beam from the mirror at point _B_ to point _A._ When the light beam reflected from the mirror is returning to point _A_ , point _A_ is racing toward the oncoming light beam with the velocity _v._ The total distance the light beam travels in its return journey is the length of the rod, _r_ AB, minus the distance that point _A_ has traveled toward the returning light beam in the time it takes the light beam to complete its return journey, _v(t'_ A – _t_ B _)_. The total distance the light beam travels is _[r_ AB – _v(t'_ A – _t_ B _)]_ , the velocity of the rod is _v_ , and the time it takes the light beam to return from point _B_ to point _A_ is _(t'_ A – _t_ B _)_. The observer's now can use a similar method to the one just described to determine _(t'_ A – _t_ B _) =r_ AB _/(V_ \+ _v)_. Their results will again be the same as Einstein's.

The observers co-moving with the rod abandon Einstein's definition and determine that their clocks are synchronized if the following conditions are met: when the light beam is traveling in the same direction as the rigid rod, the time the light beam takes to travel from point _A_ to point _B_ is _r_ AB _/(V –_ _v),_ and when the light beam is traveling in the direction opposite to that of the rigid rod the time the light beam takes to travel from point _B_ to point _A_ is _r_ AB _/_ _(V_ \+ _v)_.

Therefore, the observers co-moving with the rigid rod could determine that their clocks are synchronized. Since they are not at rest in the rest system, but rather a system in motion, they conclude they are not bound by Einstein's definition for "the time of the rest system," which is defined "by means of clocks at rest in the rest system." Both of the sets of observers could determine that the clocks at either end of the moving rigid rod are synchronized. The set of observers co-moving with the rigid rod could use a test for clock synchronization appropriate for their circumstances, and the set of observers in the rest system could use a test for clock synchronization appropriate for their circumstances.

Step 6: Another Interpretation

Another interpretation is possible for Einstein's definition of a common time for distant events. It dispenses with the notion that synchronized clocks agree in the time they keep. Einstein writes, "The two clocks are synchronous by definition if _t_ B – _t_ A = _t'_ A _– t_ B _._ " This definition allows two clocks to be considered synchronized even if they do not keep the same time. This notion is counter to the accepted definition of synchronized clocks.

An example should make this clear. We will begin with a rigid rod with a clock at either end. The rod is three light seconds in length or 558,000 miles long. The rod's velocity is . _5V_ or 93,000 miles/second and it is moving along the _X_ -axis in the direction of increasing _x_. Point _B_ is in the forward position and point _A_ is in the rear position. A light beam emanating from point _A_ takes six seconds to reach point _B_ because the effective velocity of the light beam is _(V –_ . _5V)_ or 93,000 miles/second. A light beam returning from point _B_ will take two seconds to reach point _A_ because the effective velocity of the light beam is _(V_ \+ . _5V)_ or 279,000 miles/second. If a light beam emanates from point _A_ when the clock at point _A_ reads noon, 12:00:00 p.m., and the clock at point _B_ reads 11:59:58 a.m., the light beam will reach point _B_ when the clock at point _B_ reads 12:00:04 p.m.—that is after six seconds have elapsed. The clock at point _A_ will read 12:00:06 p.m. because it is two seconds ahead of clock _B._ When the light beam returns to point _A_ , the clock at point _A_ reads 12:00:08 p.m. Thus, 12:00:04 p.m. – 12:00:00 p.m. = 12:00:08 p.m. – 12:00:04 p.m. or _t_ B – _t_ A = _t'_ A _– t_ B and the two distant clocks are synchronized according to Einstein's definition although they do not keep the same time.

In the above example, we are given the velocity of the rigid rod, which is 93,000 miles/second along the _X_ -axis in the direction of increasing _x._ If we did not know the absolute velocity of the rigid rod, we could calculate it once we determined that clock _A_ is two seconds ahead of clock _B._ This would tell us that the light beam requires six seconds to travel from point _A_ to point _B._ The velocity of light is constant, therefore 6 sec. × (186,000 miles/sec.) = 1,116,000 miles = 558,000 miles + _v_ × 6 sec. The total distance the light beam traveled to get from point _A_ to point _B_ is 1,116,000 miles. The length of the rigid rod, which is 558,000 miles, added to the distance the rigid rod traveled in 6 sec., which is calculated by multiplying its speed times 6 sec. is equal to 1,116,000 miles. This gives us the following equation: 558,000 miles + _v_ × 6 sec. = 1,116,000 miles. If we subtract 558,000 miles from each side of the equation, it gives us 558,000 miles = _v_ × 6sec. and dividing by 6 sec. gives us _v_ = 93,000 miles/sec. This calculation gives us the magnitude or the speed of the velocity vector. We already know its direction. It is moving along the X-axis in the direction of increasing _x_.

We should note that in the example above it takes a light beam eight seconds to complete its round-trip journey. The length of the round-trip journey is 1,116,000 miles or twice the length of the rod. When the length of the rod measured in a state of absolute rest is 558,000 miles. Since light has a constant velocity, it should take the light beam only six seconds to complete the round-trip journey. The dilation of time experienced by moving objects and the contraction of moving objects along the axis aligned with the direction of motion can eliminate this discrepancy.

Since the rod is traveling with a velocity of _.5V,_ its clocks will run slower than our clocks on earth. When one second passes on our clocks only _¾_ of a second will have passed on the rod's clocks. Therefore, when four seconds pass on our clocks only three seconds will have passed on the rod's clocks. Thus, the total time of the light beam's round-trip journey as measured by the rod's clocks will be six seconds. This means the first leg of the light beam's journey will take 4.3 seconds and the final leg will take 1.5 seconds. Under these conditions the rod's velocity will be calculated to be 63,000 miles/second. If we conclude that the rod contracts along the axis of motion by a factor of _¼_ due to its velocity of _.5V_ our calculation of the rod's velocity returns to 93,000 miles/second.

It is unclear whether or not, according to Einstein, we can ever determine that clock _A_ is two minutes ahead of clock _B._ If we cannot determine the true relationship between the clocks, we cannot determine the velocity of the rigid rod. Einstein requires that the rod is, "set into uniform parallel translational motion (with velocity _v_ ) along the _X_ -axis in the direction of increasing _x_."38 This requirement seems to bar us from transporting clock _B_ to point _A_ and comparing it to clock _A._ Once we move the clock _B_ it is no longer in uniform translational motion and it is no longer part of the moving system of the rigid rod. We cannot overcome this difficulty by placing clock _A_ and clock _B_ on a large turntable and spinning the turntable until the clocks exchange positions. We would then note the times of a round-trip journey of a beam of light starting at point _B_ and compare them to our previous results. This would allow us to determine that clock _B_ did not keep the same time as clock _A_ , but this experiment is not allowed because the motion of the turntable would remove the clocks from the uniform translational motion of the moving system.

Step 7: Einstein's Dilemma

At this point we should try to summarize Einstein's dilemma. Einstein's definition for the synchronization of distant clocks is _t_ B – _t_ A = _t'_ A – _t_ B. This definition is only accurate for distant clocks located on objects in a state of absolute rest, and it is incorrect for distant clocks located on moving objects, provided we employ the standard definition of synchronization. But, we should recall that Einstein claims there is no need for the concept of a state of absolute rest in his theory. When we say his definition is accurate, we mean it agrees with our notion of the synchronization of distant clocks, which is that at the same instant each clock displays the same time. This means regardless of the distance separating the clocks, if there was some being that could have instantaneous knowledge of the time displayed by each of the separated clocks, the time displayed by each clock at any particular moment would be the same.

To demonstrate the relativity of simultaneity Einstein employs a thought experiment in which the clocks on a moving rod are synchronized with the clocks in a system at rest. He states that for the two distant clocks located at either end of the moving rod (with clock _B_ in the forward position, clock _A_ in the rear position and the distance between the two clocks denoted as _r_ AB) the value for _t_ B – _t_ A is _r_ AB _/(V –_ _v)_ and the value for _t'_ A _– t_ B is _r_ AB _/(V +_ _v)_. Thus since _r_ AB _/(V –_ _v)_ is larger than _r_ AB _/(V +_ _v)_ , Einstein argues that _t_ B – _t_ A is larger than _t'_ A _– t_ B and thus _t_ B – _t_ A **≠** _t'_ A _–t_ B. Therefore, according to Einstein, observers co-moving with the rod conclude their clocks are not synchronized while observers in the rest system claim the clocks on the moving rod are synchronized. Thus, Einstein demonstrates the relativity of simultaneity.

Since Einstein dispenses with the concept of a state of absolute rest, we can assume the rest system is actually in motion. Let's assume the rest system was moving through space and its pilots decided they wanted to maneuver alongside the moving rod and match its velocity exactly. They wanted to come to "rest" beside the moving rod. Perhaps, the rest system should not be referred to as the "rest system" until it has come to "rest" beside the moving rod. Once the pilots accomplished this task, they decided to synchronize two of their clocks (clocks _Y_ and Z). Clock _Y_ happened to be in very close proximity to clock _A_ on the moving rod, and clock _Z_ happened to be in very close proximity to clock _B_ on the moving rod. Once the pilots on the rest system synchronized their two clocks, they decided to synchronize clock _A_ with clock _Y_ and also to synchronize clock _B_ with clock _Z_ since each pair of clocks was only separated by an infinitesimal distance _._ If the clocks on the moving rod were synchronized with the clocks of the rest system in this manner, the observers on the moving rod would conclude their two clocks were synchronized and the pilots of the rest system would agree. There would be no relativity of simultaneity.

This is so because we have demonstrated that by setting the forward clock slow by a specific amount of time we can force _t_ B – _t_ A to equal _t'_ A _– t_ B. The specific amount of time the forward clock must clock must be set slow is determined by dividing the total duration for the round-trip journey of the beam of light by two, which equals _t'_ A _/2_. Then we subtract _t'_ A _/2_ from the duration of the first leg of the light beam's round-trip journey as calculated by using the formula _t_ B – _t_ A = _r_ AB _/(_ ( _V –_ _v)_. Thus we set clock _B_ back by the amount of time equal to _[r_ AB _/(V –_ _v)]_ – _t'_ A _/2_.

There are many ways to question Einstein's definition of distant synchronized clocks, which is _t_ B – _t_ A = _t'_ A _– t_ B. We can question whether the definition is appropriate for distant clocks located on moving objects. If the definition is applied to moving objects, we can point out that clocks synchronized by this method will not agree in the time that they keep. We can also point out that the clocks in a moving system can be synchronized with the clocks in a "rest system" in a way that allows both observers in the "rest system" and observers in the moving system to agree the clocks in the moving system are synchronized according to Einstein's definition although they do not agree in the time that they keep.

In the next section we will see that Einstein changes the form of his definition of distant synchronized clocks from the form _t_ B – _t_ A = _t'_ A _–t_ B to the form _½(τ_ 0 \+ _τ_ 2 _)_ = _τ_ 1. This form states that one-half of the duration of the light beam's round-trip journey is equal to the duration of the first leg of the light beam's journey. Einstein succinctly describes the situation, "Suppose that at time _τ_ 0, a light ray is sent from the origin of the system _k_ [the moving system] along the _X-_ axis to _x'_ and reflected from there toward the origin at time _τ_ 1, arriving there at time _τ_ 2; we then must have _½(τ_ 0 \+ _τ_ 2 _)_ = _τ_ 1. . . ."39 It should be noted that the starting time _τ_ 0 is added to the completion time _τ_ 2 instead of being subtracted from it. This must mean the reading of the clock at time _τ_ 0 is zero. With only a slight explanation Einstein further modifies this form of the equation until he arrives at an extended version of the equation, which is the following: _½{τ[0_ , _0, 0, t]_ \+ _τ[0, 0, 0, t_ \+ _x'/(V – v)_ \+ _x'/(V_ \+ _v)]}_ = _τ[x', 0, 0, t_ \+ _x'/(V – v)]._ His explanation is that the extended version includes the arguments of the function _τ_ and applies the principle of the constancy of the velocity of light in the rest system. Einstein attempts to mathematically manipulate this equation to produce the Lorentz transformation equations. He does not succeed.

Chapter 2: The Twisted Path to the Transformation Equations and Beyond

Chapter Summary: An Analysis of "Section 3. Theory of Transformations of Coordinate and Time from the Rest System to a System in Uniform Translational Motion Relative to It" from _On the Electrodynamics of Moving Bodies,_ and an analysis of additional related materials.

Step 1: Metal Rods Turned Into Cartesian Coordinates

In the third section of _On the Electrodynamics of Moving Bodies_ , Einstein's equations become quite numerous and quite complex. In order to make referring to them less cumbersome, it would be helpful if most of the equations were given a specific number. This poses a slight difficulty because the editors of _Einstein's Miraculous Year_ did not number the equations. Since Einstein did not number the equations in his original paper, the editors were being true to the original text.

The publisher's preface to _Einstein's Miraculous Year_ notes that the five papers produced by Einstein in 1905, "reappeared in the original German, with editorial annotations and prefatory essays, in volume 2 of the _Collected Papers of Albert Einstein,_ an ongoing series of volumes being prepared by the Einstein Papers Project at Boston University under the sponsorship of Princeton University Press and the Hebrew University of Jerusalem. _Einstein's Miraculous Year_ draws heavily from this volume _(The Swiss Years: Writings, 1900—1909),_ which remains the definitive and authoritative text of all Einstein's writings of those years. . . ."40

The English translation of that text, which appears in _Einstein's Miraculous Year,_ is by Trevor Lipscombe, Alice Calaprice, Sam Elworthy and John Stachel. As we have noted, their translation does not attach a number to any of Einstein's equations to make references to them less cumbersome. For the numbering of Einstein's equations we have referred to Arthur I. Miller's translation that appears in the appendix of his book _Albert Einstein's Special Theory of Relativity._ However, it should be noted that, without providing any explanation, A. I. Miller discontinues numbering the equations in the middle of the fourth section and in the fifth section no equations are numbered. He begins numbering the equations again in _Part B,_ but the numbering appears to be haphazard. Therefore, we have extended A. I. Miller's numbering sequence to include the final half of the fourth section and the fifth section in its entirety.

Early on in the third section we are introduced to an unusual equation that mixes coordinates designated by numbers with coordinates designated by variables. The equation _½{τ[0_ , _0, 0, t]_ \+ _τ[0, 0, 0, t_ \+ _x'/ (V – v)_ \+ _x'/ (V_ \+ _v)]}_ = _τ[x', 0, 0, t_ \+ _x'/ (V – v)]_ , which is Eq.(3.1) according to A. I. Miller's numbering scheme, is complicated. The equation is derived from a complicated thought experiment involving a moving system and a resting system. In the moving system a beam of light makes a round-trip journey. The beam of light begins its journey at a point denoted as _(ξ, η, ζ)_ = _(0, 0, 0)_ that is the origin point of a Cartesian coordinate system. The moving system is essentially a Cartesian coordinate system constructed out of metal rods. In fact, in this thought experiment, both the moving system _k_ and the resting system _K_ are material constructions of a Cartesian coordinate system, plus these structures are equipped with certain experimental equipment.

The light beam travels along the _ξ_ \- axis in the direction of increasing _ξ_ until it strikes a mirror at a distance _x'_ from the starting point. Einstein does not provide a numerical value for _x'_. A numerical value for _x'_ would tell us the distance from the starting point, _(ξ, η, ζ)_ = _(0, 0, 0)_ , to the mirror. In the Greek alphabet " _ξ_ ," " _η_ " and " _ζ"_ are lowercase letters; their English counterparts are " _x_ ," " _e_ " and " _z"_ respectively. The spellings for " _ξ_ ," " _η_ " and " _ζ"_ are _xi_ , _eta_ , and _zeta_ respectively; their pronunciations are z _ī_ , _ā_ t _' ə_ and z _ā_ t _' ə_ respectively. There is no equivalent for the letter " _y_ " in the Greek alphabet.

Once the light beam strikes the mirror, it is reflected back to its starting point. Einstein's Eq.(3.1) informs us that the total time of the light beam's round-trip journey, multiplied by _½_ is equal to the time it takes the light beam to travel from its starting point to the mirror located at a distance _x'_ from the origin of system _k_.

To calculate the total time of the round-trip journey of the light beam Einstein adds the starting time reading to the return time reading. This may seem odd. To subtract the starting time reading from the return time reading seems more natural. For instance, if I went on a walk, in which I left my home at 8:00 a.m. and returned home at 10:00 a.m., the total time of my walk is determined by subtracting the starting time reading from the return time reading. The result is that the round-trip time for my walk is two hours. If I used a stop watch to measure the duration of my walk, the starting time reading would be 00 hours: 00 minutes. In this instance, I could add the starting time reading to the return time reading to calculate the total time of my walk. I could also subtract the starting time reading from the return time reading since the starting time reading is set to zero. In determining how much time elapsed from my starting point to my turnaround point the same type of calculations would apply. Of course, if the starting time of any journey is set to zero the duration of any segment of the journey can be calculated by taking the elapsed-time reading at the moment of the completion of that particular segment. These considerations will be important when we use certain mathematical rules involving partial derivatives. Now, we should return to our description of the resting system _K_ and the moving system _k._

The place and time of any event, which is at rest in the resting system _K_ , are described by the coordinates _x, y, z, t_. Time in the resting system _K_ is denoted by _t._ The place and time of any event, which is at rest in the moving system _k_ , are described by the coordinates _ξ, η, ζ, τ_. Time in the moving system is denoted by _τ._

The moving system _k_ is moving with a constant velocity, which we will denote with the letter _v._ The moving system _k_ is traveling in the direction of increasing _x_ of the resting system _K._ The "X-axis" of the moving system is called the _ξ-axis_ , and it coincides with the X-axis of the resting system. The "Y-axis" and the "Z- axis" of the moving system are called the _η-axis_ and the _ζ-axis_ respectively. They are parallel to their counterparts, the _Y_ and _Z_ axes, of the system at rest.

Because of the way the two systems were aligned before system _k_ began to move and because of the nature of their new alignment since system _k_ began moving, we can say _x'_ = _x_ − ( _v_ × _t_ ), where the sign "×" indicates _multiplied by_. This means that a point with the value _x_ on the X-axis of the resting system will coincide with a point with the value _x'_ on the _ξ-axis_ of the moving system at a given time _t_ of the resting system. The value of _x'_ will be smaller than the value of _x_ by an amount equal to _v_ × _t._ Because of the nature of the alignment of the moving system _k_ with the resting system _K_ , a point at rest in the _k_ system belongs to the system of values _x', y, z,_ independent of time. Einstein defines _τ_ (time in the moving system _k_ ) as a function of _x', y, z,_ and _t._ Therefore, _τ_ is a function with four variables, _τ(x', y, z, t)_.

In mathematics a function of four variables would typically be written in this manner: _ƒ(x, y, z, t)_ = _x² y z − 3z + t³_ or some other combination of variables, where variables such as _yz_ indicates _y multiplied by z_. In this instance, the right side of the equation, _x² y z − 3z + t³_ , is the rule that associates to each foursome of values for the variables a number or numbers. Einstein does not give us the rule, i.e., the right side of the equation that associates to each foursome of values for the variables a particular number or numbers. Instead, he tells us the relationship between three sets of values for the variables _x', y, z,_ and _t._ The three sets of values are the following: _(0, 0, 0, t)_ and _(0, 0, 0, t + x'/ (V − v) + x'/ (V + v))_ and _(x', 0, 0, t + x'/ (V − v))._

If Einstein set the function _τ(x', y, z, t)_ equal to some grouping of variables, we could find the partial derivative of function _τ(x', y, z, t)_ for each of the variables _x', y, z,_ and _t._ We could then evaluate each of the four partial derivatives for each of the three sets of values for the variables _x', y, z,_ and _t._ It should be noted that the three sets of values for the variables are a mixture of numbers and variables. So it could be a situation such that even if Einstein gave us an equation for the function _τ(x', y, z, t)_ we still could not satisfactorily evaluate the partial derivatives for the three sets of values he has given us.

According to Einstein if the term _x'_ in Eq.(3.1) is allowed to become "infinitesimally small,"41 the result is: _½[(1/(V − v)_ \+ _1/(V_ \+ _v))∂τ/∂t]_ = _∂t/∂x'_ \+ _[1/(V − v)]∂τ/∂t_ , which is Eq.(3.2). In Arthur I. Miller's book _Albert Einstein's Special Theory of Relativity,_ he states, on page 209, "Einstein took _x'_ to be infinitesimal and expanded both sides of Eq. (3.1) as a series in _x'_. Neglecting terms higher than first order, the result is Eq. (3.2)."42 The exact mathematical steps indicated by A. I. Miller's explanation are difficult to ascertain.

Step 2: The Misapplication of Mathematical Rules

However, it is possible that Einstein misapplied certain mathematical rules in his analysis of Eq.(3.1). These rules are used in calculations involving the partial derivatives of functions with two or more variables. Unfortunately, the following explanation of the contention that Einstein did misapply certain mathematical rules is extremely lengthy.

To understand how Eq.(3.1) generates Eq.(3.2) through the misapplication of mathematical rules, we need to rewrite Eq.(3.1). In Einstein's version of Eq.(3.1) there is a slight inconsistency. The left side of the equation, which denotes the duration of a round-trip journey of a beam of light, consists of a term representing the starting time, to which a term representing the return time is added. However, the right side of the equation, which denotes the duration of a light beam's one-way journey from the staring point to the mirror, consists only of the term representing the time reading when the light beam arrives at the mirror. To be consistent the right side of the equation should consist of the term representing the starting time and to this term we should add the term representing the time reading when the light beam arrived at the mirror. This inconsistency is not an error because the starting time has been set to zero. In fact, since the starting time is set to zero it does not need to appear on either side of the equation.

However, if the starting time was not set to zero, it would need to appear on both sides of the equation. The starting time would be subtracted from the return time on the left side of the equation, and the starting time would be subtracted from the time the light beam arrived at the mirror on the right side of the equation. We are going to rewrite Eq.(3.1) in this more generalized form in order to make the application of certain mathematical rules for partial derivatives more straightforward. The rewritten Eq.(3.1) is as follows: _½{ τ[0, 0, 0, t_ \+ _x'/ (V − v)_ \+ _x'/ (V_ \+ _v)] −_ _τ[0, 0, 0, t]_ _}_ = _τ[x', 0, 0, t_ \+ _x'/ (V − v)] −_ _τ[0, 0, 0, t]_. Rewritten in this form it is easier to apply two specific mathematical rules for partial derivatives. To make the mathematics clearer, we are going to concern ourselves with only the variables _x'_ and _t_ from Eq.(3.1).

The two mathematical rules come from the textbook _Calculus and Its Applications_ the 4th edition by Larry J. Goldstein, David C. Lay and David I. Schneider. In their descriptions of the rules, they use functions of two variables, but the rules can be generalized to apply to functions of any number of variables.

The authors state the first rule we are going to use on page 351. We can call it _Rule A,_ and we should note the sign " _≈_ " indicates approximately equal _._ The authors state the portion of the rule we are concerned with as follows: "Let _ƒ(x, y)_ be a function of two variables. Then if _k_ is small we have _ƒ(a, b_ \+ _k) – ƒ(a, b) ≈ ∂ƒ/∂y(a, b)_ × _k_."43 We will apply _Rule A_ to the left side of the rewritten Eq.(3.1). _Rule A_ allows us to manipulate Eq.(3.1) without knowing the right side of the equation of the function _τ(x', y, z, t)_ , which is the particular rule associated with the function _τ(x', y, z, t)._ It allows us to use two of the three sets of values for the variables that Einstein has provided in Eq.(3.1). There is also a similarity between _Rule A'_ s requirement that _k_ be small and Einstein's requirement that _x'_ must become infinitesimally small.

_Rule A_ is also similar to the method used to determine the duration of a light beam's round-trip journey. The duration of a light beam's round-trip journey is calculated as follows: the time reading at the end of the journey minus the time reading at the beginning of the journey. The time reading at the end of the journey is similar to the term _ƒ(a, b \+ k)_ , and the time reading at the beginning of the journey is similar to the term _ƒ(a, b)_. The constant _a_ would represent the point on the _ξ-axis_ that coincides with both the beginning and end of the light beams round-trip journey. Although the _ξ-axis_ is an axis of the moving system _k_ , the _ξ_ coordinate of the point is the same for both events, and according to Einstein's designation its numerical reading on the _ξ-axis_ is zero. The constant _b_ would represent the starting time for the light beam's journey, and by Einstein's designation it is also zero. It is represented in Eq.(3.1) by the quantity _t._ The sum _(b + k)_ would represent the starting time plus the addition of the return time, which would be represented by _k_ _._ The duration of the light beam's round-trip journey, i.e., _k_ is represented in Eq.(3.1) by the sum _[ x'/ (V − v)_ \+ _x'/ (V_ \+ _v)]_.

As we have noted, in Eq.(3.1) Einstein gives us three sets of values for the variables _(x', y, z, t)_ : _(0, 0, 0, t)_ and _(0, 0, 0, t_ \+ _x'/ (V − v)_ \+ _x'/ (V_ \+ _v))_ and _(x', 0, 0, t_ \+ _x'/ (V − v))_. Let us apply _Rule A_ to the left side of the rewritten Eq.(3.1). For convenience we will exclude, for the moment, the quantity _½_ , which is located outside these parentheses _{_. . . _}_. The quantities within these parentheses include the first and second set of values from Eq.(3.1). We must match up the quantities in _Rule A_ to their counterparts on the left side of the rewritten Eq.(3.1) taking into account that _Rule A_ is written for a function with two variables. In this case _ƒ_ = _τ_ , _ƒ(x, y)_ = _τ(x', t)_ , _a_ = _0_ , _b_ = _0_ , _k_ = _(x'/ (V − v)_ \+ _x'/ (V_ \+ _v)), ƒ(a, b_ \+ _k)_ = _τ(0, t_ \+ _x'/ (V − v)_ \+ _x'/ (V_ \+ _v))_ and _ƒ(a, b)_ = _τ(0, 0)_.

