Professor Charles
Bailyn: Okay,
here's the plan for today.
I want to do one last foray
into relativity theory.
And this is going to be a
tricky one, so I hope you're all
feeling mentally strong this
morning.
If not, we--gosh,
we should have ordered coffee
for everyone.
And, in so doing,
I want to introduce one key
concept, and also answer at
least three of the questions
that you guys have asked before
in a more--in more depth,
and also relate the whole thing
back to black holes.
And then, having done that,
we'll have some more questions.
And then, having done that,
I want to get back to
astronomy;
that is to say,
to things in the sky that
actually manifest these
relativistic effects.
So, that's where we're going
today.
And along the way,
as I said, we'll deal with some
of the questions you've been
asking in a deeper kind of way.
In particular--so, questions.
Watch out for the answers to
these questions.
Somebody asked,
"What's special about special
relativity, and what's general
about general relativity?"
How do they relate?
So, we'll come back to that one.
Somebody also asked,
"Why use the speed of light to
convert time into space and vice
versa, to get them in the same
coordinate system?"
So, why use c to convert
time to space and vice versa?
And then, also,
there was the question of,
you know, "What is the
mathematical formulation of
general relativity?"
So, how to express general
relativity in some kind of
equation.
And we'll get to the key
equation, which is something
called a metric,
for general relativity,
and then we're going to stop.
Because to go on from there is
fairly heavy calculus and we're
just not going to do that.
But I want to get at least that
far.
Okay, so let's go back to
special relativity for a minute.
So, special relativity.
Flat space-time, no gravity.
And you'll recall what happens.
As you get close to the speed
of light, all sorts of things
that you thought were kind of
constant and properties of
objects,
like mass and length and
duration, and duration of time,
and things like that,
all start to get weird and
change.
So, length, time,
mass, all these things,
vary with the velocity of the
person doing the measuring.
And so, you could ask the
question, is there anything that
doesn't vary?
Is there anything that's an
invariant?
And the answer is, yes.
There are some things that
don't vary.
So, some things are invariant.
And Einstein actually said
later in his career that it's
actually the invariants that are
important, not the things that
change.
And so, he should have called
his theory invariant theory
instead of relativity theory.
Think of what that would have
done to pop philosophy.
Instead of saying,
"everything is relative," all
this stuff, you would have had
the exact same theory.
You would have called it
invariance theory.
And the pop philosophy
interpretation of this would be,
"some things never change."
And it would have been a whole
different concept in three in
the morning dorm room
conversations.
Okay, so some things are
invariant, what things?
Now, let me first give you a
little bit of a metaphor and
then come back to how this
really works in space-time.
Supposing you're just looking
at an xy-coordinate
system and you have two points
in a two-dimensional space.
So, here's a point and here's a
point.
Now, if you arrange for some
kind of coordinate
system--here's a coordinate
system.
This is x,
this is y--and you ask
how far apart these points are.
Well, you can do that--let's
see.
They're separated in x
by this amount here,
which we'll call delta [Δ]
x. And they're separated
in y by this amount here,
and that's Δ y.
But of course,
those quantities depend on the
orientation of your coordinate
system.
If I now take this coordinate
system and I shift it like this,
now it's going to be totally
different.
Now I'm going to have x
look like this and I'm going to
have y,
Δ y look like that.
So Δ y has gotten a
whole lot smaller.
Δ x has gotten a whole
lot bigger.
And all I did was twist the
coordinate system.
Yeah?
Student: [Inaudible.]
Professor Charles
Bailyn: You still get the
same distance,
thank you very much,
that's exactly right.
The distance is the invariant.
The x-coordinate and the
y-coordinate,
those vary with the coordinate
system, but the distance is the
same.
That's exactly the point.
And so, the quantity--let me
summarize this.
For points on a 2-D space,
Δ x varies.
Δ y varies.
But there is a quantity that is
invariant and that is--well,
let's call it (Δ x^(2))
+ (Δ y^(2)),
which is the distance squared,
(Δ D^(2)).
And this is invariant.
It doesn't matter which way you
twist things around,
that will--that quantity will
remain the same.
So, now, imagine that you've
got events in space-time.
So an event in space-time has
three spatial coordinates and
one time coordinate.
So, it's basically a point in a
four-dimensional space.