_Rule A_ states that _k_ must be small. This is not a precise definition for the numerical value of _k._ In the examples provided in the textbook _Calculus and Its Applications_ the 4th edition the value of _k_ often equals one. The authors also state on page 378, "the approximation improves as _k_ approaches zero."44 In Eq.(3.1) if we let _x'_ assume the numerical value of one unit of length (for example, one mile), then for all values of _v_ that are at least one unit of distance per unit of time (for example, one mile per second) less than the velocity of light, the value of _k_ will be either approximately one or less than one. For example, if we let _v_ equal the orbital velocity of the earth, the value for _k_ would be approximately 0.000011 seconds. However, as _v_ exceeds 99.999% the velocity of light, _k_ will grow progressively larger than one. Thus, when _x'_ assumes the value of one unit of distance, _k_ assumes the values of either approximately one or less than one for values of _v_ less than 99.999% the velocity of light. Therefore, if _x'_ assumes the value of one unit of distance, then under most circumstances, when _Rule A_ is applied to Eq.(3.1), _k_ assumes values that are typically associated with the requirement that _k_ should be small. This may explain why Einstein allows _x'_ to assume the value of one although his stated requirement is that _x'_ must become infinitesimally small. This is evidence that Einstein is applying _Rule A._

Taking this lengthy explanation into consideration, we apply _Rule A_ to the left side of the rewritten Eq.(3.1) gives us the following:

Formula A

_τ(0, t_ \+ _1/(V − v)_ \+ _1/(V_ \+ _v)) − τ(0, t)_ _≈_ _∂τ/∂t(0, t)_ × _(1/(V − v)_ \+ _1/(V_ \+ _v))_

The right side of Formula A is very similar to the left side of Einstein's Eq.(3.2). We can multiply the entire left side and the entire right side of the Formula A by _½_ , since the _½_ in Eq.(3.1) was separated from the terms to which we applied _Rule A_ by parentheses. This gives us the following:

Formula B

_½[(0, t_ + _1/(V − v)_ + _1/(V_ \+ _v)) − τ(0, t)] ≈ ½[∂τ/∂t(0, t)_ × _(1/(V − v)_ \+ _1/(V_ \+ _v))]_

Thus, the only differences between the right side of Formula B and the left side of Einstein's Eq.(3.2) are the term _(0, t)_ and the approximately equal sign, **≈** , which replaces an = sign.

The right side of Formula B, _½[∂τ/∂t(0, t)_ × _(1/(V − v)_ \+ _1/(V_ \+ _v))]_ , can be expressed as follows: The partial derivative of the function _τ(x', y, z, t)_ with respect to the variable _t_ is evaluated using the following values for _x'_ and _t: x'_ = _0_ and _t_ = _t._ Then this result is multiplied by _[1/(V – v)_ \+ _1/(V_ \+ _v)]_ and finally the entire result is multiplied by _½_

To further familiarize ourselves with _Rule A,_ we should review an example where we evaluate partial derivatives at certain values. Once we familiarize ourselves with evaluating partial derivatives in the two paragraphs that follow, we will turn our attention to a review of an example that applies _Rule A_ in a situation that is more conventional than the complex behavior exhibited by a light beam, when its motion is described from both a moving and resting frame of reference.

As a more conventional example of evaluating partial derivatives, let us evaluate the partial derivatives of the following function: _τ(x', t)_ = _x' ²_ \+ _3t x' ²_ \+ _t³_. To find the partial derivative of a function, the function must be expressed in the above manner. The partial derivative of function _τ(x', t)_ with respect to variable _t_ is as follows: _∂τ/∂t_ = _3x' ²_ \+ _3t²._ The term _x' ²_ is treated as a constant, and the derivative of a constant is zero. The term _3t x' ²_ is treated as the variable _t_ multiplied by the constant _3x' ²_ , and the derivative of _t_ is one, which is multiplied by the constant _3x' ²._ The derivative of the term _t³_ is _3t²._ If we evaluate the partial derivative _∂τ/∂t_ for the coordinates _(0_ , _t)_ , the result is _3(0²)_ \+ _3t²_ = _3t²._

Let us continue in our evaluation of the partial derivatives from this example, the partial derivative of function _τ(x', t)_ with respect to the variable _x'_ is _∂τ/∂x'_ = _2x'_ \+ _6t x._ If we evaluate the partial derivative _∂τ/∂x'_ for the coordinates _(0, t),_ the result is _2(0)_ \+ _6t(0)_ = _0._

Let us apply _Rule A_ in a conventional example. In this example we will find the approximate value of the following: _48_ • _(125)_ 1/3 • _(255)_ 3/4. The sign "•" indicates _multiplied by._ The solution is to let _ƒ(x, y)_ = _48x_ 1/3 _y_ 3/4. Since _ƒ(a, b_ \+ _k) - ƒ(a, b) ≈ ∂ƒ/∂x(a, b) • k,_ we let _a_ = _125_ , _b_ = _256_ and _k_ = – _1_. This gives us _ƒ(125, 256 –1) – ƒ(125, 256) ≈ ∂ƒ/∂x(125, 256) • (–1)._ The partial derivative of the function _ƒ(x, y)_ with respect to the variable _y_ is _∂ƒ/∂y_ = _48(1/3)_ _x_ −2/3 _y_ 3/4 = _16x_ −2/3 _y_ 3/4 _._ Evaluating the partial derivative at _∂ƒ/∂x(125, 256)_ yields _16 • 1/25 • 64 ,_ which equals _40.96._ We multiply this value by _k,_ which we recall in this example is –1, and the result is – _40.96._ Thus we have _ƒ(125, 256 - 1) - ƒ(125, 256)_ ≈ _-40.96._ Since we want to solve the equation for _ƒ(125, 256 – 1),_ we have _ƒ(125, 256 – 1)_ ≈ _ƒ(125 , 256)_ – _40.96._ This gives us _(48• 125_ 1/3 • _255_ 3/4 _)_ ≈ _(48 • 5 • 64) – 40.96,_ which we can calculate to be _15,319.04._ This gives us, _48_ • _(125)_ 1/3 • _(225)_ 3/4 ≈ _15,319.04._ Using a calculator to determine _48_ • _(125)_ 1/3 • _(225)_ 3/4 yields _15,314.97799._

The above calculation is merely an example of the way _Rule A_ can be applied to a function with two variables. There are many types of mathematical calculations for which we can employ _Rule A_ to arrive at a reliable approximation of the correct answer. Of course, there are many practical applications, as well, in which _Rule A_ can give us reliable approximations. The important point, for our purposes, is that when _Rule A_ is applied to the left side of rewritten Eq.(3.1), _Rule A_ generates a quantity that is strikingly similar to the left side of Eq.(3.2).

Now, we will apply _Rule B_ to the right side of the rewritten Eq.(3.1). The rewritten Eq.(3.1) is as follows: _½{ τ[0, 0, 0, t_ \+ _x'/(V − v)_ \+ _x'/(V_ \+ _v)] − τ[0, 0, 0, t]}_ = _τ[x', 0, 0, t_ \+ _x'/(V − v)] − τ[0, 0, 0, t]._

_Rule B_ is given on page 378 of _Calculus and Its Applications_ the 4th edition, "Therefore, it is not surprising that when both coordinates are changed, the change in _ƒ(x, y)_ is approximated by the sum of these terms. Precisely, we have as follows: _ƒ(a_ \+ _h, b_ \+ _k) – ƒ(a, b) ≈ [∂ƒ/∂x (a, b)]_ • _h_ \+ _[∂ƒ/∂y (a, b)]_ • _k_ , where the approximation improves as _h_ , _k_ approach zero. The expression on the right side is usually called a _total differential._ Its value depends on _h_ and _k_ , as well as the partial derivatives at _x_ = _a_ , _y_ = _b."_ 45

We must match the terms of _Rule B_ to the appropriate terms from the right side of the rewritten Eq.(3.1). The term _ƒ(x, y)_ = _τ(x', t)_ , the term _ƒ(a, b)_ = _τ(0, t)_ , and the term _ƒ(a + h, b_ \+ _k) = τ(0_ \+ _x', t_ \+ _x'/ (V – v))._ From this we can determine that _a_ = 0, _b_ = _t_ , _h_ = _x'_ , and _k_ = _x'/ (V – v)_. The term _∂ƒ/∂x_ = _∂τ/∂x'_ and the term _∂ƒ/∂y_ = _∂τ/∂t_. When we match up the terms from _Rule B_ to the right side of the rewritten Eq.(3.1), there is an insignificant discrepancy. The term _(a_ \+ _h)_ is matched to the term _(0_ \+ _x')_ while in the rewritten Eq.(3.1) it appears as the term _(x' )_.

Applying _Rule B_ to the right side of Eq.(3.1) gives us the following:

Formula C

_τ(0_ \+ _x', t_ \+ _x'/ (V – v)) – τ(0, t) ≈ [∂τ/∂x' (0, t)]_ • _x'_ \+ _[∂τ/∂t (0, t)]_ • _x'/ (V – v)_.

If we follow the convention we established previously that _x'_ should be set equal to one so that both _h_ and _k_ are small, we can transform the right side of Formula C so that it is very similar to the right side of Eq.(3.2). As we stated previously Einstein required that _x'_ become infinitesimally small, yet on both the left and right sides of his Eq.(3.2) the term _x'_ was changed into one. This is evidence that Einstein is applying _Rules A_ and _B._ If we allow _x'_ to become one, the formula becomes the following:

Formula D

_τ(0_ \+ _1, t_ \+ _1/(V – v)) – τ(0, t) ≈ [∂τ/∂x' (0, t)]_ • _1_ \+ _[∂τ/∂t (0, t)]_ • _1/(V – v)_.

The right side of the formula is stated in straightforward words in the following paragraph.

The partial derivative of the function _τ(x', t)_ with respect to the variable _x'_ is evaluated for the coordinates _(0_ , _t)_. The result of the evaluation is multiplied by one. To this result, we add another quantity. That quantity is the following: the partial derivative of the function _τ(x', t)_ with respect to the variable _t_ , evaluated for the coordinates _(0_ , _t)_ , and with the result of this evaluation is multiplied by _1/(V – v)._

Perhaps, using another pair of parentheses would make the right side of the formula clearer. Thus, it would appear as follows: _{[∂τ/∂x' (0, t)]_ • _1}_ \+ _{[∂τ/∂t (0, t)]_ • _1/(V – v)}_. We should note that multiplying the term _[∂τ/∂x' (0, t)]_ by one, leaves the term unchanged, giving us _[∂τ/∂x' (0, t)]_. Thus, generating the following formula:

Formula E

_τ(0_ \+ _1, t_ \+ _1/(V – v)) – τ(0, t) ≈ ∂τ/∂x' (0, t) + ∂τ/∂t (0, t)_ • _[1/(V – v)]_.

Except for the appearance of the coordinates _(0_ , _t)_ and the ≈ sign, the right side of Formula E is the same as the right side of Einstein's Eq.(3.2) which is as follows: = _∂τ/∂x'_ \+ _[1/(V – v) ] ∂τ/∂t_. Of course, the right side of Eq. (3.2) can be rewritten as = _∂τ/∂x'_ \+ _∂τ/∂t [1/(V – v) ]_ to accent its similarity to the right side of Formula E. The similarity between the right side of Eq.(3.2) and the right side of Formula E, which is produced by applying _Rule B_ to the right side of Eq.(3.1), is striking.

There are three remaining areas that need to be expanded upon to give a complete picture of the application of _Rules A_ and _B_ to Eq.(3.1) _._ First, _Rules A_ and _B_ must be expanded from rules that apply to functions with two variables to rules that apply to functions with four variables. Secondly, since the coordinates _(0_ , _t)_ , which appear once when _Rule A_ is applied and twice when _Rule B_ is applied, are the only terms that separate the above results from Einstein's results, we must determine whether these terms can be deleted from right sides of Formulas B and E so that they more closely replicate Einstein's results. Thirdly, the significance of the use of the approximately equal sign, ≈, as opposed to an = sign must be explored.

First, the _Rules A_ and _B_ can be expanded to accommodate functions with four variables. The expanded _Rule A_ is as follows: Let _ƒ(x, y, z, t)_ be a function of four variables. If _k_ is small, we have _ƒ(a, b, c, d_ \+ _k) - ƒ(a, b, c, d) ≈ ∂ƒ/∂t (a, b, c, d)_ • _k_. The expanded _Rule B_ is as follows: Let _ƒ(x, y, z, t_ ) be a function of four variables. If both _h_ and _k_ are small, we have _ƒ(a_ \+ _h, b, c, d_ \+ _k) – ƒ(a, b, c, d) ≈ [∂ƒ/∂x(a, b, c, d)]_ • _h_ \+ _[∂ƒ/∂t(a, b, c, d)]_ • _k._ The result of expanding _Rules A_ and _B_ is that the term _(0_ , _t)_ is enlarged from two coordinates to four coordinates, and it becomes _(0_ , _0_ , _0_ , _t)_.

Secondly, the term ( _0_ , _t_ ) and the term in its expanded form _(0_ , _0_ , _0_ , _t)_ cannot be deleted from Formulas B and E. If Einstein applied _Rules A_ and _B_ to his Eq.(3.1) to produce Eq.(3.2), he deleted the term _(0, 0, 0, t)_ to suit his own purposes.

Thirdly, the introduction of the approximately equal sign, ≈, undercuts Einstein's argument that the left side of Eq.(3.2) is equal to the right side of Eq.(3.2). By applying _Rule A_ , we produce a quantity that is approximately equal to the left side of Eq.(3.1). By applying _Rule B_ , we produce a quantity that is approximately equal to the right side of Eq.(3.1). These two approximations are not equal to each other. It is true that the approximation of the left side of Eq.(3.1) improves as _k_ approaches zero. In this instance _k_ is equal to _(x'/(V_ \+ _v)_ \+ _x'/(V – v))_. It is also true that the approximation of the right side of Eq.(3.1) improves as both _h_ and _k_ approach zero. In this case _h_ is equal to _x'_ and _k_ is equal to _x'/(V – v)_. It may be confusing that the term _k_ appears both in _Rule A_ and _Rule B_. It does not mean that the term _k_ must assume the same value in both _Rule A_ and _Rule B._ The use of the term _k_ in both rules was merely a matter of convenience for the authors of _Calculus and Its Applications_ 4th edition.

The term _k_ employed for _Rule A_ , which was applied to the left side of Eq.(3.1), produces a much more accurate approximation than the term _h_ employed for _Rule B_ , which was applied to the right side of Eq.(3.1). Therefore, the approximation of the left side of Eq.(3.1) is much more accurate than the approximation of the right side of Eq.(3.1). Thus, the left side approximation of Eq.(3.1) is not equal to the right side approximation of Eq.(3.1). The formula these approximations generate, Eq.(3.2), is not an equality. The left side of the formula is only approximately equal to the right side of the formula. Einstein's requirement that _x'_ must become infinitesimally small does not alter the fact that Eq.(3.2) is not an equality. No matter how infinitesimally small _x'_ becomes the value of the term _k_ , which is generated for the left side of the Eq.(3.1), will be much closer to zero than the term _h_ , which is generated for the right side of Eq.(3.1). Therefore, the approximation produced of the left side of Eq.(3.1) will be more accurate than the approximation of the right side of Eq.(3.1) because the right side approximation employs the term _h_ , which is not as close to zero as the term _k._ This remains so even if the term _h_ is infinitesimally small.

The following example will illustrate the points made in the previous argument. Let _v_ represent the orbital velocity of the earth, which is about 19 miles/second. The term _V_ represents the velocity of light, which is about 186,000 miles/second. The term _x'_ represents the distance the beam of light in Einstein's thought experiment travels from the starting point until it strikes the mirror. For convenience, let's set _x'_ equal to one mile.

We applied _Rule A_ to the left side of Eq.(3.1); the term _k_ for _Rule A_ is as follows: _k_ = _x'/(V_ \+ _v)_ \+ _x'(V – v)_ = 1 mile /(186,000 miles/sec. + 19 miles/sec.) + 1mile /(186,000 miles/sec. – 19miles/sec. ) = 0.000011 seconds.

We applied _Rule B_ to the right side of Eq.(3.1); the term _h_ for _Rule B_ was as follows: _h_ = _x'_ = 1 mile.

The value for the term _k_ for _Rule A_ is about 90,000 times smaller than the value for the term _h_ for _Rule B._ Since the value for the term _k_ is smaller than the value for the term _h_ , the approximation of the left side of Eq.(3.1) is more accurate than the approximation of the right side. This is an instance where you can mix apples and oranges or at least seconds and miles. However, the role that term _k_ for _Rule B_ plays in these calculations should be examined.

As we noted before, we applied _Rule B_ to the right side of Eq.(3.1), using the term _k_ for _Rule B._ The term _k_ for _Rule B_ is as follows: _k_ = _x'/(V – v_ ) = 1mile/(186,000miles/sec.– 19miles/sec.) = 0.0000054 seconds.

The value for the term _k_ for _Rule B_ is about twice as small as the value of the term _k_ for _Rule A._ However, this is overshadowed by the fact that the value of the term _k_ for _Rule A_ is about 90,000 times smaller than the value of the term _h_ for _Rule B._ Nothing in this example suggests that there is equality between the value of the term _k_ for _Rule A_ and the values of the terms _h_ and _k_ for _Rule B._ Such equality would be needed to suggest that the approximation of the left side of Eq.(3.1) is equal to the approximation of the right side of Eq.(3.1).

The value of the term _k_ for _Rule A_ will always be about 90,000 times smaller than the value of term _h_ for _Rule B._ This is the case even if the term _h_ for _Rule B_ , i.e., _x'_ , becomes infinitesimally small. This shows that Eq.(3.2), _½[1/(V – v)_ \+ _1/(V_ \+ _v)]∂τ/∂t_ = _∂τ/∂x'_ \+ _[1/(V – v) ] ∂τ/∂t_ , is not an equality; the equation is only an approximate equality. The equation should be expressed as follows: _½[1/(V – v)_ \+ _1/(V_ \+ _v)]∂τ/∂t ≈ ∂τ/∂x'_ \+ _[1/(V – v) ] ∂τ/∂t._ The right side of the equation is about 90,000 times more accurate an approximation than the left side of the equation. Further, the approximate equality is only valid if we accept that Eq.(3.1) is valid. Also, the approximate equality is only valid for the value of _x'_ specified in the approximate equality. The specified value of _x'_ in this instance is _x'_ = _1_ unit of distance.

The conclusion of this entire analysis is that Einstein's transformation of Eq. (3.1) into Eq. (3.2) is flawed. The mathematics, he most likely employs, do not generate formulas that are equalities, instead they are designed to generate formulas that are approximate equalities. His requirement that _x'_ _be chosen infinitesimally small_ is remarkable because that requirement is clearly not satisfied.

His argument is analogous to the kind of mistake a bicyclist could make. Let us imagine that a bicyclist is going on a ten-mile round-trip journey. During the outbound half of his journey, he is heading west and riding into a steady, 20 mph wind from out of the west. During the return half of his journey, he is traveling east, and riding now with the same, steady wind at his back.

To test his bicycle's brakes and tire inflation, before he begins his journey, he travels back and forth over a 30-yard distance several times. He travels west 30 yards then he returns traveling east 30 yards. He concludes that his bicycle is in good condition, and he also concludes that the wind will not affect his journey. However, a few miles into his journey, he realizes his second conclusion is wrong. The wind will affect his journey. When he was traveling back and forth over a 30-yard distance, the slight distance he traveled gave him the wrong impression that the wind would not affect his journey. He concludes that when he was traveling back and forth over the 30-yard distance the wind did affect him, but its affect was too small for him to notice.

This is the kind of argument Einstein is making; however, Einstein never acknowledges that his conclusion is incorrect. He maintains that if the round-trip distance, which a light beam travels, is infinitesimally small, then the motion of the of the body, on which the light beam experiment is taking place, will have no effect on the duration of the outbound segment or the return segment of the light beam's journey.

We can make an argument that supports Einstein's conclusion. If we say that _infinitesimally small_ refers to a distance that is zero units in length or effectively zero units in length for all possible calculations, then Einstein's argument is correct, but all this means is the following: on a moving body, when a beam of light makes a round-trip journey of zero units in length the duration of each leg of the journey is equal. The statement may be true, but more importantly it is pointless because the statement claims that the light beam has made a round-trip journey when in fact it has made no journey, whatsoever. This kind of argument should not serve as a template for analyzing the behavior a light beam making a round-trip journey on a moving body.

Step 3: A Change in Direction

Is there another way to derive Eq.(3.2) from Eq.(3.1)? If a way could be found to derive Eq.(3.2) so that its equal sign was legitimate, it would strengthen Einstein's argument. A significant barrier to overcome is the fact that Eq.(3.1) is not in form that can undergo partial differentiation, yet Eq.(3.2) contains two partial derivatives, _∂τ/∂x'_ and _∂τ/∂t_. Eq.(3.1) consists of three quartets (groups of four) of coordinates for the function _τ(x', y, z, t)_. The function _τ(x', y, z, t),_ which is composed of the four variables _x', y, z,_ and _t_ , needs a rule (an equation) that associates to each quartet of values for the variables a number in order for the function to be partially differentiated. Einstein does not supply us with such a rule.

As we mentioned before, on page 209 of Arthur I. Miller's _Albert Einstein's Special Theory of Relativity_ he provides an explanation of the derivation of Eq.(3.2) from Eq.(3.1). He states, "Toward obtaining differential equations whose solutions provide the functional dependence of _τ_ on _(x', y, z, t),_ Einstein took _x'_ to be infinitesimal and expanded both sides of Eq.(3.1) as a series in _x'_. Neglecting terms higher than first order the result is _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)_ • _∂τ/∂t_ = _0."_ 46

First, we should explain why the equation _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)_ • _∂τ/∂t_ = _0_ does not look like the Eq.(3.2) we are familiar with. The equation _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)_ • _∂τ/∂t_ = _0_ is Einstein's Eq.(3.3), which is derived from Eq.(3.2) by finding a common denominator for the fractions and grouping the partial derivatives on the left side of the equation. The procedure is as follows: Eq.(3.2), _½[1/(V – v)_ \+ _1/(V_ \+ _v)]_ • _∂τ/∂t_ = _∂τ/∂x'_ \+ _[1/(V – v) ]_ • _∂τ/∂t_ , with a common denominator for the fractions becomes, _½[2V/(V²_ _– v²)]_ • _∂τ/∂t_ = _∂τ/∂x'_ \+ _[(V_ \+ _v)/(V²_ _– v²)]_ • _∂τ/∂t_. Next, we subtract the term _[V/(V²_ _– v²)]∂τ/∂t_ from both sides of the equation, and we obtain the following: _0_ = _∂τ/∂x'_ \+ _[(V_ \+ _v)/(V²_ _– v²)] ∂τ/∂t –_ _V/(V²_ _– v_ 2 _)]∂τ/∂t_ _(Note_ : _½ • 2_ = _1)._ The next step is to group together the terms that contain _∂τ/∂t,_ which gives us the following: _0_ = _∂τ/∂x' + [v/(V²_ _– v²)] ∂τ/∂t_. Finally, we express the equation with the terms containing partial derivatives placed on the right side of the equation, which gives us the following Eq.(3.3), _∂τ/∂x' + [ v/(V²_ _– v²)]_ • _∂τ/∂t =0_.

With that explanation concluded, we can explore the notion of expanding both sides of Eq.(3.1) as a series in _x'_. Is it mathematically legitimate to expand both sides of Eq.(3.1) as any series in _x'_ or are we limited to a certain group of series in _x'_ ? Is there only one series in _x'_ in which we can legitimately expand both sides of Eq.(3.1)?

We could expand both sides of Eq.(3.1) in many different series in _x'_ , and this would produce many different outcomes. We assume A. I. Miller requires that we only expand both sides of Eq.(3.1) in those series in _x'_ that will produce the desired outcome. This is not a mathematically legitimate procedure, unless he acknowledges that the outcome he generates is only one among a great many possible outcomes. Or, he could be referring to a specialized series such as the Taylor's series.

For simplicity, if we ignore all the terms in Eq.(3.1) that do not directly contain the value _x'_ , we could expand the left side of Eq.(3.1) as a series in _x'_ as follows: _[x'/(V_ \+ _v)_ \+ _x'/(V – v)]_ = _2Vx'/ (V²_ _– v²)_ , which could produce the following series, _2Vx'/ (V²_ _–v²), 4Vx'/ (V²_ _– v²), 6Vx'/ (V²_ _– v²), . . . 2nVx'/ (V²_ _– v²),_ where _n = 1, 2, 3, . . .._

Again for simplicity, if we ignore all the terms in Eq.(3.1) that do not directly contain the value _x'_ , we could expand the right side of Eq.(3,1) as a series in _x'_ as follows: _x'_ could produce the following series, _1x', 2x', 3x', . . . nx'_ , where _n_ = _1, 2, 3, . . .,_ and _[x'/(V – v)]_ could produce the following series, _1x'/(V – v), 2x'/(V – v), 3x'/(V – v), . . . nx'/(V – v)_ , where _n_ = _1, 2, 3, . . ._

A. I. Miller's next requirement is that we neglect terms higher than the first order. According to the _Mathematics Dictionary_ edited by Glenn James and Robert C. James the word _order_ has as many as twenty different mathematical definitions. Of these definitions, two seem to be applicable to Miller's scenario. The _Mathematics Dictionary_ defines the "order of an algebraic curve or surface as the degree of its equation. . . ."47 While this definition may be applicable, we will ignore it in favor of the definition for differences of the first order because that definition involves sequences. We will implicitly return to the concept of the degree of an equation when we discuss a Taylor's series. The _Mathematics Dictionary_ defines the "differences of the first order or first-order differences as the sequence formed by subtracting each term of a sequence from the next succeeding term. The first-order differences of the sequence (1, 3, 5, 7, . . .) would be (2, 2, 2, . . .)."48 We begin by subtracting the first term of a series from the second term of a series. When we apply that procedure to the series we generated for the left side of Eq.(3.1), the result is as follows: _4Vx'/(V²_ _– v²) – 2Vx'/ (V²_ _– v²)_ = _2Vx'/ (V²_ _– v²)_. When we apply the same procedure to the two series we generated for the right side of Eq.(3.1), the result is as follows: _2x' – 1x'_ = _1x' and 2x'/(V – v) – 1x'/(V – v)_ = _1x'/(V – v)._

This simplified version of Eq.(3.1) is as follows: _2Vx'/(V²_ _– v²)_ = _1x'_ \+ _1x'/(V – v)_. The entire left side of this simplified version of Eq.(3.1) is multiplied by _½_ because of the _½_ outside of all the parentheses of the original Eq.(3.1). This gives us the following: _Vx'/(V²_ _– v²)_ = _1x'_ \+ _1x'/(V – v)_. Subtracting _Vx'/(V²_ _– v²)_ from each side and finding a common denominator for the fractions gives us the following: _0_ = _1x'_ \+ _vx'/(V²_ _– v²)_. If we assume that the term _vx'/(V²_ _– v²)_ is a term denoting time, which it is, and therefore arbitrarily replace the term _x'_ with the term _t_ , the result is the following: _0_ = _1x'_ \+ _vt/(V²_ _– v²)_. If we further assume _τ(x', t)_ = _1x'_ \+ _vt/(V²_ _– v²)_ , we can find the partial derivatives _∂τ/∂x'_ and _∂τ/∂t_. They are the following: _∂τ/∂x'_ = 1 and _∂τ/∂t_ = _v/(V²_ _– v²)_. This outcome is not the result we desired, but the process does highlight some of the difficulties in following A. I. Miller's strategy.

If we expand both sides of Eq.(3.1) as a series in _x'_ , how do we determine the precise form of that series? Is the Eq.(3.1) expanded into two series or three series? The series should include the term _t_ with a coefficient that is similar to _v/(V²_ _– v²)_ , but this does not appear to be possible. Somehow the series should be set equal to the function _τ(x', t)_ , but this does not appear to be possible. Somehow the terms _∂τ/∂x'_ and _∂τ/∂t_ should appear in the same equation, but this does not appear to be possible.

A. I. Miller may be referring to a Taylor's series when he uses the phrase _expanded both sides of Eq. (3.1) as a series in x'_. A Taylor's series is a complicated mathematical device that could produce from Eq. (3.1) an equation that does bear some resemblance to Eq. (3.2). Without going into a great deal of complicated mathematics we can demonstrate Einstein is not using a Taylor's series or at least not correctly using a Taylor's series. First, we need to define Taylor's theorem in order to understand a Taylor's series. According to the _Mathematics Dictionary_ Taylor's theorem is, "A theorem which defines a polynomial whose graph runs very close to that of a given function throughout a certain interval, and a remainder which supplies a numerical limit to the difference between the ordinates of the two curves; the approximate representation of a given function on a certain interval (in the neighborhood of a certain point) by a polynomial."49 Taylor's theorem is the approximate representation of a given function. Taylor's theorem and Taylor's series can be extended to functions of any number of variables. Taylor's theorem for a function of two variables does bear some resemblance to the results produced by Einstein.

The definition of Taylor's series dispels any notion that Einstein could use it to obtain his results. The _Mathematics Dictionary_ gives the following definition of Taylor's series, "If _n_ be allowed to increase without limit in the polynomial obtained by _Taylor's theorem,_ the result is called a Taylor's series. The sum of such a series represents the expanded function if, and only if, the limit of _R_ n as _n_ becomes infinite is zero."50 The key is that only the _sum_ of a Taylor's series can represent a given function, not the first member of the series. The important point of the quotation is the following: The _sum_ of such a series represents the expanded function.

Step 4: Return of the Misapplication of Mathematical Rules

We now have completed our analysis of Eqs. (3.1) and (3.2), except for the fact that Einstein utilizes two other thought experiments that are nearly identical to the one we have just analyzed _._ He introduces the two other thought experiments directly following Eq. (3.2). To reiterate, both of these thought experiments are similar to the one already described. They both take place on the moving system _k_ , and the moving system _k_ is moving along the _X_ -axis in the direction of increasing _x_ of the rest system _K_. Both thought experiments involve a beam of light traveling from a starting point to a mirror and back to the starting point. In one thought experiment the light beam travels along the _η-axis_ , and in the other thought experiment it travels along the _ζ-axis_. When the light beam travels along either the _η-axis_ or the _ζ-axis_ , it is observed from the rest system _K_. The velocity of the light beam when observed from the rest system is _(V²_ _– v²)_ ½. The reason for this is that although the light beam is traveling along either the _η-axis_ or the _ζ-axis_ it does not appear this way to the observers in the rest system.

The observers in the rest system observes the light beam originating from a starting point on the _X_ -axis and traveling diagonally to a point on either the _η-axis_ or _ζ-axis_. Since the _X-axis_ of the resting system _K_ and the _ξ-axis_ of the moving system _k_ coincide, the starting point is both _x_ and _x'_ = _x –_ _vt_. The velocity with which the light beam appears to travel along the diagonal line is _V_ , when observed from the rest system. The diagonal line forms the hypotenuse of a right triangle. The segment of either the _η-axis_ or the _ζ-axis_ from the starting point in the moving system _k_ to the location of the mirror forms one leg of the right triangle. The other leg of the right triangle is the segment of the _X-axis_ from the starting point in the resting system _K_ to the point on the _X-axis_ where either the _η-axis_ or the _ζ-axis_ intersect the _X-axis_ at the moment the light beam strikes the mirror. Thus using the Pythagorean Theorem, Einstein concludes, "that light always propagates along these axes with the velocity _(V²_ _– v²)_ ½ when observed from the rest system."51

The reasoning seems to be forced or, at least, inconsistent when we recall Einstein's second postulate. The speed of light is constant and independent of the velocity of the source of the light. An observer on the rest system _K_ observes, for instance, a light beam traveling "up" the _η-axis_ of the moving system _k_. The observer mistakenly assumes the light beam is traveling along a line directed diagonally upward from the rest system with the velocity _V._ The observer calculates that the vertical or "up" component of the light beam's velocity is _(V²_ _– v²)_ ½ _._ The observer makes this mistaken assumption for every light beam he observes traveling "up" the _η-axis._ Therefore, a light beam travels "up" the _η-axis_ with the velocity _(V²_ _– v²)_ ½, at least according to our mistaken observer.

Perhaps, it is misleading to refer to the observer as a "mistaken observer." It may be more accurate to say the observer observes a triangular sheet of light or a wide light beam—a composite light beam formed from many narrow light beams. The observer incorrectly interprets this composite light beam as a single diagonal light beam.

Einstein applies the same kind of reasoning that he applied to a beam of light making a round-trip journey along the _ξ-axis_ of the moving system _k_ , to a beam of light making a round-trip journey either along the _η-axis_ or _ζ-axis_ of the moving system _k._ He states, "Analogous reasoning —applied to the _η_ and _ζ_ axes—yields, remembering that light always propagates along these axes with the velocity _(V²_ _– v²)_ ½ when observed from the rest system, _∂τ/∂y_ = _0_ and _∂τ/∂z_ = _0."_ 52

Einstein goes into no greater detail to explain the derivation of the partial derivatives listed above. Arthur I. Miller's _Albert Einstein's Special Theory of Relativity_ does provide a more detailed explanation of their derivation. He rewrites Eq. (3.1) so that it represents a light beam making a round-trip journey along the _η-axis_ of the moving system _k._ The round-trip journey of the light beam is viewed from the rest system _K._ The light beam begins its journey when system _k_ and system _K_ are superposed on each other, and the light beam begins its journey from the coordinate _(0, 0, 0, t)_. The light beam travels along the _η-axis_ until it strikes a mirror at a distance of _y'_ from the starting point. Since the _y-axis_ of _K_ is parallel to the _η-axis_ of _k_ , the distance _y'_ = _y._ When the light beam is observed from the _K_ system its velocity is _(V²_ _– v²)_ ½ so the duration of the first leg of its journey is calculated by dividing the distance the light beam travels by the apparent speed of the light beam or _y/(V²_ _– v²)_ ½. Likewise, the total duration of the light beam's journey is _2y/(V²_ _– v²)_ ½. When observed from system _K_ the light beam not only travels along the _η-axis_ of the _k_ system, it also has a component of motion along the _X-axis_ of the _K_ system because system _k_ is in motion along the _X-axis_. For the first leg of the light beam's journey, the distance it travels along the _X-axis_ is the velocity of system _k_ multiplied by the duration of the first leg of the journey or _v_ • _[y/(V²_ _– v²)_ ½ _]_. Likewise, for the light beam's round-trip journey, the distance it travels along the _X-axis_ is _v_ • _[2y/(V²_ _– v²)_ ½ _]_.

Therefore, Eq.(3.1) can be reformulated to describe the round-trip journey of a light beam along the _η-axis_ of the moving system _k_ as follows: _τ[ vy/(V²_ _– v²)_ ½ _, y, 0, t_ \+ _y/(V²_ _– v²)_ ½ _]_ = _½[ τ(0, 0, 0, t)_ \+ _τ(2 vy/(V²_ _– v²)_ ½ _, y, 0, t_ \+ _2y/(V²_ _– v²)_ ½ _]_. Arthur I. Miller states, "The result of an expansion of Eq.(6.17) [the equation above] for _y_ considered as infinitesimal is _∂τ/∂y_ = _0."_ 53 For the reasons given before this does not seem likely. The reasons were put forth in "Step 3: A Change in Direction" in which we analyzed the derivation of Eq. (3.2) from Eq. (3.1).

If we apply _Rule B_ to the left side of the equation and _Rule A_ to the right side of the equation, we can obtain results very similar to Einstein's results.

We must rewrite the left side of the equation as follows: _τ[vy/(V²_ _– v²)_ ½ _, y, 0, t_ \+ _y/(V²_ _– v²)_ ½ _]– τ(0, 0, 0, t)._ We ignore the _x_ and _z_ coordinates (noting that the _x_ coordinate is _vy/(V²_ _– v²)_ ½ ). Next, we rewrite the _y_ coordinate of the first term—the term within the _[ . . . . ]_ parentheses— as _0_ \+ _y_ and apply _Rule B_ , which gives us the following: _τ(0_ \+ _y, t_ \+ _y/(V²_ _– v²)_ ½ _) – τ(0, t) ≈ ∂τ/∂y(0, t)_ • _y_ \+ _∂τ/∂t (0,t)_ • _y/(V²_ _– v²)_ ½ _._

Now, we turn our attention to the right side of the reformulated Eq. (3.1). We must rewrite the right side of the reformulated Eq. (3.1) as follows: _½[τ(2 vy/(V²_ _– v²)_ ½ _, 0, 0, t_ \+ _2y/(V²_ _– v²)_ ½ _) – τ(0, 0, 0, t)]._ We ignore the _x_ and _z_ coordinates (noting the _x_ coordinate is _2 vy/(V²_ _– v²)_ ½ ). We apply _Rule A_ , which gives us the following: _½[τ(0, t_ + _2 y/(V²_ _– v²)_ ½ _– τ(0, t)] ≈ ½ [∂τ/∂t(0, t)_ • _2y/(V²_ _– v²)_ ½ _]._

Next, we recombine the approximation of the left side of the equation and the approximation of the right side of the equation, which gives us the following: _∂τ/∂y(0, t)_ • _y_ \+ _∂τ/∂t (0,t)_ • _y/(V²_ _– v²)_ ½ _≈ ∂τ/∂t (0,t)_ • _y/(V²_ _– v²)_ ½. We should note that the first term of the approximation of the right side of the equation was multiplied by _½_ , and thus, the 2 (from the quantity _2y/(V²_ _– v²)_ ½ )and the _½_ itself disappear from the right side of the formula. We now subtract _∂τ/∂t (0,t)_ • _y/(V²_ _– v²)_ ½ from each side, which gives us the following: _∂τ/∂y(0, t)_ • _y_ ≈ _0._ We assume that _y_ does not equal zero because it represents the distance the light beam travels on the first leg of its journey, and therefore _∂τ/∂y(0, t)_ ≈ _0._ As before, our result is similar to Einstein's result of _∂τ/∂y_ = _0_.

The same considerations can be applied to a light beam traveling along the _ζ-axis_ of the moving system _k_ , and the result is _∂τ/∂z(0, t)_ ≈ _0_ , which again is similar to Einstein's result of _∂τ/∂z_ = _0._

The conclusion of the preceding analysis presented in Steps 2, 3, and 4— very lengthy though parts of it were—can be stated quite succinctly. The three partial derivative equations, _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)_ • _∂τ/∂t_ = _0, ∂τ/∂y_ = _0,_ and _∂τ/∂z_ = _0_ , are invalid. Einstein incorrectly derived the first partial derivative equation from Eqs. (3.1) and (3.2). Einstein incorrectly derived the second and third partial derivative equations from equations analogous to Eqs. (3.1) and (3.2). We can now turn our attention to Einstein's further interpretation of these three equations.

Step 5: More Strange Equations

According to Einstein, the following three equations _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)_ • _∂τ/∂t_ = _0, ∂τ/∂y_ = _0,_ and _∂τ/∂z_ = _0_ , are the partial derivatives of a specific equation. Einstein claims the above equations "yield, since _τ_ is a _linear_ function, _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ , where _a_ is a function _φ(v)_ as yet unknown, and where we assume for brevity that at the origin of _k_ we have _t_ = _0_ when _τ_ = _0._ "54 The function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ is denoted as Eq. (3.4). We should note some of the differences between _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)∂τ/∂t_ = _0_ and _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ _._ The term + _v/(V²_ \+ _v²)_ from the partial derivative equation undergoes a transformation, and it reappears in the linear function as the term – _v/(V²_ − _v²)._ It changes from a positive to a negative term. Also, in a change that is more difficult to account for, the term _v_ 2, which we find in the denominator of + _v/(V²_ \+ _v²)_ changes from a ( _+v_ 2 ) into a (− _v_ 2 ). In the partial derivative equation, the term + _v/(V²_ \+ _v²)_ is multiplied by the partial derivative _∂τ/∂t_ while in the linear equation the term – _v/(V²_ − _v²)_ is multiplied by _x'_ .

It is not clear that an equation with four independent variables can be considered a linear equation. We should recall that Einstein described _τ_ as a function of the independent variables _x', y, z,_ and _t,_ which can be expressed _τ(x', y, z, t)._

In fact it is difficult to conceive that an equation with as few as two variables, _ƒ(x, y),_ could be a linear equation. The equation _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ can be considered a function of two independent variables _x'_ and _t_. We can make this claim because in the last paragraph of "Section Three" of _On the Electrodynamics of Moving Bodies_ Einstein determines that _a_ is equal to one, and thus, it is no longer represents the function _φ(v)_ with the independent variable _(v)._ For now though _a_ is still a function _φ(v)_ as yet unknown. The claim that _τ(x', t)_ is a linear function is similar to claiming that a function with one variable could refer to a single point. For example, the linear equation _y_ = _3_ has a _y_ value of _3_ for every value of _x_ found on the _X-axis._ For every value of _x_ on the _X-axis_ we must raise perpendicularly three units to find the value of _y_ , and thus, a line is generated. The authors of _Calculus and Its Applications_ write, "A function _ƒ(x, y)_ of two variables may be graphed in a manner analogous to that for functions of one variable. It is necessary to use a three-dimensional coordinate system, where each point is identified by three coordinates _(x, y, z)._ For each choice of _x, y,_ the graph of _ƒ(x, y)_ includes the point _(x, y, ƒ(x, y))._ The graph of _ƒ(x, y)_ is thus a surface in three-dimensional space."55 Thus for each choice of the coordinates _(_ _t, x' )_ the graph of _τ(t, x')_ would include the point _(t, x', τ(t, x'))_ and so it would be a surface in three-dimensional space and not a line _._

The preceding analysis is flawed, but it shows just how much care must be taken when analyzing Einstein's words. The preceding analysis applies to the term _linear equation_ and is unlikely to apply to the term _linear function_ unless we are using linear function in an informal sense. It is unclear if Einstein was deliberately trying to confuse his readers. Perhaps his elliptical and confusing style was the natural outcome of the revolutionary character of his ideas.

The correct interpretation of the phrase _linear function_ is given by Glen James and Robert James in their _Mathematics Dictionary._ Their entry for the phrase _linear function_ refers the reader to the second definition of _linear transformation_. The following definition employs the mathematical term for summation, which is denoted by the uppercase Greek letter sigma. Since many e-book platforms cannot reproduce the Greek letter sigma, it has been replaced with the phrase _[the summation of the following term from i to n]._

"A transformation which takes _ax_ \+ _by_ into _ax'_ \+ _by'_ for all _a_ and _b_ if it takes vectors _x_ and _y_ into _x'_ and _y'_. It is sometimes also required that the transformation be continuous. Here _x_ and _y_ may be vectors in n-dimensional Euclidean space or in vector space, or in particular they may be ordinary real or complex numbers. The numbers _a_ and _b_ may be real, complex, or of any field for which multiplication with elements of the vector space is defined. In an n-dimensional vector space (or Euclidean space of dimension n), such a transformation is _y_ i = _[the summation of the following term from i to n]a_ i j _x_ j _( i= 1, 2, 3, . . .,n)_ or _y_ = _Ax_ , where _x_ and _y_ are one-column matrices (vectors) with elements _( x_ 1 _, x_ 2 _, . . . , x_ n _,)_ and _(y_ 1 _, y_ 2 _, . . . , y_ n _,), A_ is matrix _(a_ i j _),_ and multiplication is matrix multiplication."56

It does not seem likely that any type of linear transformation could generate the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ from the three partial derivative equations, even if the three partial derivative equations first underwent some form of integration. However, there are some superficial similarities. The term _a_ from the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ is superficially similar to the terms _a_ i j and _A_ from the equations _y_ i = _[the summation of the following term from i to n]a_ i j _x_ j _( i= 1, 2, 3, . . .,n)_ and _y=Ax,_ respectively. The similarities are the use of the letter _a_ and the placement of the letter _a_ at the beginning of the right hand side of their respective equations. The terms _a_ i j and _A_ are two different terms for the same matrix, and the term _a_ is not a matrix.

There are other superficial similarities between the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ and linear functions, i.e., linear transformations. For example, linear transformations are often denoted by the term _T,_ which is superficially similar to the term _τ_ from the above function. If we carefully examine an example from Tom M. Apostol's _Linear Algebra_ we will see the coefficients of certain variables switch their alliance to other variables. This is superficially similar to the fact that the term _\+ v/(V² +_ _v²)_ is multiplied by the partial derivative _∂τ/∂t_ in the partial derivative equation while in the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ the term _− v/(V² −_ _v²)_ is multiplied by _x'_. The coefficient _\+ v/(V² +_ _v²)_ seems to switch from the variable _∂τ/∂t_ to the variable _x'_ ( after undergoing a subtle transformation itself).

To make T. M. Apostal's example easier to follow, we must understand that the basis elements _i_ is associated with the variables x1 and y1, and the basis element _j_ is associated with the variables x2 and y2.

"Find a matrix representation relative to the basis _(_ _i_ _,_ _j_ _)_ of the linear transformation _T_ : **R** ² → **R** ² that maps (1, 1) onto (1, ½), and maps (1, −1) onto (½, 1). From the given information we can determine the action of _T_ on the basis elements _i_ and _j_. The given information tells us that _T(_ _i_ \+ _j_ _)_ = _i_ \+ _½_ _j_ and _T(_ _i_ − _j_ _)_ = _½_ _i_ \+ _j_. By linearity we have _T(_ _i_ \+ _j_ _)_ = _T(_ _i_ _)_ \+ _T(_ _j_ _)_ and _T(_ _i_ − _j_ _)_ = _T(_ _i_ _)_ − _T(_ _j_ _)_ , so the given information implies _T(_ _i_ _)_ \+ _T(_ _j_ _)_ = _i_ \+ _½_ _j_ and _T(_ _i_ _)_ − _T(_ _j_ _)_ = _½_ _i_ \+ _j_. By adding and then subtracting these two equations we find _T(_ _i_ _)_ = _¾_ _i_ + _¾_ _j_ and _T(_ _j_ _)_ = _¼_ _I_ − _¼_ _j_. Using the coefficients of _i_ and _j_ as columns, we find that the matrix of _T_ [is a square matrix consisting of two rows and two columns. The first row consists of the terms 3/4 and 1/4. The second row consists of the terms 3/4 and -1/4. [It would be convenient to express these terms in their familiar matrix formulation, but many e-book platforms cannot reproduce matrixes.]

The corresponding linear equations describing _T_ are y1 = ¾x1 \+ ¼x2 and y2 = ¾x1 − ¼x2."57

When we carefully compare linear equations relating the components to the linear equations describing _T_ we see certain coefficients have switched positions. We must recall that the basis element _i_ is associated with the variables x1 and y1. The basis element _j_ is associated with the variables x2 and y2. We compare the following equations: _T(_ _i_ _)_ = _¾_ _i_ + _¾_ _j_ and y1 = ¾x1 \+ ¼x2. The coefficient of both _i_ and x1 is ¾, but the coefficients of _j_ and x2 are ¾ and ¼ respectively. The coefficient of x2 has switched. The coefficient of x2 is the coefficient of _i_ from the other linear equation that relates these components, which is _T(_ _j_ _)_ = _¼_ _i_ − _¼_ _j_. Next, we compare the following equations: _T(_ _j_ _)_ = _¼_ _i_ − _¼_ _j_ and y2 = ¾x1 − ¼x2. The coefficient of both _j_ and x2 is −¼, but the coefficients of _i_ and x1 are ¼ and ¾ respectively. The coefficient of x1 has switched. The coefficient of x1 is the coefficient of _j_ from the other linear equation that relates these components, which is _T(_ _i_ _)_ = _¾_ _i_ + _¾_ _j_ .

This switching of the coefficients in the above example is superficially similar to the switching of the coefficients between the partial derivative equation _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)∂τ/∂t_ = _0_ and the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]._ The above example has several differences from the partial derivative equations that transform into the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]._ The partial derivative equations are not in the form of linear equations relating the components of a linear transformation. They are not similar to _T(_ _i_ _)_ = _¾_ _i_ + _¾_ _j_ and _T(_ _j_ _)_ = _¼_ _I_ − _¼_ _j._ In order to get certain coefficients to switch alliances in the above example a pair of linear equations relating the components of a linear transformation were needed. This pair was used to produce a matrix that in turn produced a pair of linear equations describing the transformation _T._ Einstein produces only one function, _τ = a[t – v/(V² −_ _v²)_ • _x' ],_ from the three partial derivative equations _._ If the partial derivative equations were manipulated in some manner to produce a matrix, the matrix would need at least 2 rows and 2 columns to produce the switching of certain coefficients. A matrix of this kind would yield two functions not the one function produced by Einstein.

We can now turn our attention from linear transformations and investigate the connection between the function _φ(v)_ and multiple integration. We recall that Einstein defined the term _a_ from the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ as a function of _φ(v),_ but when he introduced the function _φ(v)_ he did not immediately define it. The term _φ(v)_ has a superficial similarity to a term used in the process of multiple integration. The _Mathematics Dictionary_ refers to the multiple integral as the iterated integral and provides the following definition. Since many e-book platforms cannot reproduce either the symbol that indicates the calculation of the indefinite iterated integral or the symbol that indicates the calculation of the indefinite integral, these symbols have been replaced with the phrases _[the indefinite iterated integral of]_ and _[the indefinite integral of],_ respectively.

"An indicated succession of integrals in which integration is to be performed first with respect to one variable, the others being held constant, then with respect to a second, the remaining ones being held constant, etc.; the inverse of successive partial differentiation, if the integration is indefinite integration. . . . The iterated integral, _[the indefinite iterated integral of] xydydx,_ may be written _[the indefinite integral of]{ [the indefinite integral of]x y dy} dx._ Integrating the inner integral gives _{ ½ x y² + C_ 1 _}_ where _C_ 1 is any function of _x,_ only. Integrating again gives _¼ x² y²_ + _[the indefinite integral of]C_ 1 _dx_ \+ _C_ 2 where _C_ 2 is any function of _y_. The result may be written in the form _¼ x² y²_ + _φ_ 1 _(x)_ \+ _φ_ 2 _(y) ,_ where _φ_ 1 _(x)_ and _φ_ 2 _(y)_ are any differentiable functions of _x_ and _y_ respectively."58

The term _φ_ _(v)_ has a superficial similarity to terms _φ_ 1 _(x)_ and _φ_ 2 _(y)_ used in multiple integration. If Einstein was using multiple integration, the terms _φ_ 1 _(x')_ and _φ_ 2 _(t)_ would seem to be the proper terms to use _, instead of φ_ _(v)._ Although, perhaps, using the term _φ(v)_ is Einstein's profound way of saying distance _(x')_ and time _(t)_ are functions of velocity or stated another way the measurement of distance and time are relative to velocity at which one is traveling.

We can conclude that the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ has superficial similarities with the equations employed in linear transformations, and it also has superficial similarities with some of the terms employed in multiple or iterated integration.

If we calculate the partial derivatives of the function _τ = a[t – v/(V² −_ _v²)_ • _x' ]_ , they do not match the three partial derivative equations that, according to Einstein, are integrated in some fashion to yield Eq. (3.4). The partial derivatives of Eq. (3.4) are _∂τ/∂t_ = _a_ and _∂τ/∂x' =av/(V²_ \+ _v²)_. Since Eq. (3.4) does not contain the variables _y_ and _z,_ is it legitimate to conclude _∂τ/∂y_ = _0, and ∂τ/∂z_ = _0?_

However, calculating the partial derivatives of the following ungainly and highly questionable equation, _τ_ = _[–v/(V²_ \+ _v²)∂τ/∂t_ • _x'] – [v/(V²_ \+ _v²)_ • _∂τ/∂x'_ • _t]_ would yield the three equations. The two partial derivatives on the right side of this equation should be treated as constants when calculating the partial derivatives of the equation. Also, for convenience the first of Einstein's three partial derivative equations, _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)_ • _∂τ/∂t_ = _0_ , should be rewritten as two separate equations, _∂τ/∂x'_ = – _v/(V²_ \+ _v²)_ • _∂τ/∂t_ and _∂τ/∂t_ = _v/(V²_ \+ _v²)_ • _∂τ/∂x'_. We should recall that these three partial derivative equations were integrated in some fashion and underwent linear transformations, and thus, according to Einstein, produced Eq. (3.4). Although Einstein has not given us a specific value for the function _a_ = _φ(v)_ , at this point in his paper, it is unlikely the partial derivatives of Eq. (3.4) can be made to match the partial derivative equations from which Einstein claims it was generated by some form of integration and linear transformation.