And as your velocity changes,
the distance and time also
change.
That's the equivalent of
rotating the coordinate system.
But there is something that
doesn't change,
and let me write that down.
This is usually given the Greek
letter Tau [T]
squared.
And this is equal to (Δ
x^(2)) + (Δ
y^(2)) + (Δ
Z^(2)) - c^(2)
(ΔT)^(2).
And this is invariant.
This is an invariant interval,
sometimes called proper
distance.
And as you change your
velocity--as the space,
as the mass,
as the time all change--this
quantity,
describing the separation of
two events--so,
this describes the separation
of two events,
that quantity changes--doesn't
change.
That quantity is invariant.
Okay, so now,
this answers one of the
questions that was asked before.
Why does one use c^(2)
or c to transform the
space coordinate into the time
coordinate and back?
It's because you need the
c^(2) out here in order
to make this invariant.
If you're calculating the
distance, if you use
x^(2) plus 1/2
y^(2) or some other
constant times y^(2),
you're not going to get
something that's invariant.
And it's only when you use the
c, here,
that you end up with something
that's invariant.
And so, if you think about
these as representing the four
coordinates of the system,
it's clear this one coordinate
is x, one is y,
one is z, just as you
would expect.
And then, there's this other
coordinate, which is c
times T,
but it's negative so it has to
be times the square root of -1.
So, the four coordinates in
space-time can be thought of as
x, y, Z and i cT,
if you want to think of it that
way.
And the time coordinate is
imaginary, because when you
square it, you have to end up
with a negative number.
Don't worry about the details
of that.
But the presence of the
c^(2) here is why you
have to use c, in
particular, to get from time to
space and back.
And that's necessary because
this is the thing that doesn't
vary with velocity.
All right.
So, this is actually kind of a
weird expression.
Because unlike the distance
between two points--and you'll
notice, these three terms put
together,
that's just the distance
squared, ordinarily,
but that is not invariant
anymore,
because there's this other term
here, which can vary.
Unlike the distance,
this doesn't have to be
positive.
You've got three different
cases here.
This interval can be zero for
different points.
In ordinary distance it can
only be zero if the two points
are the same,
but this can be zero for
different points,
for different events.
It can be zero,
it can be negative,
and it can be positive.
So, what does that mean?
What happens when it's zero?
So, if the interval is zero,
that means that the distance
between the events in
light-years,
for example,
is equal to the time separation
in years, because
that's--because this term has to
be exactly equal to that term
there,
in order for them to subtract
out and get zero.
And the c^(2) converts
from light-years to years and
back again.
And so, what does that mean?
That means if you emit a photon
at one event,
that same photon can,
if it's going in the right
direction, be present at the
second event.
So, if you ride along with
light you'll see both--you'll
participate in both these
events.
So, you sit at event one.
You flash a light.
You ride along with the
expanding light waves from that
event and you get to something
one light-year away,
exactly a year later.
And so, if the second event is
one light-year away in distance
and a year later in time,
that same photon will be
present at the second event,
as at the first event.
So, things that have one of
these intervals of zero are
separated by an appropriate
amount so that the same ray of
light can participate in both of
them.
So, if the interval--let's keep
that up there for a minute.
If the interval is negative,
what does that mean?
The distance is less than the
light travel time.
So, the photon is already past
the second event.
So, if you were to emit a ray
of light at event number one,
it would have passed the second
event by the time it occurred.
The photon has already gone by.
And similarly,
if the interval is positive,
then the light photon hasn't
reached event two--hasn't yet
reached event two.
Now, this is important,
because this means that you
can't communicate from event one
to event two.
So, if you're at event two,
you don't know what happened at
event one.
Because even if you'd sent out
a signal, a radio signal or
whatever, it would not have
reached you by the time event
two takes place.
So you can't communicate from
event one to event two.
And similarly,
you can't travel from event one
and reach event two,
because you'd have to go faster
than the speed of light to do
it.
These kinds of intervals,
these negative intervals,
these are called time-like,
because the time term is larger
than the distance turn.
And these kinds of intervals
are called space-like intervals.
And you can only travel or
communicate over time-like
intervals.
Yes?
Student: What are these
so-called events?
Professor Charles
Bailyn: So,
they're events--you can think
of them as points in space-time.
So, they have a particular
position in space and a
particular point in time.