The fact that two partial derivatives occur in the same equation i.e., _∂τ/∂x'_ \+ _v/(V²_ \+ _v²)∂τ/∂t_ = _0_ , is very strong evidence that Einstein is using _Rules A_ and _B._ Normally, when calculating the partial derivatives of an equation with several variables there is a _separate_ partial derivative equation for each variable. The form it takes will be the following: a single partial derivative sign occurs on the left side of the equation, such as _∂τ/∂x',_ and the right side of the equation consists of the derivative of the variable denoted on the left side of the equation, in our example it would be the derivative of the variable _x'_ in whatever form it occurred such as _4y² x'³_ while all the other variables in the equation would be treated as constants.

Step 6: The Transformation Equations

We will now turn our attention to four equations that are called the transformation equations. According to Einstein, all four of these equations can be produced using Eq. (3.4), in combination with several simple equations that describe the distance a beam of light travels as it makes three separate journeys. A light beam makes a separate journey along each of the three axes of the moving system _k_. Also, we should keep in mind that Eq. (3.4) allows us to transform the time values, _τ_ , of the moving system _k_ into the time values, _t_ , of the rest system _K._ It should be noted that from the outset we are dealing with a special case. We are not dealing with an object that could be moving with any given velocity, which would be represented by _v;_ instead we are concerned with a light beam with the velocity _V._ There are other special conditions as well.

Einstein claims that Eq. (3.4), _τ_ = _a[ t_ – _v/(V²_ _– v²)_ • _x']_ , can be transformed into Eq. (3.11), _τ_ = _φ(v)β(t – vx/V²)_ , where _β_ = _1/(1 – v²_ _/V²)_ ½ and _x'_ = _x –_ _vt_ and _a_ = _φ(v)_ ; however, this claim is false. Eq. (3.11) is the first of the four transformation equations. Eq. (3.11) transforms the time values of the rest system _K,_ which are represented by _t_ into the time values of the moving system _k,_ which are represented by _τ._ And, it does this in a way such that only the coordinates of the rest system _K_ appear on the right side of the equation. In other words, _x'_ in Eq. (3.4) is replaced with _x − vt_. The remaining three transformation equations transform the spatial coordinates _x, y,_ and _z_ of the rest system _K_ into their respective counterparts in the moving system _k,_ which are _ξ, η,_ and _ζ._

The error in the transformation of Eq. (3.4) into Eq. (3.11) occurs when the square root is taken of only one side of the equation and not the other. For clarity we should perform the entire calculation. It will be helpful to note that _β_ can be expressed in the following forms: _β_ = _1/(1 – v²_ _/V²)_ ½ or _β_ = _[V²/(V²_ _– v²)]_ ½ or _β_ = _V/(V²_ _– v²)_ ½ . We begin with _τ_ = _a[t – v/(V²_ − _v²)_ • _x' ]_ and since _x'_ = _x –_ _vt_ , we have _τ_ = _a[t – v(x– vt) /(V²_ − _v²)]_. Next, we find a common denominator for the terms inside the parentheses _[_. . . _]_ , which is _(V²_ − _v²)_. This gives us _τ_ = _a[(V²t – v²t −vx + v²t)/(V²_ − _v²)]_ and we find the terms _vt²_ and – _vt²_ cancel out giving us _τ_ = _a[(V²_ _t – vx)/(V²_ − _v²)]_. Now, in a subtle manipulation, we manufacture a common factor of _V²/(V²_ − _v²)_ , which acts as a common factor for the terms in the numerator of the fraction, despite the fact that they are _V²t_ and _−vx_. The manufactured term acts as a common factor for the terms in the numerator although the terms in the numerator do not actually share a common factor. This gives us _τ_ = _a[V²_ _/(V² −_ _v²)_ • _{t – (vx)/V²}]_. Next, we rewrite the two terms inside _{. . . .}_ as a single fraction _(_ _tV² – vx)/V² ,_ and then we isolate the manufactured common factor on the right side of the equation, which gives us _(τ/a)_ • _[V²/(tV²_ _–vx)]_ = _V²/(V²_ _– v²)_. Now, we incorrectly take the square root of only the right side of the equation, which gives us _(τ/a)_ • _[V²/(tV²_ _–vx)]_ = _[V²/(V²_ _– v²)]_ ½. Next, we isolate the term _τ_ on the left side of the equation, and replace the second form of _β, [V²/(V²_ _– v²)]_ ½ _,_ with the third form of _β , V/(V²_ _– v²)_ ½ _,_ which gives us _τ_ = _a[V/(V²_ − _v²)_ ½ • _(t – vx/V²)]_. It should be noted the two terms inside the curved parentheses, _(. . . .)_ , that were rewritten as a single fraction _(tV² – vx)/V²_ now resume their original form as the two terms _(t – vx/V²)_. Now, since _a_ = _φ(v)_ and _β_ = _V/(V²_ _– v²)_ ½ _,_ we generate Einstein's Eq. (3.11) _, τ = φ(v)β(t – vx/V²)._

Einstein makes the same mistake with Eq. (3.12), _ξ_ = _φ(v)β(x – vt)._ Eq. (3.12) is the second transformation equation. It transforms the spatial coordinate _x_ of the rest system _K_ into the spatial coordinate _ξ_ of the moving system _k._ Again, Einstein's mistake is that he takes the square root of one side of the equation but not the other. First, we should explain the derivation of Eq. (3.12). Einstein writes, "For a light ray emitted at the time _τ_ = _0_ in the direction of increasing _ξ_ , we have Eqs. (3.5), _ξ_ = _Vτ_ , or _ξ_ = _aV[t – v/(V² – v²)_ • _x']_. But as measured in the rest system, the light ray propagates with velocity _V – v_ relative to the origin of _k,_ so that _x' /(V – v)_ = _t._ Substituting this value of _t_ in the equation for _ξ_ , we obtain Eq. (3.7), _ξ_ = _a[V²/(V² – v²)]_ • _x'."_ 59 This much is correct, but note that the term _V²/(V² – v²)_ it is equal to _β²_ not _β._ Einstein concludes by claiming that if we substitute _(x – vt)_ for _x'_ we will obtain _ξ_ = _φ(v)β( x – vt ),_ but actually we obtain this equation _ξ_ = _φ(v)β²( x – vt )_. Because, as we just noted, _V²/(V² – v²)_ it is equal to _β²_ not _β._

A more detailed description of the erroneous calculation will be helpful. Eq. (3.5), _ξ_ = _Vτ_ , is a version of the equation distance equals velocity multiplied by time. Since _τ_ = _a[ t_ – _v/(V²_ _– v²)_ • _x']_ , the equation _ξ_ = _Vτ_ can be rewritten as _ξ_ = _aV[ t_ – _v/(V²_ _– v²)_ • _x']._ The term _t_ in the equation _ξ_ = _aV[t – v/(V² – v²)_ • _x']_ can be replaced with _x'/(V – v)_ because, as measured from the rest system, the light beam travels with velocity _(V –_ _v)_ relative to the origin of _k_ , and the distance it travels is _x'_ so that _x'/(V – v)_ = _t_ , Eq. (3.6). Substituting this value of _t_ in the equation _ξ_ = _aV[t– v/(V² – v²)_ • _x']_ gives us the following: _ξ_ = _a[V²/(V² – v²)]_ • _x'._

The substitution of the left side of Eq. (3.6), which is _x'/(V – v),_ for the term _t_ in Eq. (3.5) is strikingly fortuitous because of the way it simplifies the equation. The denominator of Eq. (3.6), _(V – v),_ is one of the two factors of the denominator of Eq. (3.5), _(V² – v²)_. The other factor is _(V_ \+ _v)._ With the substitution for _t,_ Eq. (3.5) becomes _ξ_ = _aV[x'/(V – v) – v/(V² – v²)_ • _x']_. The equation with a common denominator for the terms inside the _[. . . . ]_ parentheses is _ξ_ = _aV[Vx'_ \+ _vx' – vx']/(V² – v²)_. The terms + _vx'_ and – _vx'_ cancel out and multiplying the _V_ outside the parentheses times _V_ inside the parentheses gives us _ξ_ = _a[V²_ _/(V² – v²)_ • _x']_. We should note this simplification is only possible because Einstein chose to analyze a light beam heading in the direction of increasing _ξ._ If Einstein had analyzed a light beam heading in the opposite direction, that is to say heading in the direction of decreasing _ξ_ , the results would be different. Its velocity as measured from the rest system _K_ relative to the origin of the moving system _k_ would be _(V_ \+ _v)_. In this case, we would not have the terms + _vx'_ and − _vx'._ Instead, we would have the terms − _vx'_ and − _vx'_ , which do not cancel each other out.

To obtain Eq. (3.12) we need to isolate the term _V²_ _/(V² – v²)_ on one side of the equation because the term _V²_ _/(V² – v²)_ is equal to _β²_. Next, we incorrectly take the square root of only that side of the equation, which yields _ξ /(ax')_ = _[V²/(V² – v²)]_ ½ or rewritten using the third version of _β_ , _ξ /(ax')_ = _V/(V² – v²)_ ½. Now, we isolate _ξ_ on one side of the equation by multiplying each side of the equation by _ax'_. Since _a_ = _φ(v)_ and _β_ = _V/(V² – v²)_ ½ and x' = _x –_ _vt_ , we obtain Eq. (3.12), _ξ_ = _φ(v)β(x – vt)_.

Einstein makes a mistake that involves the term _β_ in both Eq. (3.13), _η_ = _φ(v)y_ , and Eq. (3.14), _ζ_ = _φ(v)z_. Eqs. (3.13) and (3.14) are the third and fourth transformation equations respectively. Eq. (3.13) transforms the _y_ coordinates of the rest system _K_ into the _η_ coordinates of the moving system _k_. Eq. (3.14) transforms the _z_ coordinates of the rest system _K_ into the _ζ_ coordinates of the moving system _k_.

Einstein's mistake is that he completely ignores the term _β._ Also he introduces two special conditions: _x'_ = _0_ and _t_ = _y/(V² – v²)_ ½ . The equation _x'_ = _0_ limits the moving system _k_ so that it travels no distance at all or zero distance. Since moving system _k_ is moving with a constant velocity along the _X-_ axis of the rest system _K_ , to travel zero distance would take zero time. In zero time the light beam traveling along the _η-_ axis would travel zero distance. This is similar to the situation presented in Eq. (3.1); the light beam makes a round-trip journey that is infinitesimally small—the distance it travels is zero. The equation _t_ = _y/(V² – v²)_ ½ is a version of the equation time = distance/velocity. According to Einstein, as measured from the rest system _K,_ the height to which the light beam rises is y. We should recall Einstein states that a light beam traveling along the _η-_ axis in the direction of increasing _η_ in the moving system _k_ will be seen by observers in the rest system _K_ to have originated in the rest system _K_ and to travel along an upward moving diagonal path. Thus, according to Einstein, the measure of the component of velocity that the light beam displays along the _η-_ axis as observed from the rest system _K_ is _(V² – v²)_ ½. Because of the ambiguous nature of the scenario Einstein has set up, we can say either the component of velocity that the light beam displays along the _η_ -axis or the velocity the light beam displays along the _η_ -axis.
It will be helpful to explain the derivation of Eqs. (3.13) and (3.14). Einstein refers to Eq. (3.5), _ξ_ = _Vτ_ and writes, "Analogously, by considering light rays moving along the two other axes, we get Eq. (3.8), _η_ = _Vτ_ = _aV[t – v/(V²_ _– v²)_ • _x']_ , where _y/(V² – v²)_ ½ = _t_ and _x'_ = _0;_ hence [we obtain] Eqs. (3.10), _η_ = _a[V/(V²_ _– v²)_ ½ • _y]_ and _ζ_ = _a[V/(V²_ _– v²)_ ½ • _z]_."60 Since Einstein has stated that _x'_ = _0_ the final term in Eq. (3.8), _v/(V²_ _– v²)_ • _x',_ is eliminated because under Einstein's conditions it equals zero. We are left with the first term _aV[ t ]_ , and since _t_ = _y/(V² – v²)_ ½ we have the following _ξ_ = _a[ V/(V² – v²)_ ½ • _y]._ To obtain Eq. (3.13), _η_ = _φ(v)y_ , from the previous equation we must completely ignore the term _V/(V²_ _– v²)_ ½, which is equivalent to _β_ , and since _a_ = _φ(v)_ , we obtain Eq. (3.13), _η_ = _φ(v)y_. To obtain Eq. (3.14), _ζ_ = _φ(v)z_ , we follow the same procedure. Of course, in this instance _t_ = _z/(V² – v²)_ ½ _._

Arthur I. Miller acknowledges that the four equations we have discussed, Eqs. (3.11), (3.12), (3.13) and (3.14) cannot be generated in the manner Einstein describes, but he does not go so far as to call it a mistake. Arthur I. Miller's explanation is, "Then, without prior warning Einstein replaced _a(v)_ with _φ(v)(1 – v²_ _/ V²_ _)_ ½ to obtain Eqs. (3.11), (3.12), (3.13) and (3.14)."61 It should be noted for clarity that Einstein denoted _a(v)_ as merely _a_ and with only a few exceptions Miller follows Einstein's example. Miller's explanation does explain away Einstein's mistakes, but it is odd that Einstein did not introduce this replacement of _a_ with _φ(v)(1 – v²_ _/ V²_ _)_ ½ himself.

Most likely there is a good reason why Einstein never introduced the notion that suddenly _a_ could be replaced with _φ(v)(1 – v²_ _/ V²_ _)_ ½. Einstein states that _a_ is an unknown function _φ(v)_. All indications are that _a_ was obtained by integration. A quantity obtained by integration cannot shift from _φ(v)_ to _φ(v)(1 – v²_ _/ V²_ _)_ ½. Many readers are familiar with the symbol that indicates we are to find the indefinite integral of a function with a single variable. It is shaped somewhat like a large, italicized letter _f_ without the crosspiece. Many e-book platforms would have difficulty reproducing this symbol. So for the following examples, the phrase _the indefinite integral of_ will be employed instead of the more familiar symbol indicating we are to calculate the integral of a function. _The indefinite integral of 2x dx_ = _x²_ \+ _C_ , and we cannot suddenly claim _the indefinite integral of 2x dx_ = _2x²_ \+ _C_ because _the indefinite integral of 4x dx_ = _2x² + C_. Thus, if _the indefinite integral of ∂τ/∂x'_ = _the indefinite integral of_ – _v/(V²_ \+ _v²)∂τ/∂t_ produces _τ_ = _a[t – v/(V²_ \+ _v²_ _)_ • _x']_ it cannot also produce _τ_ = _a_ _(1 – v²_ _/ V²_ _)_ ½ _[t – v/(V²_ \+ _v²)_ • _x']_.

As we have noted, Einstein refers to the four equations we have discussed as _transformation equations,_ and he uses them to demonstrate that there is no contradiction between a constant velocity of light and the principle of relativity. To demonstrate that there is no contradiction between a constant velocity of light and the principle of relativity he introduces the notion of a spherical wave. An example of a spherical wave of light is the light produced from a match, a candle, or the sun.

"Now we have to prove that, measured in the moving system, every light ray propagates with the velocity _V,_ if it does so, as we have assumed, in the rest system; for we have not yet proved that the principle of the constancy of the velocity of light is compatible with the relativity principle. Suppose that at time _t_ = _τ_ = _0_ a spherical wave is emitted from the coordinate origin, which at that time is common to both systems, and that this wave propagates in the system _K_ with the velocity _V_. Hence, if ( _x, y, z_ ) is a point reached by this wave, we have _x²_ + _y²_ + _z²_ = _V²t²_ , (Eq.3.16). We transform this equation using our transformation equations and, after a simple calculation, obtain _ξ²_ \+ _η²_ \+ _ζ²_ = _V_ ² _τ²_ , Eq. (3.17).Thus, our wave is also a spherical wave with propagation velocity _V_ when it is observed in the moving system. This proves that our two fundamental principles are compatible."62

Step 7: Not So Simple Calculations

Einstein claims that by using his transformation equations and some simple calculation we can transform the equation _x²_ + _y²_ + _z²_ = _V²t²_ into the equation _ξ²_ \+ _η²_ \+ _ζ²_ = _V_ ² _τ²_. Einstein does not provide us with the details of this "simple calculation" in his paper _On the Electrodynamics of Moving Bodies_. But, in "Appendix One" of his book _Relativity: the Special and the General Theory_ he does provide a version of this "simple calculation" that is seven pages in length. One of the errors that invalidate his calculations occurs on the second page so it is worthwhile to reproduce his calculations up to that point. Several small detail should be noted, the term _X-_ axis is denoted as the _x_ -axis and the moving system _k_ is denoted as the moving system _K'_ in the following quotation.

" _Simple Derivation of the Lorentz Transformation"_

"For the relative orientation of the co-ordinate systems indicated in Fig. 2, the _x_ -axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localized on the _x_ -axis. Any such event is represented with respect to the co-ordinate system _K_ by the abscissa _x_ and the time _t_ , and with respect to the system _K'_ by the abscissa _x'_ and _t'_ when _x_ and _t_ are given. A light-signal, which is proceeding along the positive axis of _x,_ is transmitted according to the equation _x_ = _ct_ or _x_ – _ct_ = _0_ . . . . (1). Since the same light-signal has to be transmitted relative to _K'_ with the velocity _c,_ the propagation relative to the system _K'_ will be represented by the analogous formula _x' – ct'_ = _0_ . . . . (2). Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation ( _x'_ – _ct'_ ) = _λ_ ( _x_ – _ct_ ). . . . (3) is fulfilled in general, where _λ_ indicates a constant; for according to (3), the disappearance of ( _x –_ _ct_ ) involves the disappearance of ( _x'_ – _ct'_ ). If we apply quite similar considerations to light rays which are being transmitted along the negative _x_ -axis, we obtain the condition ( _x'_ \+ _ct'_ ) = _μ_ ( _x_ \+ _ct_ ). . . . (4)."63

Equation (4) is incorrect. The correct equation is the following: (– _x'_ \+ _ct'_ ) = _μ_ (– _x_ \+ _ct_ ). Since the light rays, as Einstein writes "are being transmitted along the negative _x-axis_ " of both the _K_ and _K'_ systems the values for _x_ and _x'_ are – _x_ and – _x'_ respectively. As Einstein writes, "A light-signal, which is proceeding along the positive axis of _x,_ is transmitted according to the equation _x_ = _ct_."64 When a light-signal travels along the positive _x-axis_ , the distance the light-signal travels is expressed by the term + _x_. Therefore, when a light-signal travels along the negative _x-axis_ , the distance the light-signal travels must be expressed by the term - _x._ Thus, in the _K_ system a light-signal traveling along the negative _x-axis_ is transmitted according to the equation – _x_ = – _ct_ or – _x_ \+ _ct_ = _0._ In the _K'_ system a light-signal traveling along the negative _x-axis_ is transmitted according to the equation – _x'_ = – _ct'_ or – _x'_ \+ _ct'_ = _0._ Therefore, equation (4) in its correct version is the following: (– _x'_ \+ _ct'_ ) = _μ_ (– _x_ \+ _ct_ ). This error occurs near the beginning of Einstein's demonstration that the equation _x²_ \+ _y²_ \+ _z²_ = _V²t²_ is equivalent to the equation _ξ²_ \+ _η²_ \+ _ζ²_ = _V²τ²_ , and the error invalidates his proof. The term − _ct_ from the equation − _x_ = − _ct_ may seem odd as though a light wave was traveling with a strange negative velocity. The term − _c_ is introduced merely to take into account that the distance the light wave travels is − _x_.

Eight paragraphs further on there is a second error. Interestingly, this second error is similar to one we encountered in our analysis of the transformation equations. Instead of taking the square root of one side of an equation but not the other side, now, Einstein's error is that one side of an equation is squared while the other side is not squared.

To understand Einstein's second error we should begin with Eq. (3), _x'_ − _ct'_ = _λ(x_ − _ct),_ and Eq. (4), _x'_ \+ _ct'_ = _μ(x_ \+ _ct)_ , which, as we have noted before, is incorrect. We must add Eq. (3) to Eq. (4), and then we must subtract Eq. (4) from Eq. (3). The following is the addition of the equations:

x' − c t' = λ(x − c t)

+ _x' + c t' = μ(x + c t)_

2x' = (λ +μ)x − (λ − μ)c t

or

_x' =_ _(λ +μ)/2_ ∙ _x − (λ − μ)/2_ ∙ _c t_

The following is the subtraction of the equations:

x' − c t' =λ(x − c t)

− _x' + c t' = μ(x + c t)_

− _2c t = (λ − μ)x −(λ + μ)c t_

or

_c t' = (−λ + μ)/2_ ∙ _x + (λ +μ)/2_ ∙ _c t_

Now, we let _a_ = _(λ_ \+ _μ)/2_ and _b_ = _(λ_ − _μ)/2_. We substitute _a_ and _b_ into the two equations we have produced by addition and subtraction, and we produce the equations Einstein denotes as Eqs. (5), _x'_ = _ax_ − _bct_ and _ct'_ = _act_ − _bx_.

Einstein reminds us that the coordinate system _K'_ is moving with a constant velocity relative to coordinate system _K_. He also reminds us that at the origin of _K'_ the term _x'_ will always equal zero. If we substitute _0_ for _x'_ in the first of the Eqs. (5), we obtain _0_ = _ax_ − _bct_ or adding _bct_ to each side of the equation _bct_ = _ax._ Solving the equation for _x_ we obtain _x_ = _[(b c)/a] t_. Distance equals velocity multiplied by time. Since _x_ = distance and _t_ = time, the term _(b c)/a_ must equal velocity. Thus, Einstein produces Eq. (6) _v_ = _(b c)/a_. It is the velocity with which the origin of _K'_ is moving relative to _K_. Through further analysis, Einstein designates _v_ as the relative velocity of the two systems.

Next, Einstein introduces the notion of taking a "snapshot" of the coordinate system _K'_ from the coordinate system _K._ The "snapshot" is a kind of thought experiment because Einstein describes it as an instantaneous photograph as opposed to a real photograph. The "snapshot" will allow Einstein to compare the relative lengths of the units of measure on coordinate system _K'_ compared to coordinate system _K._ For instance, two points separated by 1 foot on coordinate system _K'_ will appear to be separated by less than 1 foot in the instantaneous photograph taken of _K'_ from _K._ Also, taking the instantaneous photograph allows Einstein to set _t_ = _0._

Now, we have examined enough of the background of Einstein's second error to understand it more fully.

"Furthermore, the principle of relativity teaches us that, as judged from K, the length of a unit measuring-rod which is at rest with reference to _K'_ must be exactly the same as the length, as judged from _K',_ of a unit measuring-rod which is at rest relative to _K_. In order to see how the points of the _x'_ -axis appear as viewed from _K_ , we only require to take a "snapshot" of _K'_ from _K_ ; this means that we have to insert a particular value of _t_ (time of _K_ ), _e.g. t_ = _0_. For this value of _t_ we then obtain from the first of the equations (5) _x'_ = _ax._ Two points of the _x'-_ axis which are separated by the distance Δ _x'_ = _1_ when measured in the _K'_ system are thus separated in our instantaneous photograph by the distance Δ _x_ = _1/a_. But if the snapshot be taken from _K'_ ( _t'_ = 0), and if we eliminate _t_ from the equations (5), taking into account the expression (6), we obtain _x'_ = _a(1_ – _v²_ / _c²)x_."65

We recall that the Eqs. (5) are the following: _x'_ = _ax –_ _bct_ and _ct'_ = _act –_ _bx_. We follow Einstein's instructions by letting _t'_ = _0_ and letting _t_ = _0_ , as well, since this eliminates _t_ from equations (5). As we just noted, letting _t_ = _0_ allows us to eliminate _t_ from the equations. This gives us the following: _x'_ = _ax –_ _0_ and _0 =_ _0 –_ _bx_. Next, we add the two equations together, and the result is the following: _x' =_ _ax –_ _bx_. We recall that Eq. (6) is the following: _v_ = _bc_ / _a_ , and solving for _b_ gives us _b_ = _av_ / _c_. Thus, we can rewrite _x' =_ _ax –_ _bx_ as _x'_ = _ax –_ _av/c_ • _(x)_ or taking into account the common factor _ax_ the equation becomes _x' =_ _a (1_ – _v/c) x_. To isolate the term _v/c_ on one side of the equation, we divide each side of the equation by _ax_ , next, we subtract 1 from each side of the equation, and then we multiply each side of the equation by (−1). Thus, we obtain _1 – x'/ax_ = _v/c._ Next, we square the right side of the equation, but do not square the terms on the other side of the equation, and we obtain _1 – x'/ax_ = _v²/c²._ To obtain Einstein's equation _x'_ = _a (1− v²/c²)_ • _x_ , of course, we need to rewrite the equation so that the term _x' was_ isolated on one side of the equation. To do this we multiply each side of the equation by −1. Next, we add 1 to each side of the equation. Finally, we multiply each side of the equation by _ax_ , and we obtain _x'_ = _a (1− v²/c²)_ • _x_.

Apart from these two errors Einstein accomplishes his task of demonstrating that the equation _x²_ \+ _y²_ \+ _z²_ = _V²t²_ is equal to the equation _ξ²_ \+ _η²_ \+ _ζ²_ = _V_ ² _τ²_. Although these two errors may seem inconsequential, they are significant enough to invalidate his proof.

Step 8: Another Version of the Not So Simple Calculations

Lillian R. Lieber tries to accomplish the same task in her book _The Einstein Theory of Relativity_. She encounters difficulties similar to those encountered by Einstein. She introduces the imaginary number _i_ = (–1)½ to assist her in her efforts, but it is to no avail. Her terminology is slightly different from the terminology Einstein employed in his paper. She uses the following terms that are different from Einstein's terms: _t' instead_ of _τ_ , _x' instead_ of _ξ, c_ instead of _V_ and Einstein's term _φ (v)_ is entirely absent from her Lorentz transformation equations. (This absence will be fully explained in our analysis of the final portion of the third section of Einstein's paper.) Taking these changes into consideration, her versions of the transformation equations are similar to Einstein's versions. In order for L. R. Lieber to accomplish her task, she must set the speed of light, _c_ , equal to one with apparently no accompanying units of measure. As she writes below this is done to introduce "simplicity" into the equations. L. R. Lieber's use of "simplicity" is similar to Einstein's use of "naturalness" and "logical simplicity."

"Let us examine the similarity between [equation] (20) and the Lorentz transformation a little more closely, selecting from the Lorentz transformation only those equations involving _x_ and _t,_ and disregarding those containing _y_ and _z,_ since the latter remain unchanged in going from one coordinate system to the other. Thus we wish to compare [equation] (20) with: _x' =_ _β(x – vt)_ and _t' =_ _β (t – vx/c²)_. Or, if, for simplicity, we take _c_ = 1, that is, taking the distance traveled by light in one second, as the unit of distance, we may say that we wish to compare [equation] (20) with _x' =_ _β (x – vt)_ and _t' =_ _β (t – vx_ ). . . . (21)" 66

The term _vx/c²_ from the equation _t' =_ _β (t – vx/c²)_ is a measure of time and this is appropriate because it is subtracted from _t,_ which is a measure of time, as well. For instance, _v_ could be measured in _miles/second_ , _x_ could be measured in _miles_ and _c²_ could be measured in _miles squared / seconds squared_. This gives us the complex fraction ( _miles squared/second_ )/ ( _miles squared/seconds squared_ ), which simplifies to seconds, a measure of time. The term _vx_ from equation (21) is not a measure of time. It is a measure of _lengths squared / unit of time_. It would not be appropriate to subtract this term from a measure of time. Lieber continues her demonstration.

"Let us first solve [equation] (21) for _x_ and _t,_ so as to get them more nearly in the form of [equation] 20. By ordinary algebraic operations, we get the following: _x_ = _β(x' + vt')_ and _t_ = _β (t' +vx') . . . ._ (22)"67

There are no ordinary algebraic operations that can produce the two equations above. Solving for _x_ and _t_ we obtain the following: _x_ = _x'/β + vt_ and _t_ = _t'/β + vx._ Notice that when we solve for _x_ and _t_ respectively the final terms in the respective equations are _vt_ and _vx_ as opposed to _vt' and_ _vx'._

Lieber next introduces the square root of negative one into the proceedings written in the form _i_ = (–1)½.

"Let us now return to the comparison of [equations] (22) and (20): Minkowski pointed out that if, in (22), _t_ is replaced by _i_ _τ_ {where _i_ = (–1)½ }, and _t'_ by _i_ _τ',_ then (22) becomes _x_ = _β x'_ _+_ _i_ _βvτ'_ and _i_ _τ_ = _i_ _βτ' + βvx'._ Or (by multiplying the second equation by – _i_ ): _x_ = _βx' +_ _i_ _βvτ'_ and _τ_ = _βτ' –_ _i_ _βvx'_ ."68

The replacement of _t_ and _t'_ with _i_ _τ_ and _i_ _τ'_ respectively is arbitrary. It is a clever way to manufacture formulas where certain inconvenient terms will cancel out, thus allowing the demonstration of the desired point. She is not following the accepted practice of multiplying each side of an equation by the same quantity such as five or seven. Instead she is multiplying a selected term of an equation or selected terms of an equation by _i_.

Also, if we examine the equation _i_ _τ_ = _i_ _βτ' + βvx'_ to see if it complies with the rules which govern the formation of complex numbers, we discover some troubling implications. According to the _Mathematics Dictionary,_ "A complex number is any number, real or imaginary, of the form _a_ \+ _bi_ , where _a_ and _b_ are real numbers and _i²_ = −1. . . . Two complex numbers are defined to be equal if and only if they are identical, _i. e._ , _a_ \+ _bi_ = _c_ \+ _di_ means _a_ = _c_ and _b_ = _d_."69 If we believe that the equation _i_ _τ_ = _i_ _βτ' + βvx'_ equates two complex numbers, it must take the form of 0 + _i_ _τ_ = _i_ _βτ' + βvx'._ This would mean that 0 = _βvx'_. The most straightforward conclusion would be that either the distance, _x'_ , traveled by the moving system or the velocity, _v_ , of the moving system is zero. It seems reasonable that if _x'_ = 0, then _v_ = 0, as well, and vice versa.

Lieber continues with her demonstration.

"Finally, substituting cos _θ_ for _β_ and sin _θ_ for – _i_ _βv_ these equations become _x_ = _x'cosθ – τ'sinθ_ and _τ_ = _x'sinθ + τ'cosθ_. . . . (23)"70

These substitutions are justified by Lieber in the following clever manner.

"Note that _sin_ ² _θ_ \+ _cos_ ² _θ_ = 1 holds for imaginary angles as well as for real ones; hence the above substitutions are legitimate, thus _β²_ \+ (– _i_ _βv_ )² = _β²_ – _β²_ _v²_ = _β²(1_ – _v²_ _)_ = 1 since _β²_ = _1/(1_ – _v²)_ , _c_ being taken equal to one."71

The mathematical sequence above can be confusing. It is helpful to rewrite the term (– _i_ _βv_ )² as (– _1_ _i_ _βv_ )² or (– _1_ )² _i²_ _β²_ _v²_. Since _(–1)_ ² = _1_ and _i²_ = _–1_ the term (– _i_ _βv_ )² is correctly presented as – _β²_ _v²_.

Although Lieber's substitution of cos _θ_ for _β_ and sin _θ_ for − _i_ _βv_ is clever, it is not mathematically correct. It is quite similar to the polar form of a complex number in which _x_ \+ _y_ _i_ = _r_ (cos _θ_ \+ _i_ _sinθ_ ). The complex number _β_ − _i_ _βv_ expressed in its polar form is _r cosθ − r_ _i_ _sinθ_. If we accept Lieber's substitutions, and then examine the term _tan θ_ , the result is highly questionable. Since _tanθ_ = _sinθ / cosθ_ , the term _tanθ_ is equal to _−_ _i_ _βv / β_ , which equals − _i_ _v_. The tangent of an angle is a ratio. A ratio has no units of measure attached to it. The velocity, _v_ , has units of measure attached to it— _miles/second in a specific direction_.

Lieber's next step is to square each side of the two equations denoted as equation (23).

"And since, from [equation] (23)

_x²_ = ( _x'_ )² _cos²θ –_ _2x' τ' sinθ cosθ +_ ( _τ'_ )² _sin²_ _θ_

_τ²_ = ( _x'_ )² _sin²θ +_ _2x' τ' sinθ cosθ +_ ( _τ'_ )² _cos²_ _θ_

then, obviously,

_x²_ _\+ τ²_ = ( _x'_ )² _+_ ( _τ'_ )²

or (since _y_ = _y'_ and _z_ = _z'_ ),

_x²_ _\+ y² + z² + τ²_ = ( _x'_ )² _+_ ( _y'_ )² + ( _z'_ )² _+_ ( _τ'_ )² _._ "72

When squared versions of equations (23) are added together, it is important to recall that _sin²θ + cos²θ_ = 1. The addition of the left sides of the two equations is straightforward, _x²_ \+ _τ²_ = _x²_ \+ _τ²._ The addition of the right sides of the two equations is more complicated.

( _x'_ )² _cos²θ +_ ( _x'_ )² _sin²_ _θ_ = ( _x'_ )² _(cos²θ + sin²_ _θ )_ = ( _x'_ )²

_+_ _2x' τ' sinθ cosθ_ – _2x' τ' sinθ cosθ_ _=_ 0

_+_ ( _τ'_ )² _sin²_ _θ_ _+_ ( _τ'_ )² _cos²_ _θ_ = _+_ ( _τ'_ )² _(_ _sin²_ _θ + cos²_ _θ )_ = _+_ ( _τ'_ )²

( _x'_ )² _+_ ( _τ'_ )²

Despite Lieber's best efforts she does not succeed in demonstrating that _x²_ _\+ y² + z² + τ²_ = ( _x'_ )² _+_ ( _y'_ )² + ( _z'_ )² _+_ ( _τ'_ )² _._ She fails to explain convincingly how _c²_ may be replaced by the number one, especially since it is the number one without any units of measure attached to it. Also, she fails to explain the algebraic operations she employs to obtain _vt'_ and _vx'_ from equations that contain _vt_ and _vx_ respectively. Finally, she does not justify the replacement of _τ_ and _τ'_ with _i_ _τ_ and _i_ _τ'_ respectively.

We can now turn our attention from "Appendix One: Simple Derivation of the Lorentz Transformation" from Einstein's book, _Relativity: the Special and General Theory_ as well as "Chapter VII: The Four-Dimensional Space-time Continuum" from Lillian R. Lieber's book, _The Einstein Theory of Relativity_. We can return our attention to "Section Three" of Einstein's paper _On the Electrodynamics of Moving Bodies_.

Step 9: Another Metal Rod Coordinate System

We can now turn our attention to the final portion of the third section of _On the Electrodynamics of Moving Bodies_. In this portion of the third section, Einstein determines that _a_ , which represents the as yet unknown function _φ(v)_ has a well defined value, in fact _φ(v)_ = 1. His method has a certain amount of arbitrariness to it. He introduces a third coordinate system denoted as _K'_ that turns out to be at rest relative to the rest system _K_.

"The transformation equations we have derived also contain an unknown function _φ_ of _v,_ which we now wish to determine. To this end we introduce a third coordinate system _K',_ which, relative to the system _k,_ is in parallel-translational motion, parallel to the axis Ξ, such that its origin moves along the Ξ-axis with velocity – _v._ Let all three coordinate origins coincide at time _t_ = 0, and let the time _t'_ of system _K'_ equal zero at _t_ = _x_ = _y_ = _z_ = 0. We denote the coordinates measured in the system _K'_ by _x', y', z'_ and by twofold application of our transformation equations, we get

_t'_ = _φ(–v)β [τ_ \+ _v/V²_ • _ξ ]_ = _φ(v)φ(–v)t,_.........(3.18)

_x'_ = _φ(–v)β [ξ_ \+ _vτ]_ = _φ(v)φ(–v)x,_.................(3.19)

_y'_ = _φ(–v)η_ = _φ(v)φ(–v)y,_................................(3.20)

_z'_ = _φ(–v)ζ_ = _φ(v)φ(–v)z._................................(3.21)"73

In the Greek alphabet "Ξ" is the upper case representation of the English letter "x." As Einstein states the Eqs. (3.18)—(3.21) are produced by a "twofold application of our transformation equations." The variables on the far left, _t', x', y'_ and _z'_ are the coordinates of the new coordinate system Einstein has just introduced, which he denotes as _K'_. The velocity of system _K'_ relative to the moving system _k_ is _−v_. This indicates that the new system _K'_ is at rest relative to the rest system _K_. This is so because the velocity of the moving system _k_ relative to the rest system _K_ is + _v_. The velocity of both the rest system _K_ and the new system _K'_ relative to the moving system _k_ is − _v._ Thus, the rest system _K_ and the new system _K'_ are both at rest relative to each other.

To reiterate, the coordinates of the new system _K'_ are _t', x', y'_ and _z'_. Each of these coordinates is expressed using the appropriate transformation equation and the coordinates of the moving system _k_ , which are _τ, ξ, η,_ and _ζ_. For example, _t'_ is expressed using the appropriate transformation equation and the coordinates of the moving system _k_ , and we obtain _t' = φ(–v)β [τ_ \+ _v/V²_ • _ξ ]._ This procedure is followed for all the coordinates of the new system _K'_.

Next, we express the coordinates of the moving system _k,_ which are _τ, ξ, η,_ and _ζ_ using the coordinates of the rest system _K,_ which are _t, x, y_ and _z,_ and the appropriate transformation equations. Thus, _t'_ = _φ(–v)β [τ_ \+ _v/V²_ • _ξ ]_ is equal to _t'_ = _φ(v)φ(–v)t_ when the coordinates _τ_ and _ξ_ are expressed using the coordinates of the rest system _K_ , which are _t, x, y_ and _z,_ and the appropriate transformation equations.

The procedure can be confusing so it is useful to go through the intermediate steps. We begin with _t'_ = _φ(–v)β [τ_ \+ _v/V²_ • _ξ ]_ and then we substitute the rest system _K_ values for _τ_ and _ξ,_ which gives us the following: _t'_ = _φ(–v)β {_ _φ(v)β (t_ – _v/V²_ • _x)_ \+ _v/V²_ _[_ _φ(v)β (x_ – _vt)]}._ The term _φ(v)β_ is a common factor for the terms inside _{. . . .}_ so we can rewrite the equation as follows: _t'_ = _φ(–v)β {_ _φ(v)β [t_ – _v/V_ 2 • _x_ \+ _v/V_ 2 • _x –_ _v_ 2 _/V_ 2 • _t ]}_ . This lengthy equation is confusing. When we make _φ(v)β_ a common factor the term _v/V²_ _[_ _φ(v)β (x_ – _vt)]_ becomes _v/V_ 2 ( _x –_ _v_ _t)_ or _v/V_ 2 • _x –_ _v_ 2 _/V_ 2 • _t_ . As we study the right side of the lengthy equation for _t'_ , we note that the terms − _v/V_ 2 • _x_ and _\+ v_ 2 _/V_ 2 • _x_ cancel out, and the two remaining terms inside _[. . .]_ have _t_ as a common factor. Thus, they can be expressed as _t(1 – v_ 2 _/ V_ 2 _)._ This gives us _t'_ = _φ(-v)β {_ _φ(v)β t(1 –_ _v_ 2 _/ V_ 2 _)}._ Now we recall that _β_ = _1/(1 –_ _v_ 2 _/ V_ 2 _)_ ½ so we can rewrite the equation as the following: _t'_ = _φ(–v)_ _φ(v) t_ • _1/(1 – v_ 2 _/ V_ 2 _)_ ½ • _1/(1 –_ _v_ 2 _/ V_ 2 _)_ ½ • _(1 –_ _v_ 2 _/ V_ 2 _)._ The last three terms cancel out because in the denominator of the first and second terms in question we have the square root of a quantity times the square root of the same quantity. This gives us the quantity _(1 –_ _v_ 2 _/ V_ 2 _)_ in the denominator. The third term in question consists of the quantity itself _(1 –_ _v_ 2 _/ V_ 2 _)_ so we have _(1–_ _v_ 2 _/ V_ 2 _)_ / _(1 –_ _v_ 2 _/ V_ 2 _)_ , which equals one. For example, _1/(25)_ ½ • _1/(25)_ ½ • _25_ = 1. Thus, we have _t'_ = _φ(–v)_ _φ(v) t_ .

The procedure is the same for the equation _x'_ = _φ(–v)β (ξ_ \+ _vτ)_ so we will list the five steps without any commentary.

_x'_ = _φ(–v)β [_ _φ(v)β (x_ – _vt)_ \+ _v(_ _φ(v)β ( t_ – _v/ V_ 2 • _x)]_

_x'_ = _φ(–v)β [_ _φ(v)β (x_ – _vt_ \+ _vt –_ _v_ 2 _/ V_ 2 • _x)]_

_x'_ = _φ(–v)β φ(v)β x (1_ – _v_ 2 _/ V_ 2 _)_

_x'_ = _φ(–v) φ(v) x_ • _1/(1_ – _v_ 2 _/ V_ 2 _)_ ½ • _1/(1_ – _v_ 2 _/ V_ 2 _)_ ½ • _(1_ – _v_ 2 _/ V_ 2 _)_

_x'_ = _φ(–v) φ(v) x_

In order to transform the equation _y'_ = _φ(–v)η,_ we recall that _η_ = _φ(v)y,_ which gives us the following: _y'_ = _φ(v)_ _φ(–v)y._ In order to transform the equation _z'_ = _φ(–v)ζ,_ we recall that _ζ_ = _φ(v)z,_ which gives us the following: _z'_ = _φ(v) φ(-v)z._

If we compare _t'_ = _φ(–v)β [τ_ \+ _v/V²_ • _ξ ]_ to _τ_ = _φ(v)β [t_ − _v/V²_ • _x ]_ and _x'_ = _φ(–v)β (ξ_ \+ _v_ _τ)_ to _ξ_ = _φ(–v)β (x_ − _v t)_ , we discover that when the function _φ(v)_ has a _(−v)_ the terms _v/V²_ • _ξ_ and _vτ_ are positive, and when the function _φ(v)_ has a _(_ + _v)_ the terms _v/V²_ • _x_ and _v t_ are negative. The change in the status of _(v)_ from the function _φ(v)_ has no effect on the status of _(v)_ from _β_.

Following the equations above Einstein informs us that _φ(v)_ • _φ(–v)_ = 1. This is true for the special circumstances engendered by the fact that the system _K'_ is at rest relative to the system _K_.

"Since the relations between _x', y', z'_ and _x, y, z_ do not contain the time _t,_ the systems _K_ and _K'_ are at rest relative to each other, and it is clear that the transformation from _K_ to _K'_ must be the identity transformation. Hence, _φ(v)_ • _φ(–v)_ = 1,..........(3.22)."74

With regard to the term _identity transformation,_ the _Mathematics Dictionary_ refers the reader to the term _identity function_. The _Mathematics Dictionary_ gives the following definition for _identity function:_

"An identity function is the function from a set into itself that maps each member of the set to itself. For a set _S,_ the identity function is the function _f_ defined by _f(x)_ = _x_ for each _x_ in _S_. For example, the identity function for the set of real numbers is the function _f_ whose graph is the line _y_ = _x, i.e.,_ the function that associates with each member _x_ the number itself."75

Since _t' =_ _φ(−v)_ • _φ(v)_ • _t, x' =φ(−v)_ • _φ(v)_ • _x, y' = φ(−v)_ • _φ(v)_ • _y_ and _z' =φ(−v)_ • _φ(v)_ • _z,_ it follows that _φ(v)_ • _φ(–v)_ = 1. The identity transformation is the function that associates with each member _x_ the number itself. To meet the requirement that each member of the set _t'_ is associated with the number itself, we must produce the equation _t' = t_. In order for the function _t' =φ(−v)_ • _φ(v)_ • _t_ to allow _t'_ to equal _t_ , the term _φ(−v)_ • _φ(v)_ must equal one, hence _φ(−v)_ • _φ(v) =_ 1 _._ Since the transformation from _K_ to _K'_ is an identity transformation, would it be more accurate to state that under special circumstances _φ(−v)_ • _φ(v) =_ 1?

Einstein continues his investigation of the function _φ(v)_. To facilitate the understanding of the following quotation it should be pointed out that the letter " _H"_ is the upper case version of the letter " _η"_ in the Greek alphabet. Thus, when Einstein refers to the _H_ -axis of the system _k_ , he is referring to the axis typically denoted as the _y_ -axis in Cartesian coordinates.

"Let us now explore the meaning of _φ(v)_. We shall focus on that portion of the _H-_ axis of the system _k_ that lies between _ξ_ = 0, _η_ = 0, _ζ_ = 0, and _ξ_ = 0, _η_ = _length l, ζ_ = 0. This portion of the _H-_ axis is a rod that, relative to the system _K,_ moves perpendicular to its axis with a velocity _v_ and its ends have the coordinates in _K_ : x1 = _vt_ , y1 = _l/φ(v)_ , z1 = 0....................(3.23) and x2 = _vt_ , y2 = 0, z2 = 0,....................(3.24).

The length of the rod, measured in _K,_ is thus _l/φ(v)_ ; this gives us the meaning of the function _φ_. . . . Thus, the length of the moving rod measured in the rest system does not change if _v_ is replaced by – _v_. From this we conclude: _l/φ(v)_ = _l/φ(–v) or φ(v)_ = _φ(–v)_ ,.................(3.25). From this relation and the one found earlier it follows that _φ(v)_ = 1. . .."76

The explanation for the formula y1 = _l/φ(v)_ is that _l_ is substituted for _η_ in the transformation formula _η_ = _φ(v)y_ thus becoming _l_ = _φ(v)y,_ which can be rewritten as y1 = _l/φ(v)_. The explanations for the formulas _x_ 1 = _vt_ and x2 = _vt_ are that 0 is substituted for _ξ_ in the transformation formula _ξ_ = _φ(v)β(x – vt)_. Zero divided by _φ(v)β_ is zero, which gives us 0 = _x – vt_ or _x_ = _vt._

The relation found earlier is _φ(−v)_ • _φ(v)_ = 1. Since Eq. (3.25) states that _φ(v) = φ(−v)_ , we can replace _φ(−v)_ with _φ(v)_ in the relation found earlier, which is _φ(−v)_ • _φ(v) =_ 1 as we just noted, and we obtain _φ(v)_ • _φ(v) =_ 1 _._ Since (1) (1) =1, _φ(v) =_ 1. The term _φ(-v)_ must also equal 1 because if it were allowed to equal −1, then we could obtain from Eq. (3.25), _φ(−v)_ • _φ(v) =_ 1 _,_ the relation (−1)(1) =1, which is invalid.

The result of finding the meaning of the function _φ(v)_ is that since, according to Einstein, it equals one, the function can be eliminated from the transformation equations. Thus the absence of the term _φ(v)_ in L. Lieber's version of the transformation equations is explained.

"Since _φ(v)_ = 1, the transformation equations take the following form:

_τ_ = _β( t – vx/V_ 2 _),_............(3.26)

_ξ_ = _β(x – vt),_...................(3.27)

_η_ = _y,_...............................(3.28)

_ζ_ = _z,_................................(3.29)

where _β_ = _1/[1 – (v²/V²)]_ ½."77

Thus, the third section of _On the Electrodynamics of Moving Bodies_ ends with the listing of the transformation equations in their final form.

Chapter 3: Components of a Radius and Confusing Clocks

Chapter Summary: An Analysis of "Section 4. The Physical Meaning of the Equations Obtained as Concerns Moving Rigid Bodies and Moving Clocks" from _On the Electrodynamics of Moving Bodies_

In the fourth section of _On the Electrodynamics of Moving Bodies,_ Einstein demonstrates two concepts. The first concept is the contraction of matter along the axis aligned with the direction of motion of an object. The second concept is the dilation or slowing down of time for an object in motion.

The object he uses in his first demonstration is a sphere with radius _R._ This sphere is at rest relative to the moving system _k,_ and the center of the sphere is located at the origin of _k._ The equation for the surface of the sphere is _ξ²_ \+ _η²_ \+ _ζ²_ = _R²_ ................Eq. (4.1).

An example will help us understand this equation. Let _ξ_ = _η_ = _ζ_ = 4 feet. We start at the origin of _k_ , which is also the center of the sphere and move four feet to the right along the _ξ_ -axis. Next, we move four feet "upward" ( in the _ξ —η_ plane) in a direction perpendicular to the _ξ-axis._ Using the Pythagorean theorem _(a²_ \+ _b²_ = _c²_ _)_ we calculate we are (32)½ feet from the origin because 4² sq. ft.+ 4² sq. ft. = 32 sq. ft. The four foot lengths form the sides of a right triangle and (32 sq. ft.)½ is the length of the hypotenuse—about 5½ feet. Next, we move perpendicularly four feet above the _ξ—η_ plane. We have reached a point on the surface of the sphere. We can calculate the radius of the sphere using the Pythagorean Theorem. The four feet we moved above the _ξ—η_ plane is one leg of a right triangle. The other leg of the right triangle is the distance (32)½ feet, which is the distance we moved from the origin in the _ξ—η_ plane. Thus, 4² sq. ft. + [(32 sq. ft.)½ ]² = 48 sq. ft. and the radius of the sphere is (48 sq. ft.)½, which is about seven feet.

This sphere of about seven feet in radius is at rest relative to the moving system _k._ Considered from the rest system _K_ the sphere's dimensions are expressed in terms of _x, y,_ and _z._ Einstein states, "Expressed in terms of _x, y,_ and _z,_ the equation of this surface at time _t_ = _0_ is _x²/{[(1 – v²/_ _V²)]_ ½ _}²_ \+ _y²_ \+ _z²_ = _R²_...............Eq. (4.2)"78 The first term of Eq. (4.2), _x²/{[(1 – v²/_ _V²)]_ ½ _}²_ , is obtained by using the equation _ξ_ = _β( x – vt)_ when _t_ = 0, which gives us the following: _ξ_ = _β(x)_ and therefore _ξ²_ = _β²(x)²_. Since _β_ is equal to _1/(1 – v²/_ _V²)_ ½, we obtain _ξ²_ = _x²/{[(1 – v²/_ _V²)]_ ½ _}²_ , which can be simplified as follows: _ξ²_ = _x²/(1 – v²/_ _V²)_ or _ξ_ = _x/(1 – v²/_ _V²)_ ½ . Thus, we have expressed _ξ²_ in terms of _x._ Next, we must express _η²_ and _ζ²_ in terms of _y_ and _z_ respectively. Since _η_ = _y_ and _ζ_ = _z,_ we obtain _η²_ = _y²_ and _ζ²_ = _z²_ , and we can express all the terms of Eq. (4.1), _ξ²_ \+ _η²_ \+ _ζ²_ = _R²,_ in terms of _x, y,_ and _z._ Thus we obtain Eq. (4.2) _x²/{[(1 – v²/_ _V²)]_ ½ _}²_ \+ _y²_ \+ _z²_ = _R²_ .

If we return to our example where _ξ_ = _η_ = _ζ_ = 4 feet, we can use the transformation equations _η_ = _y_ and _ζ_ = _z_ to calculate that _y_ = 4 feet and _z_ = 4 feet. But, we cannot let _x_ = 4 feet because that would result in _ξ_ being equal to _4 feet/(1 – v²/_ _V²)_ ½, which would result in a value for _ξ_ larger than 4 feet for velocities _(v)_ greater than zero and less than the velocity of light. If we let _x_ = _4 feet • (1 – v²/_ _V²)_ ½ _,_ the resulting value of _ξ_ is 4 feet. To summarize the procedure, we know that _x_ must be divided by _(1 – v²/_ _V²)_ ½ and this result must equal _ξ,_ which in our example is 4 feet. We also know that when _x_ is divided by _(1 – v²/_ _V²)_ ½ the quotient (result) will be larger than _x_ when the velocity _(v)_ is greater than zero and less than the velocity of light. This is because _x_ will be divided by a fraction greater than zero and less than one. For example if, _(v²/_ _V²)_ = _3/4,_ the result is _(1 – 3/4)_ ½ _,_ which is equal to _(1/4)_ ½ or _½._ When _x_ is divided by _½_ the result is _2x._ To counter this we must multiply _x_ by _½_ or _(1 – 3/4)_ ½. In general we must let _x*(for calculation purposes only)_ = _ξ_ and multiply _x*(for calculation purposes only)_ by _(1 –_ _v²/_ _V²)_ ½ to obtain _x_.

When _ξ_ = _R_ where _R_ is the radius of the sphere, Einstein states, "A rigid body that has a spherical shape when measured at rest has, when in motion— considered from the rest system— the shape of an ellipsoid of revolution with axes _R(1 –_ _v²/_ _V²)_ ½ , _R, R._...............Eq. (4.3)"79 A question naturally arises. How can _ξ = R_ when _ξ_ is only one of the three components of _R_? We should recall that the other two components being _η_ and _ζ_. A sphere has a limitless number of radii, and they all have components _ξ, η,_ and _ζ_. When a particular radius lies along the _ξ_ -axis, the components _η_ and _ζ_ will both equal zero, and _ξ =R._ The equation _ξ²_ \+ _η²_ \+ _ζ²_ = _R²_ becomes _ξ²_ \+ _0²_ \+ _0²_ = _R²_ or _ξ²_ = _R²_ or _ξ_ = _R._

Now, Einstein turns his attention to his second demonstration, which is the slowing down or dilation of time for objects in motion. Einstein imagines a clock at rest in the moving system _k_. The clock is located at the origin of the moving system _k_ , and set so that it indicates the time of the moving system _k_ , which is designated as _τ_. Einstein asks the question, "What is the rate of the clock, when considered from the rest system?"

"We further imagine one of the clocks that is able to indicate time _t_ when at rest relative to the rest system and time _τ_ when at rest relative to the moving system to be placed at the origin of _k_ and set such that it indicates the time _τ._ What is the rate of this clock when considered from the rest system?

"The quantities _x, t,_ and _τ_ that refer to the position of this clock obviously satisfy the equations _τ_ = _1/(1 – v²/ V²_ _)_ ½ • _[t – (v/ V²_ • _x)]_............... (4.4) and _x_ = _vt_...............(4.5). We thus have _τ_ = _t(1 – v²/ V²_ _)_ ½ = _t – [1 – (1 – v²/ V²_ _)_ ½ _]_ • _t,_...............(4.6) from which it follows that the reading of the clock considered from the rest system lags behind each second by _[1 – (1 – v²/ V²_ _)_ ½ _]_ sec. or, up to quantities of the fourth and higher order, by _½(v²/ V²)_ sec.

"This yields the following peculiar consequence: If at the points _A_ and _B_ of _K_ there are clocks at rest that, considered from the rest system, are running synchronously, and if the clock at _A_ is transported to _B_ along the connecting line with velocity _v,_ then upon arrival of this clock at _B_ the two clocks will no longer be running synchronously; instead, the clock that has been transported from _A_ to _B_ will lag _½ t(v²_ _/ V²)_ sec. (up to quantities of the fourth and higher orders) behind the clock that has been in _B_ from the outset where _t_ is the time needed by the clock to travel from _A_ to _B. . .._

"From this we conclude that a balance-wheel clock located at the Earth's equator must, under otherwise identical conditions, run more slowly by a very small amount than an absolutely identical clock located at one of the Earth's poles."80

It will be helpful to show the intermediate steps of the above calculations beginning with Eq. (4.4). If we let _x_ = _vt_ in Eq. (4.4), _τ_ = _1/(1 – v²/ V²_ _)_ ½ • _[t – (v/ V²_ • _x)],_ the result is the following: _τ_ = _1/(1 – v²/ V²_ _)_ ½ • _[t – (v/ V²_ • _vt)]_. The final term, _[t – (v/ V²_ • _vt)]_ has _t_ as a common factor and therefore can be rewritten as _t(1 – v²/ V²)_. In its rewritten form we can see it shares a common factor with its companion term _1/(1 – v²/ V²)_ ½ . When we divide _(1 – v²/ V²)_ by its square root _(1 – v²/ V²)_ ½ the result is _(1 – v²/ V²)_ ½ _._ Thus we obtain the equation _τ_ = _t(1 – v²/ V²)_ ½. We can rewrite the equation as _τ_ = _t – [ 1 – (1 – v²/ V²)_ ½ _]_ • _t_ because the + _t_ and the – _t_ cancel out each other, as do the negative (minus) signs before the _[. . . .]_ parentheses and the _(. ._ _. . )_ parentheses _._ The introduction of a rewritten version of this equation, in which both the + _t_ and the – _t,_ and the plus sign are highlighted, should provide additional clarity. The plus sign is highlighted because it was formed from the multiplication of two minus signs. Thus we obtain the following _τ_ = + _t_ _–_ _t_ + _t (1 – v²/ V²)_ ½

We should note that the equation _x_ = _vt_ is a special case of the equation _x_ = _x'_ \+ _vt_ in which _x'_ is equal to zero.

We should also concern ourselves with the meaning of the phrase, "up to quantities of the fourth and higher order." The phrase is used twice, and its meaning is unclear. It seems to imply that for every one of the individual quantities generated by the use of a particular velocity _(v)_ in the term _[1– (1 – v²/ V²)_ ½ _]_ sec., the same quantity can be generated by using the same particular velocity _(v)_ in the term _½(v²_ _/ V²)_ sec. This is not the case. The term _½(v²_ _/ V²)_ sec. is an approximation of the term _[1– (1 – v²/ V²_ _)_ ½ _]_ sec. This can be demonstrated by the following example. Let _(v²/ V²)_ = _19/100,_ which gives us _[1– (1 – 19/ 100_ _)_ ½ _]_ sec. = _[1– (81/ 100_ _)_ ½ _]_ sec. = _(1 – 9/ 10_ _)_ sec. = _1/_ 10 sec. The term _½( v²/ V²)_ sec. = _½(19/100)_ sec. = 19/200 sec., which is _1/200_ sec. less than _1/10_ sec.

As we noted the phrase occurs twice, and A. I. Miller's first translation of the phrase is, "neglecting magnitudes of fourth and higher order,"81 and his second translation of the phrase is, "up to magnitudes of fourth and higher order."82 These two different translations only add to the ambiguity of the phrase.

A. I. Miller also offers an interesting explanation of the term _½ t(v²_ _/ V²)._ According to Miller, the term _½( v²/ V²)_ is multiplied by _t_ because for each second that passes the clock lags behind _½( v²/ V²)_ seconds when considered from the rest system. Therefore, if _t_ seconds pass the clocks lags behind _½t( v²/ V²)_ when considered from the rest system. He does not conclude that _½t( v²/ V²)_ is an approximation. A. I. Miller's further explanation of the term _½t( v²/ V²)_ is, at least, consistent with explanation of Eq. (3.2). It is difficult to imagine that it is correct, but it is consistent. He invokes an explanation remarkably similar to his explanation of Eq. (3.2). Miller states, "Expanding the square root in the equation _[1 – (1 – v²/ V²)_ ½ _]_ , and retaining terms up to second order in _v/V,_ gives _½_ _t(v²_ _/ V²)._ "83 We should note that the expression _[1 – (1 – v²/ V²)_ ½ _],_ which according to Miller, produces the term _½t(v²_ _/ V²)_ when its square root is expanded does not include the variable _t,_ which is necessary for the term _½ t(v²/ V²)_ ½ _._ The absence of the variable _t_ from the expression _[1 – (1 – v²/ V²)_ ½ _]_ may be a debilitating condition regarding A. I. Miller's explanation of the term _½t(v²_ _/ V²),_ but, perhaps, we are to conclude that the variable _t_ can be inferred from the expression's context, i.e., Einstein's statement, "the rest system lags behind _each second by [1 – (1 – v²/ V²)_ ½ _]_ . . .."

In the third paragraph of the quotation with endnote 78, Einstein discusses two clocks _A_ and _B_ located at points _A_ and _B_ respectively in the rest system _K._ These two clocks are running synchronously. When clock _A_ is transported to the location of clock _B_ the two clocks are no longer running synchronously. Clock _A_ will be running behind clock _B_ by the amount of _½ t(v²_ _/ V²)_ sec. where _t_ is the time it takes to transport the clock from point _A_ to point _B_ and _v_ is the velocity with which the clock is transported. It is interesting to note that Einstein sees no difficulty in applying an argument developed for clocks in the rest system _K_ to clocks located on the Earth, a moving system.

With that in mind, we will construct a thought experiment and apply his arguments to a rod moving with the velocity of . _5c._ The rod is 6 light seconds in length or 1,116,000 miles. The rod is moving such that point _B_ is in the forward position and point _A_ is in the rear position. There is a clock at point _A_ that is synchronized with the clock at point _B._ Clock _A_ is transported to the location of clock _B_ with a velocity of 100,000 miles/second relative to the rod. The time it takes to transport clock _A_ to clock _B_ is 11.16 seconds. The amount of time clock _A_ should lag behind clock _B_ after clock _A_ is transported to clock _B_ is _½ t(v²_ _/ V²)_ sec., which is equal to _½(11.16 sec.)_ • _100,000²_ _(miles/second)²_ _/3.46_ × _10_ 10 _(miles/second)²_ = _1.6 seconds._ The final "sec." of the term _½ t(v²_ _/ V²)_ sec. is unnecessary. Also, we should note that _(186,000miles/sec.)²_ is equal to _3.46_ × _10_ 10 _(miles/second)²_. According to our calculations when clock _A_ is transported to clock _B,_ clock _A_ will lag _1.6 seconds_ behind clock _B._

This result implies that distant synchronized clocks agree in the time they keep. As we have seen according to Einstein's definition of distant synchronized clocks, which is _t_ B _– t_ A = _t_ A' _– t_ B, these clocks do not have to agree in the time they keep in order to be considered synchronized. Einstein's definition states that the time it takes a light beam to travel from clock _A_ to clock _B_ is equal to the time it takes a light beam to travel from clock _B_ to clock _A._ Since the velocity of our rod is . _5c_ in a direction that makes point _B_ the forward point, the effective velocity of a light beam traveling from point _A_ to point _B_ is _186,000 miles/second_ _– 93,000 miles/second_ = _93,000 miles/second ._ Therefore, a light beam takes 12 seconds to travels from point _A_ to point _B._ The effective velocity of a light beam traveling from point _B_ to point _A_ is _186,000_ _miles/second_ \+ _93,000_ _miles/second_ = _279,000 miles/second._ Therefore, a light beam takes 4 seconds to travel from point _B_ to point _A._ If the clock at point _B_ is set 4 seconds slow, the clocks will be synchronized according to Einstein's definition. The light beam begins its journey from point _A_ at _t_ A = _00:00,_ or _0 seconds_ and it strikes the mirror at point _B_ at _t_ B = _00:08_ or _8 seconds_ later _,_ of course, at this same instant the clock at point _A_ reads _00:12_ or _12 seconds_ later _._ The light beam returns to point _A,_ and the clock at point _A_ reads _t_ A' = _00:16_ or _16 seconds_ later. Thus, _t_ B _– t_ A = _t_ A' _– t_ B since _00:08 sec. – 00:00_ sec. = _00:16 sec. – 00:08 sec._

In our thought experiment clock _B_ is 4 seconds behind clock _A,_ when the clocks are located on the ends of a rod 1,116,000 miles in length, and the rod is traveling at _.5c._ with clock _B_ in the forward position. When clock _A_ is transported to clock _B_ and the clocks are placed side by side, clock _B_ will be only 2.4 seconds behind clock _A_ instead 4 seconds. This is because during its journey clock _A_ slows by 1.6 seconds. But, according to Einstein's determination clock _A_ should lag behind clock _B_ _,_ but actually the situation is the opposite clock _B_ lags behind clock _A_. According to Einstein clock _A_ should lag behind clock _B_ by 1.6 seconds, but, instead, clock _B_ lags behind clock _A_ by 2.4 seconds. From this discrepancy, it can be determined that clock _B_ runs 4 seconds behind clock _A._ With this information the absolute velocity of the rod could be determined. Of course, since it is a thought experiment we already know the absolute velocity of the rod is _.5c_ in a direction such that point _B_ is the forward point.

The calculation of the velocity of the rod begins with the fact that it takes the light beam 12 seconds to travel from point _A_ to point _B,_ instead of the 8 seconds the observers originally thought necessary for the first leg of the round-trip journey. The distance the light beam travels is _12 seconds_ × _186,000 miles/second_ = _2,232,000 miles._ If we subtract the length of the rod, we are left with the distance point _B_ rushed away from the oncoming light beam in _12 seconds_ the calculation is _2,232,000 miles_ _\- 1,116,000 miles_ = _1,116,000 miles._ The distance point _B_ rushed away from the oncoming light beam is _1,116,000_ miles. If we divide 1,116,000 miles by the 12 seconds it took the rod to travel this distance we calculate the speed of the rod, which is _93,000_ miles/second. Since we know the rod is traveling in a direction that makes point _B_ the forward position, we have calculated the velocity of the rod.

Chapter 4: The Limits of Special Cases and Running on a Train

Chapter Summary: An Analysis of "Section 5. The Addition Theorem for Velocities" from _On the Electrodynamics of Moving Bodies_

The mathematics in the fifth section of _On the Electrodynamics of Moving Bodies_ are quite involved, and a thorough explanation of the intermediate steps that occur between each equation will prove helpful. We should begin our examination with the two paragraphs with which Einstein begins the section. Einstein's scenario is that the moving system _k_ is moving with velocity _v_ along the _X_ -axis of the rest system _K_. A point in the moving system _k_ is moving in the _ξ, η_ plane. The component of its velocity along the _ξ_ -axis is _ω_ ξ, and the component of its velocity along the _η_ -axis is _ω_ η _._

"In the system _k_ moving with velocity _v_ along the _X-_ axis of the system _K,_ let a point move in accordance with the equations _ξ_ = _ω_ ξ _τ_...............(5.1), _η_ = _ω_ η _τ_................(5.2), _ζ_ = _0_...............(5.3), where _ω_ ξ and _ω_ η denote constants.

We seek the motion of the point relative to the system _K._ Introducing the quantities _x, y, z, t_ into the equations of motion of the point by means of the transformation equations derived in section 3, we obtain _x_ = _(ω_ ξ \+ _v)/[1_ \+ _(vω_ ξ _/ V²)] • t,_...............(5.4), _y_ = _[1 – (v²_ _/ V²)]_ ½ _/ [1_ \+ _(vω_ ξ _/_ _V²)] • ω_ η _t,_...............(5.5), _z_ = _0_...............(5.6)."84

The Eq. (5.4) is generated from the Eq. (5.1). In Eq. (5.1) the term _ξ_ is replaced with _β(x – vt)_ from the transformation equation _ξ_ = _β(x – vt),_ and the term _τ_ is replaced with _β(t – vx/ V²)_ from the transformation equation _τ_ = _β(t – vx/ V²)_. This gives us _β(x – vt)_ = _ω_ ξ _[β(t – vx/ V²)],_ which can be rewritten as _βx – βvt_ = _ω_ ξ _βt – ω_ ξ _β vx/ V²._ If we group the terms with _x_ in them on the left side of the equation and the terms with _t_ in them on the right side of the equation we obtain _βx_ \+ _ω_ ξ _β( vx/ V²)_ = _β t (ω_ ξ \+ _v)._ Then we divide each side of the equation by _β,_ and we obtain the following: _x_ \+ _ω_ ξ _( vx/ V²)_ = _t(ω_ ξ \+ _v)._ Since _x_ is a common factor on the left side of the equation, we can rewrite the equation as the following: _x (1_ \+ _ω_ ξ _v/ V²)_ = _t(ω_ ξ \+ _v)._ Finally, we divide each side of the equation by _(1_ \+ _ω_ ξ _v/ V²),_ and we obtain Eq. (5.4) _x_ = _t(ω_ ξ \+ _v)/ (1_ \+ _ω_ ξ _v/ V²)._

The generation of Eq. (5.5) involves the use of a complex fraction. A complex fraction is a fraction constructed of a fraction in the numerator and a fraction in the denominator. For example, _(½)/(¼)_ is a complex fraction. To simplify a complex fraction we must invert the denominator and then multiply the numerator by the inverted denominator. Thus, the complex fraction _(½)/(¼)_ = _(½)• (4/1)_ = _4/2_ = _2._ The complex fraction we are going to be involved with is more confusing. In the example above, the denominator of the numerator of the complex fraction is two. In the complex fraction we are going to generate the denominator of the numerator of the complex fraction is itself a fraction. For clarity we will separate the numerator of the complex fraction from the denominator of the complex fraction with the sign ÷, and also we should note the denominator of the complex fraction will be _V²,_ which we will denote as _V²/_ _1._

The Eq. (5.5) is generated from Eq. (5.2) _η =ω_ η _τ_ . In Eq. (5.2) the term _η_ is replaced with _y_ from the transformation equation _η_ = _y,_ and the term _τ_ is replaced with _β(t – vx/ V²)_ from the transformation equation _τ_ = _β(t – vx/V²)_. Thus we have _y_ = _ω_ η _β(t – vx/ V²)_. We replace _x_ with _t(ω_ ξ \+ _v)/ (1_ \+ _ω_ ξ _v/ V_ 2 _)_ from Eq. (5.4), _x_ = _t(ω_ ξ \+ _v)/ (1_ \+ _ω_ ξ _v/ V²)_. This gives us the following: _y_ = _ω_ η _β [ t_ – _vt(ω_ ξ \+ _v)/(1_ \+ _ω_ ξ _v/ V²)_ ÷ _V²/1]._ We can rewrite the denominator of the numerator of the complex fraction as the following: _(V²_ \+ _vω_ ξ _)/ V²._ Thus, we obtain _y_ = _ω_ η _β [ t_ – _vt(ω_ ξ \+ _v)/{(V²_ \+ _ω_ ξ _v)/ V² }_ ÷ _V²/1]._ When we invert the denominator, _V²/1_ , we obtain _1/ V²_ _._ The numerator of the complex fraction is _vt(ω_ ξ \+ _v)/[(V²_ \+ _ω_ ξ _v)/ V²]._ When we multiply the numerator of the complex fraction by the inverted denominator, we obtain _[vt(ω_ ξ \+ _v)_ _•_ _1_ _]/[(V²_ \+ _ω_ ξ _v)/ V²] •_ _V²_ _._ Thus, the complex fraction is simplified and we obtain _y_ = _ω_ η _β [ t_ – _vt(ω_ ξ \+ _v)/(V²_ \+ _ω_ ξ _v)]._

Next, we note that _t_ is a common factor for the two terms in the numerator. For clarity, we will write the numerator with the common factor _t_ highlighted, _t_ **–** _v_ _t_ _(ω_ ξ \+ _v)_. We can rewrite the equation as _y_ = _ω_ η _β { t [1_ – _v(ω_ ξ \+ _v)/(V²_ \+ _ω_ ξ _v)]}._ Now, we find a common denominator for the terms inside _[. . . .]_ parenthese _s._ But, before we do, we must pause and examine the terms within the _[. . . .]_ parentheses. There is an error. The 1 is incorrectly separated from the fraction – _v(ω_ ξ \+ _v)/(V²_ \+ _ω_ ξ _v)_. The correct representation of the terms within the _[. . . .]_ parentheses is _[1_ – _v(ω_ ξ \+ _v)]/(V²_ \+ _ω_ ξ _v)_ , but we must ignore the correct representation. We must find a common denominator for the incorrect terms, _[1_ – _v(ω_ ξ \+ _v)/(V²_ \+ _ω_ ξ _v)]_ , which is the following: _(V²_ \+ _ω_ ξ _v)._ This allows us to rewrite the terms inside _[. . . .]_ parentheses as _(V²_ \+ _ω_ ξ _v – ω_ ξ _v – v²)/(V²_ \+ _ω_ ξ _v)_. We note that the + _ω_ ξ _v_ and the – _ω_ ξ _v_ cancel out, which leaves us with _(V² – v²)/(V²_ \+ _ω_ ξ _v)_. Now, Eq. (5.5) can be rewritten as the following: _y_ = _ω_ η _β t[(V² – v²)/(V²_ \+ _ω_ ξ _v)]._ We can rewrite the term _(V² – v²)_ as _V²(1 – v²/ V²)_ and we can write _β_ as the following: _1/ (1 – v²/ V²)_ ½ _._ Thus, the equation with _β_ written as _1/ (1 – v²/ V²)_ ½ becomes _y_ = _1/ (1 – v²/ V²)_ ½ _• V²_ _(1 – v²/ V²) /_ _(V²_ \+ _ω_ ξ _v) • ω_ η _t._ Since a term is divided by its square root the equation becomes _y_ = _V²(1 – v²/ V²)_ ½ _/_ _(V²_ \+ _ω_ ξ _v) • ω_ η _t._ Next we must multiply _both_ the numerator and denominator of the fraction by _1/ V²_ _,_ which is the same as multiplying by one. The equation becomes the following: _y_ = _(1 – v²/ V² )_ ½ _/_ _[(V²_ \+ _ω_ ξ _v)/ V²_ _] • ω_ η _t._ This can be rewritten to produce the final form of Eq. (5.5) by simplifying the term _(V²_ \+ _ω_ ξ _v)/ V²_ as _(1_ \+ _ω_ ξ _v/ V²)_ , and we obtain the following: _y_ = _(1 – v²/ V²)_ ½ _/_ _(1_ \+ _ω_ ξ _v/ V²) • ω_ η _t,_ which is Eq. (5.5)

Eq. (5.6), _z_ = _0,_ is generated from Eq. (5.3), _ζ_ = _0_ by using the transformation equation _ζ_ = _z._

As we have noted, we discovered a mathematical error in the generation of Eq. (5.5). In rewriting the numerator of Eq. (5.5) with _t_ as a common factor, Einstein made an error. Besides the mathematical error we discovered, there is an insurmountable difficulty with the reasoning we have employed in the five previous paragraphs. The transformation equation _ξ_ = _β (x – vt)_ from which we substituted _β (x – vt)_ for _ξ_ in Eq. (5.1), _ξ_ = _ω_ ξ _τ,_ is generated from and is, in fact, equal to the equation _ξ_ = _Vτ._ Thus, we have the impossible situation of _Vτ_ = _ω_ ξ _τ_. The equation states that the velocity of a beam of light, traveling along the _ξ_ -axis in the direction of increasing _ξ,_ multiplied by a specific time interval _τ_ is equal to any constant velocity along the _ξ-_ axis multiplied by the same specific time interval _τ._ A succession of equations will demonstrate that, according to Einstein, _Vτ_ = _ω_ ξ _τ_. They are the following: _ξ_ = _Vτ = aV[t – vx'/(V² – v²)]_ = _φ(v)β(x – vt)= β(x – vt)= ξ_ = _ω_ ξ _τ_ or _Vτ_ = _ω_ ξ _τ_.

The equation _ξ_ = _Vτ_ is a special case of the more general equation _ξ_ = _ω_ ξ _τ_ in two fundamental ways. It may not seem, at first glance, that the equation _ξ_ = _Vτ_ represents a special case of _ξ_ = _ω_ ξ _τ_ , but it does.

The first way that the equation _ξ_ = _Vτ_ is a special case of _ξ_ = _ω_ ξ _τ_ involves the comparison of the term _ω_ ξ with the term _V._ The term _ω_ ξ represents a component velocity, which is the motion of the point as measured along the _ξ_ -axis. That component velocity can assume any particular constant velocity as long as it is directed along the _ξ_ -axis. The term _V_ represents only one specific velocity—the velocity of a light beam—in this case the direction of the light beam is along the _ξ_ -axis in the direction of increasing _ξ_. The point also has another component velocity _ω_ η , which represents the velocity of the point's motion as measured along the _η-_ axis, and that component velocity can be any particular constant velocity, as well. The terms _ω_ ξ and _ω_ η are the component velocities of a point with a constant velocity _ω._ However, the important distinction is that the term _V_ in the equation _ξ_ = _Vτ_ represents only one specific velocity and it is the velocity of a light beam traveling along the _ξ-_ axis in the direction of increasing _ξ._ The point with velocity _ω_ can have any particular constant velocity as long as it is traveling in the _ξ, η_ plane.

The second way that the equation _ξ_ = _Vτ_ is a special case is that the point with component velocities _ω_ ξ and _ω_ η is carried along by the moving system _k._ The light beam with velocity _V_ is not carried along by the moving system _k._

If we think of the moving system _k_ as a long train consisting of many flat-bed cars, which are pulled by a powerful locomotive, the point can be represented by a runner stationed on the last flat-bed car. The runner is carried along by the velocity of the train. If he starts running towards the engine, no matter how great the velocity of the train, he only needs to travel the length of the flat-bed cars to reach the locomotive. However, a light beam projected from the last flat-bed car will not be carried along by the train. The engine will be rushing away from the oncoming light beam, and the light beam will need to travel a distance greater than the length of the flat-cars to reach the locomotive.

To generate the transformation equation _ξ_ = _φ(v)β(x – vt),_ Einstein begins with the equation _ξ_ = _Vτ,_ and since _τ_ = _a[t – vx'/(V² – v²)],_ he generates the equation _ξ_ = _a_ _V_ _[t – vx'/(V²_ _– v²)]._ The term _V_ in bold type represents the velocity of a beam of light. It does not represent the velocity of a point. The transformation equation deals with a light beam with velocity _V_. The velocity of the light beam becomes combined with other terms so we can easily lose sight of it. There is no reason for equations analyzing the behavior of a point with component velocities _ω_ ξ and _ω_ η to include the velocity of a beam of light _V_ _,_ but according to Einstein they must.

As we have discussed previously, Einstein continues his analysis of the equation _ξ_ = _a_ _V_ _[t – vx'/(V²_ _– v²)]_. He states, "But as measured in the rest system, the light ray propagates with velocity _V – v_ relative to the origin of _K,_ so that _x'/(V – v)_ = _t._ "85 Since there is no light beam in the thought experiment involving the point with component velocities _ω_ ξ and _ω_ η, it is inconceivable that _x'/(V – v)_ = _t_. Instead, the time _t_ can be expressed in three ways: _x'/ω_ ξ = _t, y/ω_ η = _t,_ or _d/ω_ = _t,_ where _d_ is the distance traveled by the point and _ω_ is the velocity of the point. Since the point is carried along by the moving system _k,_ the point is like the runner running on the flat-bed cars of a train. The velocity of the point and the runner relative to the origin of _K_ is _ω_ \+ _v,_ not _ω – v._

Contrary to Einstein's claim, the equation _ξ_ = _ω_ ξ _τ_ expressed in the quantities _x_ and _t_ assumes the following form: _ξ_ = _aω_ ξ _[(x'/ω_ ξ _– v/(V²_ – _v²) • x' ]._ Since _x'_ is a common factor for the terms within _[. . . . ]_ and since _x'_ = _x – vt,_ the equation becomes _ξ_ = _aω_ ξ _(x – vt)[(1/ω_ ξ _– v/(V²_ – _v²)]._ If we replace the variable _ω_ ξ, which can represent a point with any particular velocity in the _ξ_ , _η_ plane, with the constant _V,_ which represents only one velocity— the velocity of a light beam traveling along the _ξ-_ axis in the direction of increasing _ξ_ — we will obtain Einstein's equation _ξ_ = _φ(v)β(x – vt)._ To accomplish this transformation we need to keep in mind the following particulars: a light beam travels with velocity _V – v_ relative to the origin of _K,_ _a_ represents the function _φ(v),_ the term _x'/ω_ ξ is equal to _t,_ and that in our special case instance where _V_ = _ω_ ξ the term _t_ does equal _x'/(V – v)._ Thus, we rewrite the equation _ξ_ = _aω_ ξ _[(x'/ω_ ξ _– v/(V²_ – _v²) • x' ]_ as _ξ_ = _aV[(x'/(V – v) – v/(V²_ – _v²) • x' ]._ We follow the same pattern of mathematical manipulation we followed to generate Eq. (3.12) including incorrectly taking the square root of only one side of the equation. Following this procedure will produce Einstein's equation _ξ_ = _φ(v)β(x – vt)_. And, since _φ(v) =_ 1, we also produce the equation _ξ_ = _β(x – vt)_

The equation _ξ_ = _Vτ_ is a special case of the equation _ξ_ = _ωτ_ where the term _ω_ represents a point with any particular velocity. If we prefer to adhere strictly to Einstein's thought experiments, we would say _ξ_ = _Vτ_ is a special case of _ξ_ = _ω_ ξ _τ_.

If we examine a familiar example of a special case that occurs in arithmetic, we can appreciate the limitations of a special case. In division there occurs the special case of dividing by ten or a power of ten. The divisor can be 10, 100, 1000, . . . 10n _._ The dividend can be any number. To divide any dividend by ten or a power of ten, express the divisor in the form 10n and then move the decimal point of the of dividend _n_ place values to the left to obtain the quotient. If the divisor is 10-n, the decimal point of the dividend is shifted _n_ place values to the right. For example, 6,321 ÷ 100 = 63.21; the decimal point of the dividend is moved two place values to the left to obtain the quotient. In this special case, the procedure of long division is greatly simplified, but this procedure is only valid with a divisor of ten or a power of ten. For example, we cannot obtain the quotient of 6,321 ÷ 127 by any kind of shifting of the decimal point.

Divisors of ten or a power of ten have something in common with all dividends. The place values that are used in the expression of the dividends are the following: 10, 100, 1000, . . . 10n. In a similar fashion the term _V_ in the equation _ξ_ = _Vτ_ has something in common with the term _τ_ when it is expressed with the quantities _t_ and _x'_ since _τ_ = _a[t – v/(V²_ _– v²) • x'],_ and also the term _V_ has something in common with the term _t_ when it is expressed as _t_ = _x'/(V – v)._

To divide a number by 10, we can use a shortcut. We move the decimal point of the number we are dividing one place value to the left. This shortcut works because all the place values in base 10 have something in common with the number 10. Einstein employs a system in which the velocity of light, _V_ , has something in common with both the time, _t_ , of the rest system and the time, _τ_ , of the moving system.

It is also interesting to note the way the velocity _ω_ ξ from the equation _ξ_ = _ω_ ξ _τ_ behaves in Eq. (5.4), _x_ = _(ω_ ξ \+ _v)/[1_ \+ _(vω_ ξ _/ V²)] • t._ The term _ξ_ is transformed into _β(x – vt),_ and _τ_ is transformed into _β(t – vx/V²)_ to produce Eq. (5.4), but _ω_ ξ remains unchanged. The way the equation is presented gives the impression that _ω_ ξ has been transformed but it has not. It is packaged with the terms that are used to transform the variables _ξ_ and _τ_ into the variables _x_ and _t,_ and they provide the appearance of transforming _ω_ ξ. Viewed from the rest system _K,_ the velocity _ω_ ξ would be augmented by the velocity _v_ so that _ω_ x _(of the rest system)_ = _ω_ ξ \+ _v._

If we think of the moving system _k_ as a long train of flat-bed cars pulled by a locomotive, we can picture _ω_ ξ as a runner running along the flat-bed cars. If we imagine the train was traveling slowly due east at 2 mph, and the runner running due west at 2 mph, to an observer standing on the railroad embankment, the runner would appear to be running in place. The observer standing on the railroad embankment represents an observer in the rest system _K._

This situation is represented by the formula _ω_ ξ \+ _v_ = _0_ or -2 mph due west + 2mph due east = 0 mph. When _ω_ ξ \+ _v_ = _0_ mph the value for _x_ is zero because _(0)/[1_ \+ _(vω_ ξ _/ V²)] • t_ = _x_ = _0._ If we imagine that there is a runner on every flat-bed car running due west at 2 mph, according to Eq. (5.4) the _x_ coordinate for each runner is zero, but this is clearly impossible. If the runner on the last flat-bed car was assigned the _x_ coordinate zero that value could not be assigned to the others. Eq. (5.4), _x_ = _(ω_ ξ \+ _v)/[1_ \+ _(vω_ ξ _/ V²)] • t,_ is incomplete. When the velocities _ω_ ξ and _v_ cancel out Eq. (5.4) assigns all points the _x_ coordinate of zero regardless of their true position on the _X-_ axis. This is not surprising considering that Eq. (5.4) is constructed from equations that represent special cases. Eq. (5.4) needs a term such as _x_ 0 to indicate the position of a point or a runner before they are set into motion.

Chapter 5: Scalars or Vectors or Neither?

Chapter Summary: A Continuation of the Analysis of "Section 5. The Addition Theorem for Velocities" from _On the Electrodynamics of Moving Bodies_

Let us continue our analysis of Einstein's discussion of a point in motion in the moving system _k._ As we continue our analysis, we will focus on a handful of equations, which Einstein introduces in rapid succession.

"Thus _,_ according to our theory, the vector addition for velocities holds only to first approximation. Let _U²_ = _(dx/dt)²_ \+ _(dy/dx)²_ ....................(5.7), _ω²_ = _ω_ ξ _²_ \+ _ω_ η _²_...............(5.8), and _α_ = arctan _ω_ η _/ω_ ξ..............(5.9); _α_ is then to be considered as the angle between the velocities _v_ and _ω._ After a simple calculation we obtain

_U_ = _[( v²_ + _ω²_ + _2vω cosα) – (vω sinα/ V)²]_ ½ ,...............(5.10).

_1_ \+ _vω cosα/ V²_

It is worth noting that _v_ and _ω_ enter into the expression for the resultant velocity in a symmetrical manner. If _ω_ also has the direction of the _X_ -axis _(Ξ_ _-_ axis _)_ , we get

_U_ = _v_ + _ω ,_ ...............(5.11)."86

_1_ \+ _vω/V²_

It will be helpful to examine Eqs. (5.7) through (5.11) in detail. We will examine both the intermediate steps that are employed to produce the equations and also the validity of several of the equations.

To find _dx/dt_ we treat all the terms on the right side of Eq. (5.4) as constants except for _t._ If we denote our constant by _C,_ then _C_ = _(ω_ ξ \+ _v)/[1_ \+ _(vω_ ξ _/ V²)],_ and we can rewrite the equation as _x_ = _Ct._ Therefore, _dx/dt_ = _C_ = _(ω_ ξ \+ _v)/[1_ \+ _(vω_ ξ _/ V²)]_ and _(dx/dt)²_ = _(ω_ ξ _²_ \+ _2ω_ ξ _v_ \+ _v²)/[1_ \+ _(vω_ ξ _/ V²)]²_.

To find _dy/dt_ we treat all the terms on the right side of Eq. (5.5) as constants except for _t._ If we denote our constant by _C,_ then _C_ = _[1 – (v²_ _/ V²)]_ ½ _/ [1_ \+ _(vω_ ξ _/_ _V²)] • ω_ η, and we can rewrite the equation as _y_ = _Ct._ Therefore, _dy/dt_ = _C_ or _dy/dt_ = _[1 – (v²_ _/ V²)]_ ½ _/ [1_ \+ _(vω_ ξ _/_ _V²)] • ω_ η _,_ and _(dy/dt)²_ = _[1 – (v²_ _/ V²)] • ω_ η _²/ [1_ \+ _(vω_ ξ _/_ _V²)]²_ , which equals _[_ _ω_ η _² – ω_ η _²(v²_ _/_ _V²)]/ [1_ \+ _(vω_ ξ _/_ _V²)]²._ Eq. (5.7) states _U²_ = _(dx/dt)²_ \+ _(dy/dx)²_ ,and since _(dx/dt)²_ and _(dy/dt)²_ have the same denominator, _[1_ \+ _(vω_ ξ _/_ _V²)]²_ , we have, _U²_ = _{ω_ ξ _²_ \+ _ω_ η _²_ \+ _v²_ \+ _2vω_ ξ – _(v²_ _/ V²)ω_ η _²}/ [1_ \+ _(vω_ ξ _/_ _V² )]²._ Since _ω²_ = _ω_ ξ _²_ + _ω_ η², we have _U²_ = _{ω²_ + _v²_ + _2vω_ ξ – _(v²_ _/ V²)ω_ η _²}/ [1_ + _(vω_ ξ _/V²)]²._

Eq. (5.8), _ω²_ = _ω_ ξ _²_ \+ _ω_ η², represents the vector addition of the two components of vector _ω._ The component vectors are _ω_ ξ and _ω_ η. They are added together to produce a third vector, _ω_ , which is called the resultant. The vector _ω_ ξ is the _ξ-_ component of vector _ω._ The vector _ω_ η is the _η-_ component of vector _ω._ The vectors _ω_ ξ and _ω_ η are added together by a method known as the "Parallelogram of Forces." It is also known as the "Parallelogram Law." Since vectors have both magnitude and direction, they cannot be added together in the same manner as scalars. For example, temperature is a scalar so 30°F. + 40°F. = 70°F. However, velocity is a vector so you cannot add the vectors 30 mph due east + 40 mph due north and obtain 70 mph due east and due north. The Parallelogram of Forces is a method of adding vectors together that allows you to mesh or combine the magnitude and direction of one vector with the magnitude and direction of another vector. The resulting vector has a proportional mixture of the magnitude and direction of the two vectors that have been added together. In Eq. (5.9), _α_ = _arctan ω_ η _/ω_ ξ _,_ the term _arctan_ denotes an inverse trigonometric function, the _arctan ω_ η _/ω_ ξ is an angle _α_ , whose tangent is _ω_ η _/ω_ ξ or _tan α_ = _ω_ η _/ω_ ξ _._ Since _tan α_ = _sin α /cos α,_ it follows that _sin α /cos α_ = _ω_ η _/ω_ ξ _. Sin α_ in a given right triangle is the ratio of the length of the side opposite to angle _α_ to the length of the hypotenuse, and therefore, _sin α_ = _ω_ η _/ω. Cos α_ in a given right triangle is the ratio of the length of the adjacent side of angle _α_ to the length of the hypotenuse, and therefore, _cos α_ = _ω_ ξ _/ω._

We are now ready to generate Eq. (5.10). We have already established that _U²_ = _{ω²_ \+ _v²_ \+ _2vω_ ξ – _(v²_ _/ V²)ω_ η _²}/ [1_ \+ _(vω_ ξ _/_ _V²)]²_ .

The equation _cos α_ = _ω_ ξ _/ω_ can be rewritten as _ω_ ξ = _ω_ • _cos α,_ and therefore, the term _2vω_ ξ becomes _2vω_ • _cos α._ Likewise the term _(vω_ ξ _/V²)_ becomes _(vω_ • _cos α /V²)._ The equation _sinα_ = _ω_ η _/ω_ can be rewritten as _ω_ η = _ω_ • _sin α_ , and therefore, the term _(v²_ _/ V²)ω_ η _²_ becomes _(v²_ _/ V²)(ω_ • _sin α)²_ or _(vω_ • _sin α /V)²._ With these changes the equation becomes the following: _U²_ = _{ω²_ \+ _v²_ \+ _2vω_ • _cos α_ – _(vω_ • _sin α_ _/V)²}/ [1_ \+ _vω_ • _cos α /V²]²._ If we take the square root of each side of the equation we obtain Eq. (5.10), which is as follows: _U_ = _[ω²_ \+ _v²_ \+ _2vω_ • _cos α_ – _(vω_ • _sin α_ _/ V )²]_ ½ _/ [1_ \+ _vω_ • _cos α /V²]._

To produce Eq. (5.11), _v_ \+ _w/[1_ \+ _vw/ V²],_ we assume that vector _ω_ is traveling along the _X-_ axis or as Einstein states, "If _ω_ also has the direction of the _X-_ axis _(Ξ-_ axis _)_ . . .."87 When the vector _ω_ travels along the _X-_ axis, the angle _α_ = _0°._ Therefore, the _cos α_ = _1_ and the _sin α_ = _0._ When we insert those values into Eq. (5.10) we obtain the following: _U_ = _[ω²_ \+ _v²_ \+ _2vω(1)_ – _(vω(0)_ _/ V)_ 2 _]_ ½ _/ [1_ \+ _vω(1)/_ _V²]._ The simplified version of the equation is, _U_ = _[ω²_ \+ _v²_ \+ _2vω]_ ½ _/ [1_ \+ _vω/_ _V²)]._ The square root of the term _[ω²_ \+ _v²_ \+ _2vω]_ is _v_ \+ _ω,_ and thus we have produced Eq. (5.11), _v_ \+ _w/[1_ \+ _vw/ V²]._

This concludes our examination of the intermediate steps for Eqs. (5.7) through (5.11). Now, we will examine the validity of several of the equations.

The validity of Eq. (5.9), _α_ = _arctan ω_ η _/ω_ ξ, can be questioned. Since _ω_ is a vector it has both magnitude and direction. It follows that the component vector _ω_ η expresses both the magnitude and direction that vector _ω_ has relative to the _η-_ axis. For simplicity, let us denote that the direction along the _η-_ axis that has increasingly positive _y_ values is north, and also let us denote that the direction along the _ξ-_ axis that has increasingly positive _x'_ values is east. For example, _ω_ η could be a component vector with a velocity of 3 mph due north, and _ω_ ξ could be a component vector with a velocity of 4 mph due east. If we rewrite Eq. (5.9), _α_ = _arctan ω_ η _/ω_ ξ _,_ as _tan α_ = _ω_ η _/ω_ ξ _,_ we obtain _tan α_ = 3 due north/4 due east. Since tangents are ratios with no units of measure attached to them, the value we want to obtain is _tan α_ = _3/4._

This same dilemma would be present for the values _sin α_ = _ω_ η _/ω_ and _cos α_ = _ω_ ξ _/ω._ Since _sin α_ = _ω_ η _/ω,_ it will be instructive to interpret it in terms of the component vectors from our example, where we gave each of the component vectors a specific velocity and direction. From our previous example _ω_ η is a component vector with a velocity of 3mph due north. We can calculate the value of vector _ω_ by using the parallelogram law and the specific values we gave to the component vectors _ω_ η and _ω_ ξ, which are 3 mph due north and 4 mph due east, respectively. Thus, _ω_ is a vector with the velocity 5 mph northeast by east. So using our previous example as a guide, _sin α_ = 3 due north/5 northeast by east. The units of measure represented by mph cancel each other out. But, the problem remains; _sin α_ is a ratio, and no units of measure should be attached to a ratio. This prohibition would include units that measure or define spatial directions. This dilemma may prove to be an insurmountable problem for the production of Eqs. (5.10) and (5.11).

Before we reach any conclusions about Eqs. (5.10) and (5.11), we must clarify the type of vector multiplication Einstein is employing in Eq. (5.9). Is Einstein employing the scalar multiplication of two vectors? The _Mathematics Dictionary_ offers the following explanation of the scalar multiplication of two vectors:

"The scalar product of two vectors is the _scalar_ which is the product of the lengths of the vectors and the cosine of the angle between them. This is frequently called the dot product, denoted by **A • B** , or the inner product. It is equal to the sum of the products of the corresponding components of the vectors."88

The definition above does away with the dilemma posed by the fact that vectors have direction by removing the directional component from the vector and thus producing a scalar. Unfortunately, it does introduce a complicating factor of its own by requiring that the product of the lengths of the vectors must be multiplied by the cosine of the angle between them. For example, since _tan α_ = _ω_ η _/ω_ ξ, the dot product **A • B** would be as follows: _[_ _ω_ η **•** _1/ω_ ξ _]_ • _cos 90°_ , which results in _tan α_ = _0_. This is so because the angle between _ω_ η and _ω_ ξ is 90° and cos 90° = 0.

If Einstein were to use scalar multiplication of vectors for the multiplication of all his vectors, the result would be that all his vectors that were multiplied would lose their directional quality. It seems unlikely that Einstein's resultant velocity _,_ _U,_ could be obtained from vectors that have lost their directional component through scalar multiplication. The _Mathematics Dictionary_ offers the following definition of a non-scalar method for multiplying vectors, and the method is denoted as the vector multiplication of two vectors. This method of multiplying vectors is also incompatible with Einstein's results:

"The vector product of two vectors A and B is the vector C whose length is the product of the lengths of A and B and the sine of the angle between them (the angle from the first to the second), and which is perpendicular to the plane of the given vectors and directed so that the three vectors in order A, B, C form a _right-handed trihedral_."89

Neither the scalar multiplication of two vectors nor the vector multiplication of two vectors produce results compatible with those required by Einstein. Einstein seems to treat the vectors in Eq. (5.9) as merely scalar measurements. He disregards their directional component and employs the vectors exclusively as measures of length. He does this because it serves his purpose to do so. Since we have questioned the validity of Eq. (5.9), can we produce Eq. (5.10) without the use of Eq. (5.9)? As we have shown, without the use of Eq. (5.9) an only slightly different version of Eq. (5.10) can be obtained, i. e., _U²_ = _ω²_ \+ _v²_ \+ _2vω_ ξ – _(v²_ _/V²) ω_ η _²/ [1_ \+ _(vω_ ξ _/_ _V²)]²_ . The only significant differences are _ω_ ξ instead _ω_ • _cos α_ and _ω_ η instead of _ω_ • _sin α._ To obtain Eq. (5.11) we set _ω_ η _²_ = _0._ This serves two purposes, first it rids the above equation of the term _(v²_ _/V²)ω_ η _²,_ and secondly, when substituted in Eq. (5.8), _ω²_ = _ω_ ξ _²_ \+ _ω_ η _²_ , we obtain _ω_ = _ω_ ξ _._ These two changes allow us to produce Eq. (5.11) when we take the square root of each side of the equation.

But, do they really allow us to produce Eq. (5.11)? Can we set _ω_ η _²_ = _0_ with no consequences? The answer is _no._ If we set _ω_ η _²_ = _0,_ then _y_ in Eq. (5.5) equals zero and thus, _(dy/dx)²_ equals zero. When _(dy/dx)²_ equals zero Eq. (5.10) cannot be produced.

Since Eq. (5.10), _U_ = _[ω²_ \+ _v²_ \+ _2vω_ • _cos α_ – _(vω_ • _sin α_ _/V )²]_ ½ _/ [1_ \+ _vω_ • _cos α /V²],_ can only be produced using Eq _._ (5.9), we can question the validity of Eqs. (5.10) and (5.11). The flaw with Eq. (5.10) is that we replace two component vectors with their scalar counterparts. The component vector _ω_ ξ is replaced with its scalar counterpart _ω_ • _cos α,_ and the component vector _ω_ η is replaced with its scalar counterpart _ω_ • _sin α._ As we have noted to produce Eq. (5.11), Einstein requires that _α_ = 0° _._ Thus, _sin α_ = _0_ and the term _(vω_ • _sin α /V )²_ disappears. Since _cos α_ = _1,_ the term _2vωcosα_ becomes merely _2vω._ What is the term _2vω?_ It must be _2_ multiplied by _vector v_ multiplied by _scalar ω._ This is despite the fact that _ω_ ξ from the term _2vω_ ξ is a vector. When we replace the term, component vector _ω_ ξ with the term _ωcosα_ from the equation _ω_ ξ = _ωcosα,_ we are replacing a component vector with its scalar equivalent.

Thus, we have the equation _U_ = _[ω²_ \+ _v²_ \+ _2vω]_ ½ _/ [1_ \+ _vω_ • _cos a /V²]._ The numerator of the right side of the equation should be interpreted in the following manner. The square root of the sum of the following three terms: vector _ω²_ \+ vector _v²_ \+ _2_ multiplied by vector _v_ multiplied by scalar _ω._ Since _ω_ is a scalar, the square root of the three terms is not _ω_ \+ _v._ Therefore, Eq. (5.11) is invalid because the numerator cannot be _ω_ \+ _v._

We could try to produce Eq. (5.11), _U_ = _(v_ \+ _ω)/[1_ \+ _vω/V²]_ by stating it is merely the velocity component of Eq. (5.4), _x_ = _(ω_ ξ **+** _v)/[1_ **+** _v_ _ω_ ξ _/ V²]_ • _t,_ where _ω_ = _ω_ ξ . As we have pointed out before, Eq. (5.4) is not a valid equation. We could also try to maintain the validity of Eqs. (5.10) and (5.11) by claiming all the terms are scalars and not vectors. But, if we take away the directional component from the terms, it becomes impossible to calculate angle _α,_ which is the angle between vector _v_ and vector _ω._ We should note that both vector _v_ and component vector _ω_ ξ run along the _ξ_ -axis in the direction of increasing _x'_.

The preceding discussion may have seemed unduly confusing. There is an explanation for the confusion. In Thomas H. Barr's textbook _Vector Calculus_ he states that the division of vectors is not defined. He also states that under certain circumstances the multiplication of vectors is not defined. Since his examples of the correct determination of vector products and quotients use unit vectors, we must familiarize ourselves with that term. T. H. Barr provides a concise definition of unit vectors in "Chapter 1. Coordinate and Vector Geometry" of his textbook.

"Establish a rectangular coordinate system in three-dimensional space. Any vector **a** can be put into standard position by translating it so that its tail lies at the origin. We call the coordinates ( _a_ 1, _a_ 2, _a_ 3) of the head of **a** in standard position the components of **a**. We denote by **i** the vector with components (1, 0, 0), by **j** the vector with components (0, 1, 0) and **k** the vector with components (0, 0, 1). If **a** is any vector with components ( _a_ 1, _a_ 2, _a_ 3) then **a** = _a_ 1 **i** \+ _a_ 2 **j** \+ _a_ 3 **k.** " 90

With the introduction of the unit vectors **i** , **j** , and **k** , we are ready to encounter T. H. Barr's examples, which demonstrate that vector division and certain types of vector multiplication are not defined. Since vector division is not defined, Eq. (5.9) is invalid. The type of vector multiplication that is not defined will be applicable to our analysis of Eq. (5.10), and the vector multiplication that is not defined, will invalidate Eq. (5.10). The following quotation is from question one of "Exercises 1.4," and only the answers relevant to our analysis are included.

"Let **a** = 5 **i –** **j** , **b** = 7 **i –** 3 **j** \+ 2 **k** , **c** = – 4 **i** \+ 9 **j** – 8 **k** , and **d** = –3 **i –** 11 **j** \+ 7 **k.**

**b** / **d** not defined **ad** not defined"91

According to T. H. Barr, the division of vector **b** by vector **d** is not defined. Both vectors **b** and **d** are vectors defined in three dimensional space. We can extrapolate from T. H. Barr's example and conclude that vector division is not defined for all vectors regardless of the dimensionality of the space in which they occur. The term _ω_ η _/ω_ ξ states that component vector _ω_ η , which occurs in one dimension, is divided by component vector _ω_ ξ, which also occurs in one dimension. Thus, the term _ω_ η _/ω_ ξ is not defined. Therefore, Eq. (5.9), _α_ = arctan _ω_ η _/ω_ ξ, is invalid.

Also, according to T. H. Barr, the multiplication of vector **a** by vector **d** is not defined. Vector **a** is a vector that occurs in two dimensions. Vector **d** is a vector that occurs in three dimensions. We can extrapolate from this example and conclude that the multiplication of a vector occurring in one dimension by a vector occurring in two dimensions is not defined, and thus, the term _2vωcosα_ from Eq. (5.10) is not defined. Therefore, Eq. (5.10) is invalid. The velocity vector _v_ is a vector that occurs in one dimension. The velocity vector _v_ is restricted to the _X_ -axis of the system _K._ The velocity vector _ω_ is vector that occurs in two dimensions. The velocity vector _ω_ is restricted to the _X, Y_ plane of the system _K._

We can use T. H. Barr's examples of vector operations that are not defined to analyze an ambiguous statement made by Einstein regarding Eq. (5.10). Directly following Eq. (5.10) Einstein states, "It is worth noting that _v_ and _ω_ enter into the expression for the resultant velocity in a symmetrical manner."92 The expression for the resultant velocity is Eq. (5.10), _U_ = _[ω²_ \+ _v²_ \+ _2vω_ • _cos α_ – _(vω_ • _sin α /V )²]_ ½ _/ [1_ \+ _vω_ • _cos α /_ _V²]._ According to the _Mathematics Dictionary,_ "a symmetric relation is a relation which has the property that if _a_ is related to _b,_ then _b_ is related in like manner to _a._ The equals relation of algebra is symmetric, since if _a_ = _b,_ then _b_ = _a._ "93 Since _vω_ = not defined and not defined = _vω,_ we can say _vω_ and not defined form a symmetric relation. The _Mathematics Dictionary_ also states, "A relation is asymmetric if there are no pairs _(a, b)_ such that _a_ is related to _b_ and _b_ is related to _a._ The property of _being older than_ is asymmetric; if _a_ is older than _b,_ then _b_ is not older than _a._ "94 Since vector _ω_ occurs in one more dimension than vector _v_ and vector _v_ does not occur in one more dimension than vector _ω,_ we can say the relation between vector _ω_ and vector _v_ is asymmetric with regard to their dimensionality. This analysis has not given a precise meaning to Einstein's ambiguous statement, but it has shown there are several ways with which further interpretation could precede.

This concludes are analysis of the validity of Eqs. (5.7) through (5.11). We can now turn our attention to Einstein's analysis of Eq. (5.11).

It is from an analysis of this invalid equation that Einstein determines two fundamental principles. The first principle is that adding together two velocities that are smaller than the velocity of light results in a velocity that is smaller than the velocity of light. The second is that the velocity of light cannot be changed by adding the velocity of light to any sub-light velocity.

We will continue with Einstein's examination of Eq. (5.11), _(v_ \+ _ω)/[1_ \+ _vω/ V²]_ to show how he demonstrated the two principles noted above.

"It follows from this equation that the composition of two velocities that are smaller than _V_ always results in a velocity that is smaller than _V._ For if we set _v_ = _V – κ,_ and _ω_ = _V – λ,_ where _κ_ and _λ_ are positive and smaller than _V,_ then

_U_ = _V_ • _(2V – κ – λ)_ < _V_ ,...............(5.12)."95

_(2V – κ – λ_ \+ _κλ/ V)_

Eq. (5.12) states that _U_ equals the multiplication of the velocity of light, _V,_ by some proper fraction (such as _._ 9 for instance) and thus, the resulting value is less than the velocity of light _V._

The numerator of Eq. (5.12) comes from the addition of _v_ \+ _ω,_ and since _v_ = _(V – κ)_ and _ω_ = _(V_ – _λ),_ we have _(V – κ)_ \+ _(V – λ)._ This sum can be rewritten as _2V – κ – λ._

The denominator of Eq. (5.12) is _1_ \+ _vω/V²,_ which can be rewritten as a single fraction with the common denominator _V²_ and with the same substitutions as above for _v_ and _ω._ The result is _[V²_ \+ _(V – κ)(V – λ)]/ V²_ or _[2V² – Vκ –Vλ_ \+ _κλ]/ V²._ If we divide each term separately by _V² ,_ the denominator can be rewritten in this form _2 – κ/V – λ/V_ \+ _κλ/V²._ The equation now assumes the form of _U_ = _[2V – κ – λ]/ [2 – κ/V – λ/V_ + _κλ/ V²]._ If we multiply the right side of the equation by one in the form of _V/ V,_ we obtain _U_ = _V_ _[2V – κ – λ]/ [2V – κ – λ_ \+ _κλ/V],_ which is less than _V._ This is so because _[2V – κ – λ]_ is a smaller number than _[2V – κ – λ_ \+ _κλ/V],_ and a smaller number divided by a larger number gives us a fraction that is less than one. We should recall that _κ_ and _λ_ are positive values that are smaller than the speed of light.

Einstein continues in his examination of Eq. (5.11).

"It also follows that the velocity of light _V_ cannot be changed by compounding it with a 'subluminal velocity.' For this case we get

_U_ = _V_ + _ω_ = _V_ ,...............(5.13)."96

_1_ \+ _ω/V_

Eq. (5.13) is obtained by letting _v_ = _V_ in Eq. (5.11). The numerator becomes _V_ \+ _ω._ The denominator becomes _1_ \+ _Vω/ V²_ or _1_ \+ _ω/ V._ The denominator can be written as a single fraction _(V_ \+ _ω)/ V._ Thus the equation becomes _U_ = _(V_ \+ _ω)/[(V_ \+ _ω)/V]._ If we multiply the right side of the equation by one in the form of _V/ V,_ we obtain _U_ = _V(V_ \+ _ω)/(V_ \+ _ω)._ It is apparent that the right side of the equation equals _V_ because the term _V_ is multiplied by the same quantity, _(V_ \+ _ω),_ that it is divided by, _(V_ \+ _ω)._ Thus we obtain _U_ = _V(V_ \+ _ω)/(V_ \+ _ω)_ = _V._ If we multiply the middle term of the previous equation, _V(V_ \+ _ω)/(V_ \+ _ω),_ by one in the form of _(1/ V)/(1/ V),_ we obtain Eq. (5.13), _U_ = _(V_ \+ _ω)/[1_ \+ _ω/V]_ = _V._ In Eq. (5.12) a fraction is multiplied by one in the form of _V/ V_ and in Eq. (5.13), as we have just seen, a fraction is multiplied by one in the form of _(1/V)/(1/V)_ _._

The final point Einstein makes in the fifth section involves the introduction of another coordinate system _k'._ Since the fifth section is the last section in "A. Kinematic Part," of _On the Electrodynamics of Moving Bodies_ it is the final point we will discuss. As we have noted _K_ is the rest system. The moving system _k_ moves with velocity _v_ relative to the rest system _K_ in the direction along the _X_ -axis of increasing _x._ The _X_ -axes of the two systems coincide, and the " _X_ -axis" of the _k_ system is denoted as the _ξ_ -axis. The third coordinate system _k'_ moves with velocity _ω_ relative to the moving system _k_ in the direction of increasing _ξ._ The " _X_ -axis" of the _k'_ system coincides with the _X_ -axes of the rest system _K_ and the moving system _k;_ and is denoted as the _Ξ_ -axis. All three axes coincide, the _X, ξ_ and _Ξ_. The velocity of the _k'_ system relative to the resting system _K_ is as follows: _(v_ \+ _ω)/[1_ \+ _vω/V²]._

In this scenario, three axes coincide, and each axis is from a different system. This is difficult to imagine since Einstein describes each system as consisting of, "three mutually perpendicular rigid material lines originating from one point."97 The three rigid material lines that form the _X, ξ,_ and _Ξ_ axes cannot coincide.

Apart from that difficulty, the expression itself is merely Eq. (5.11) with a slight change in nomenclature. When we were first introduced to the moving point _ω_ , it was moving in the plane formed by the _X_ and _Y_ axes. When we meet the moving point _ω_ again in Eq. (5.11), it represents a point that seems to be traveling only along the _X_ -axis in the direction of increasing _x_. In the fifth section, _ω_ represents the velocity of the coordinate system _k'_ , which is moving along the _X_ -axis in the direction of increasing _x_.

Chapter 6: Conclusion: Sorting Through a Bag of Broken Parts

Chapter Summary: A Summary of the Mathematical Inconsistencies in the First Part, "A. Kinematic Part" of _On the Electrodynamics of Moving Bodies_

The Eq. (1.1), _t_ B _– t_ A = _t'_ A _– t_ B is invalid for moving systems, if we adopt the conventional definition of distant synchronized clocks. The conventional definition of distant synchronized clocks requires the following: the clocks must agree in the time that they keep. In a moving system, when light beam makes a round-trip journey from point _A_ to a mirror at point _B,_ where it is reflected back to point _A,_ the duration of the outbound journey, represented by _t_ B _\- t_ A, is not equal to the duration of the return journey, represented by _t'_ A _– t_ B.

The Eq. (3.1), _½{τ[0_ , _0, 0, t]_ \+ _τ[0, 0, 0, t_ \+ _x'/(V – v)_ \+ _x'/(V_ \+ _v)]}_ = _τ[x', 0, 0, t_ \+ _x'/(V – v)]_ , is a complicated restatement of Eq. (1.1), and thus, it is also invalid for moving systems if we adopt the conventional definition of distant synchronized clocks. The equation states that the total duration of a light beam's round-trip journey multiplied by _½_ is equal to the duration of the outbound portion of the light beam's journey.

The Eq. (3.2), _½[1/(V – v)_ \+ _1/(V_ \+ _v)]∂τ/∂t_ = _∂τ/∂x'_ \+ _[1/(V – v) ] ∂τ/∂t_ , is invalid because it cannot be legitimately derived from Eq. (3.1). Since Eq. (3.3), _∂τ/∂x'_ \+ _v/(V²_ \+ _v²) ∂τ/∂t_ = _0_ is merely a rearranged version of Eq.(3.2), it also is invalid because it cannot be legitimately derived from Eq.(3.1).

The unnumbered equation, _∂τ/∂y_ = _0,_ and the unnumbered equation, _∂τ/∂z_ = _0_ , are both invalid because they cannot be legitimately derived from Einstein's two implicitly reformulated versions of Eq. (3.1). The first reformulated version of Eq. (3.1) fails to produce equation _∂τ/∂y_ = _0_ , and the second reformulated version of Eq. (3.1) fails to produce equation _∂τ/∂z_ = _0_.

The Eq. (3.4), _τ = a[t – v/(V²_ \+ _v²)_ • _x' ]_ , is invalid because Einstein incorrectly claims that some fashion of integration, when applied to the three partial derivative equations listed above, along with some form of linear transformation, yields this equation. The three partial derivative equations: _∂τ/∂x'_ \+ _v/(V²_ \+ _v²) ∂τ/∂t_ = _0,_ _∂τ/∂y_ = _0,_ and _∂τ/∂z_ = _0_ do not generate Eq. (3.4) through some fashion of integration along with some form of linear transformation.

The four transformation equations are invalid. Einstein incorrectly claims all four of these equations can be produced through combining Eq. (3.4) with various simple equations. Each of these simple equations describes the distance a light beam travels. The light beam makes a separate journey along each one of the three axes of the moving system _k._ The four transformation equations are the following: Eq. (3.11) _τ_ = _φ(v)β(t – vx/V²),_ Eq. (3.12) _ξ_ = _φ(v)β(x – vt),_ Eq. (3.13) _η_ = _φ(v)y,_ and Eq. (3.14) _ζ_ = _φ(v)z_.

Einstein does not demonstrate his claim that Eq. (3.16), _x²_ \+ _y²_ \+ _z²_ = _V²t²,_ can be transformed, using the transformation equations, into Eq. (3.17), _ξ²_ \+ _η²_ \+ _ζ²_ = _V_ ² _τ²_. Einstein does not provide us with the details of this calculation in _On the Electrodynamics of Moving Bodies._ In "Appendix One" of his book _Relativity: the Special and the General Theory,_ he does provide a version of this calculation, but the calculation is invalid.

The unnumbered term from the fourth section of _On the Electrodynamics of Moving Bodies,_ _½ t(v²_ _/ V²)_ sec., gives us the amount of time a clock transported from point _A_ to point _B_ will lag behind a clock located at point _B._ Before the clock at point _A_ was transported to point _B,_ the clock at point _A_ was synchronized with the clock at point _B._ The variable _t_ is the time needed to transport the clock from point _A_ to point _B._ The unnumbered term is invalid for three reasons. The first reason is a minor point. There is no need for a unit of time measured in seconds to be appended to the term since the variable _t_ is a unit of time measured in seconds. The second and third reasons also apply to a similar unnumbered term, _½ (v²_ _/ V²)_ sec., which is also from fourth section of Einstein's paper. The second reason is that both of these terms are approximations of the term _[1 – (1 – v²/ V²_ _)_ ½ _]_ sec., but Einstein does not acknowledge the terms are approximations. He seems to ambiguously imply they are not approximations. The claim that the term _½ t(v²_ _/ V²)_ sec. is a version of the term _[1 – (1 – v²/ V²_ _)_ ½ _]_ sec. can only be made by accepting the notion that the variable _t_ is implied from the context in which the term _[1 – (1 – v²/ V²_ _)_ ½ _]_ sec. occurs. The notion that text implies we should multiply the term _[1 – (1 – v²/ V²_ _)_ ½ _]_ sec. by the variable _t_ is reasonable. The third reason is that both terms do not take into account Einstein's definition of distant synchronized clocks. According to Einstein's definition, distant synchronized clocks in a moving system do not need to agree in the time they keep, and, in fact, they will not agree in the time they keep. Yet, these two terms, _½ t(v²_ _/ V²)_ sec. and _½ (v²_ _/ V²)_ sec., operate on the assumption that distant synchronized clocks in a moving system agree in the time they keep.

The Eq. (5.4), _x_ = _(ω_ ξ \+ _v)/[1_ \+ _(vω_ ξ _/ V²)] • t,_ is incomplete. The equation assigns every instance of the occurrence of _ω_ ξ \+ _v_ = _0_ to a value of _x_ = _0_ regardless of the occurrence's actual position on the _X_ -axis. The Eq. (5.4) is also invalid because it is composed of many special case instances that do not apply to a point moving with a constant velocity in the moving system _k._ For example, Eq. (5.4) is partially composed of an equation that describes a light beam traveling in a specific direction in the moving system _k,_ as viewed from the rest system _K._ In this specific instance, because of the properties of light beams and other conditions, the velocity of the light beam when viewed from the rest system _K_ is _V – v._ However, a material point, traveling under the same conditions, with the constant component of velocity of _ω_ ξ, would, when viewed from the rest system _K_ , have a constant component velocity of _ω_ ξ \+ _v._ This is because a material point is carried along by the moving system _k_ and a light beam is not carried along by the moving system _k._

The Eq. (5.9), _α_ = arctan _ω_ η _/ω_ ξ is invalid because component vectors have direction as well as magnitude. The magnitude of _ω_ η _/ω_ ξ is equal to _tan α,_ but there is no method of vector multiplication that can get rid of the directional component of the vector without causing further complications.

The Eq. (5.10), _U_ = _[ω²_ \+ _v²_ \+ _2vω_ • _cos α_ _–(vω_ • _sin α /V )²]_ ½ _/ [1_ \+ _vω_ • _cos α /V²],_ is invalid because component vectors _ω_ η and _ω_ ξ are replaced by _ω_ • _sin α_ and _ω_ • _cos α,_ respectively. Both replacement terms are produced by the scalar multiplication of two vectors, which results in a scalar quantity, and hence, these two terms have no directional component.

The Eq. (5.11), _U_ = _(v_ \+ _ω)/[1_ \+ _vω/V²],_ is not valid because _(v_ \+ _ω)_ is not the square root of _ω²_ \+ _v²_ \+ _2vω,_ although at first glance it may appear to be. This is the case for many complicated reasons. One reason is that _(v_ \+ _ω)²_ gives us the following: (vector _v²_ ) + (vector _ω²_ ) + (vector _v •_ vector _ω_ ) + (vector _ω •_ vector _v_ ). The term _(v_ \+ _ω)²_ is not equal to _ω²_ \+ _v²_ \+ _2vω_ , which gives us the following: (vector _v²_ ) + (vector _ω²_ ) + _2_ (vector _v •_ scalar _ω_ ).

If for the moment we ignore the fact that the term _ω_ is properly assigned the quantity scalar _ω_ , we can find another reason why _(v_ \+ _ω)_ is not the square root of _ω²_ \+ _v²_ \+ _2vω_. If we treat both _v_ and _ω_ as vectors, we should take into account that vector multiplication of two vectors is not commutative, in fact it is anticommutative. The sign that signifies vector multiplication is the following: ×. If (vector _v_ × vector _ω_ ) = vector _γ_ , then (vector _ω_ × vector _v_ )= −vector _γ_ , therefore (vector _v_ × vector _ω_ ) + (vector _ω_ × vector _v_ ) =0 and not _2vω._

The concluding assessment of the "Kinematic Part" of _On the Electrodynamics of Moving Bodies_ is twofold in nature. First, the thought experiments are ambiguous and inconclusive as a partial result of definitions that are ambiguous and contradictory. Secondly, the majority of the mathematical equations introduced by Einstein are invalid.

Note on Eqs. (3.18) and (3.19)

A typographic error occurs in Einstein's Eqs. (3.18) and (3.19). Previously, this typographic error has not been formally acknowledged. In the portions of both Eqs. (3.18) and (3.19) that are located between the equal signs, there is an erroneous duplication of the term _(–v)_. Einstein presents the equations as the following: _"t'_ = _φ(–v)β (–v)[τ_ \+ _v/V²_ • _ξ ]_ = _φ(v)φ(–v)t,_...............(3.18) and _x'_ = _φ(–v)β(–v) [ξ_ \+ _vτ]_ = _φ(v)φ(–v)x,_ (3.19)."98 In the text of his book _Albert Einstein's Special Theory of Relativity,_ A. I. Miller presents the equations as the following: " _t'_ = _φ(–v)γ[τ_ \+ _v/V²_ • _ξ ]_ = _φ(v)φ(–v)t,_...............(6.35) and _x'_ = _φ(–v)γ[ξ_ \+ _vτ]_ = _φ(v)φ(–v)x,_...............(6.36)."99 The term _β_ is replaced with the term _γ._ Calculating the value of _γ_ reveal that it is equal to _β,_ which equals _1/ (1 – v²/V²)_ ½. The point to note is that there is no _(–v)_ following the term _γ_ in both Eqs. (6.35) and (6.36).

The appendix of A. I. Miller's book, which contains his translation of _On the Electrodynamics of Moving Bodies,_ reproduces Einstein's erroneous versions of the equations: " _t'_ = _φ(–v)β(–v)[τ_ \+ _v/V²_ • _ξ ]_ = _φ(v)φ(–v)t,_...............(3.18) and _x'_ = _φ(–v)β(–v)[ξ_ \+ _vτ]_ = _φ(v)φ(–v)x,_ .........(3.19)."100 There is no acknowledgment of this typographic error by the use of a footnote or by any other method in any of the following texts: A. I. Miller's _Albert Einstein's Special Theory of Relativity_ , John Stachel's _Einstein's Miraculous Year_ , and the _Collected Papers of Albert Einstein_ volume 2 _The Swiss Years: Writings, 1900—1909_.

Endnotes for the Introduction, Chapters One, Two, Three, Four, Five, Six, and the Note on Eqs. (3.18) and (3.19)

1.Arthur I. Miller, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905 1911)_ (London: Addison-Wesley Publishing Company, Inc., 1981), p. 2.

2.Ibid., p. 4.

3.Ibid., p. 123.

4.Ibid., p. 124.

5.Ibid., p. 7.

6.Ibid., p. 124.

7.Albert Einstein, _Relativity: The Special and the General Theory_ (New York: Three Rivers Press, 1961), pp. 26-27.

8.Ibid., p. 27.

9.Arthur I. Miller, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905 1911)_ (London: Addison-Wesley Publishing Company, Inc., 1981), p. 7.

10.Christopher Jon Bjerknes, _Anticipations of Einstein: In the General Theory of Relativity_ (Downers Grove, Illinois:XTX Inc., 2003), p. 7.

11.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p.101.

12.Ibid., p. 129.

13.Albert Einstein, _Relativity: The Special and the General Theory_ (New York: Three Rivers Press, 1961), p .31.

14.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 101.

15.Ibid., p. 102.

16.Ibid., p. 124.

17.Ibid., p. 124.

18.Ibid., p. 125.

19.Ibid., p. 126.

20.Ibid., pp. 123-124.

21.Ibid., p. 125.

22.Ibid., p. 125.

23.Ibid., p. 124.

24.Ibid., p. 125.

25.Ibid., p. 126.

26.Ibid., p. 127.

27.Ibid., p. 127.

28.Ibid., p. 126.

29.Ibid., p. 127.

30.Ibid., p. 128.

31.Ibid., p. 129.

32.Ibid., pp. 126-127.

33.Ibid., pp. 129-130.

34.Ibid., p. 127.

35.Ibid., p. 127.

36.Ibid., p. 127.

37.Ibid., pp. 129-130.

38.Ibid., p. 128.

39.Ibid., p. 132.

40.Ibid., p. xv.

41.Ibid., p. 132.

42.Arthur I. Miller, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905 1909)_ (London: Addison-Wesley Publishing Company, Inc.,1981), p. 209.

43.Larry J. Goldstein and others, _Calculus and Its Applications_ , 4th ed. (Englewood Cliffs: Prentice-Hall Inc., 1987), p. 351.

44.Ibid., p. 378.

45.Ibid., p. 378.

46.Arthur I. Miller, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905 1909)_ (London: Addison-Wesley Publishing Company, Inc.,1981), p. 209.

47.Glenn James and Robert C. James, _Mathematics Dictionary: Students Edition,_ 2nd ed. (Princeton: D. Van Nostrand Company, Inc., 1959), p. 273.

48.Ibid., p. 113.

49.Ibid., p. 389.

50.Ibid., p. 389.

51.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 132.

52.Ibid., p. 132.

53.Arthur I. Miller, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905 1911)_ (London: Addison-Wesley Publishing Company, Inc., 1981), p. 210.

54.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 133.

55.Larry J. Goldstein and others, _Calculus and Its Applications_ , 4th ed. (Englewood Cliffs: Prentice-Hall Inc., 1987), p. 340.

56. Glenn James and Robert C. James, _Mathematics Dictionary_ , 5th ed. (New York: Chapman & Hall, 1992), pp. 252-253.

57. Tom M. Apostol, _Linear Algebra : A First Course with Applications to Differential Equations_ , (New York: John Wiley & Sons, Inc., 1997), pp. 134-135.

58.Glenn James and Robert C. James, _Mathematics Dictionary_ , 5th ed. (New York: Chapman & Hall, 1992), p. 224.

59.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 133.

60.Ibid., pp. 133-134.

61.Arthur I. Miller, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905 1911)_ (London: Addison-Wesley Publishing Company, Inc., 1981), p. 212.

62.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), pp. 134-135.

63.Albert Einstein, _Relativity: The Special and the General Theory_ (New York: Three Rivers Press, 1961), pp. 131 132.

64.Ibid., p. 131.

65.Ibid., p. 134.

66.Lillian R. Lieber, _The Einstein Theory of Relativity_ (New York: Rinehart & Company, Inc., 1936), p. 62.

67.Ibid., pp. 62-63.

68.Ibid., p. 64.

69. Glenn James and Robert C. James, _Mathematics Dictionary_ , 5th edition (New York: Chapman & Hall, 1992), p. 70.

70.Lillian R. Lieber, _The Einstein Theory of Relativity_ (New York: Rinehart & Company, Inc., 1936), pp. 64-65.

71.Ibid., p. 65.

72.Ibid., p. 66.

73.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 135.

74.Ibid., p. 136.

75.Glenn James and Robert C. James, _Mathematics Dictionary_ , 5th edition (New York: Chapman & Hall, 1992), p.210.

76.Ibid., p. 136.

77.Ibid., p. 137.

78.Ibid., p. 137.

79.Ibid., p. 138.

80.Ibid., pp. 137-138.

81.Arthur I. Miller, _Albert Einstein's Special theory of Relativity: Emergence (1905) and Early Interpretation (1905 1909)_ (London:Addison-Wesley Publishing Company, Inc., 1981), p. 402.

82.Ibid., p. 402.

83.Ibid., p. 223.

84.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 140.

85.Ibid., p. 133.

86.Ibid., p. 141.

87.Ibid., p. 141.

88.Glen James and Robert C. James, _Mathematics Dictionary: Students Edition,_ 2nd edition (Princeton: D. Van Nostrand Company, Inc., 1959), p. 263.

89.Ibid., p. 263.

90.Thomas H. Barr, _Vector Calculus_ (Upper Saddle River, NJ: Prentice-Hall Inc., 1997), pp. 45-46.

91.Ibid., p. 52.

92.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 141.

93.Glenn James and Robert C. James, _Mathematics Dictionary_ , 5th edition (New York: Chapman & Hill, 1992), p. 411.

94.Ibid., p. 411.

95.John Stachel, _Einstein's Miraculous Year: Five Papers That Changed the Face of Physics_ (Princeton: Princeton University Press, 1998), p. 141.

96.Ibid., p. 141.

97.Ibid., p. 130.

98.Ibid., p. 135.

99.Arthur I. Miller, _Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905 1909)_ (London: Addison-Wesley Publishing Company, Inc., 1981), pp. 212-213.

100.Ibid., p. 400.

About the Author

Born in 1955, Jim Spinosa remembers being entranced by the science fiction novels he perused in a small, corner bookstore in Denville, New Jersey. Those cramped confines had claimed to contain the largest selection of books in northern New Jersey. His penchant for science fiction engendered an interest in physics. Often daunted by the difficulty of physics textbooks, he questioned whether physics could be presented as clearly and concisely as science fiction, without sustaining any loss in depth. _Nuts & Bolts: Taking Apart Special Relativity_ is an attempt to answer that question.

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