So, they can be described by
four numbers,
three spatial coordinates and a
time coordinate.
They can be anything.
You know, turning on a light,
doing anything you want to do.
Receiving a photon,
whatever it is.
But they are points in a
four-dimensional space-time and
therefore require four numbers
to describe them.
And you can only get from one
to another if they're separated
from a time-like,
that is to say,
a negative, interval.
Okay, so this expression,
which I'll write down again,
this is called a metric.
And the particular metric that
I've written down here is the
metric for flat space.
Because remember,
this is special relativity.
There's no masses,
no curvature of space,
none of that stuff,
yet.
This is the metric for flat
space with no mass present.
And there are many other
metrics possible.
Any time you add mass or do
other things,
you get different kinds of
metrics more complicated than
this.
So, what's special about
special relativity is that you
use the metric appropriate for
flat space as opposed to the
many other different kinds of
metrics that you can use in
general relativity,
which has a much more general
form for the metric.
I should say,
you can also write this down.
You can write down the spatial
terms here in polar coordinates.
Remember polar coordinates?
Polar coordinates,
you describe the position in
space, instead of with x,
y, Z, you describe it with
a radius, a distance from zero,
and some angles.
And it turns out,
that's convenient to do so.
Let me write this down in polar
coordinates, or in polar.
That's an r.
Let me write that explicitly.
This is an Omega,
that's some angle.
And then the T thing
remains the same.
And I've pulled a little bit of
a notational fast one on you
here.
I've gone away from the deltas
and I've written these down as
d.
This is the differential
d.
Those of you who have taken
some calculus will remember
this.
If this were a calculus-based
course I would explain why I did
that, but I'm not going to.
So, just allow me this slight
of hand, here.
For technical reasons,
these have to be differential.
Yes?
Student: But you do need
a second angle term for the
[inaudible]
Professor Charles
Bailyn: I do need a second
angle term.
I should say--good point.
You need two angles and a
distance in three space.
This capital Omega here is
actually--Omega squared is
actually sin θ,
d θ,
d φ,
which is the correct form.
And so you could write out both
terms here, but in fact,
this one isn't going to change.
But you're absolutely right.
In principle,
you need two angles.
Okay, why have I done this?
Excellent question.
I ask myself a question at this
point.
Where am I going?
What I want to do now is write
down a different metric.
A metric that actually involves
curved space and the presence of
a mass.
And this is something called
the Schwarzschild metric.
Remember Schwarzschild?
He had a radius.
And this is the appropriate
metric for the presence of a
single point mass at the center
of the coordinate system,
at R = 0.
That's why I put it into polar
coordinates, because the
presence of the mass is going to
change the space-time as a
function of distance--from
radial distance from where the
mass is.
And so, it's much more
convenient for the Schwarzschild
metric to use this in polar
coordinates.
So, here's the Schwarzschild
metric, (d T^(2)),
that's the--this is the
interval,
is equal to (dR) / (1 -
R_s /
R).
So, that's just like the flat
term, except with something in
the denominator there.
Plus R^(2),
d Omega squared,
that's just like the flat
metric.
And then the--whoops this had
better be - c^(2) (1 -
R_S / R)
(cT) ^(2).
Where R_S is
the Schwarzschild radius,
which we've had before,
which is 2GM /
c^(2).
Okay, so this is just like the
flat metric with two exceptions.
It's got a term in the radial
part of this 1 -
R_S / R.
And it's got that same term,
but this time in the numerator,
in the time term here.
Now, what do you do with
this--with such an equation?
Well, we've done--in special
relativity, we've dealt with
these kinds of things.
What you do is you start taking
the limiting cases.
You say, okay,
what happens when it's getting
really close to flat on the one
hand,
and what's happening when it's
getting very seriously
relativistic on the other hand?
So let's do that.
If R_S /
R goes to zero,
then the metric turns into the
flat metric.
Because if R_S
/ R = 0,
this term disappears because
it's 1 - 0, and it just cancels.
This term disappears and you
recover the flat metric.
This happens in two cases.
If the mass goes to zero,
then R_S goes
to zero, and you recover the
flat metric.
Or if R gets really big,
then R_S /
R goes to zero,
and again, you recover the flat
metric.
So in--there are two situations
where Schwarzschild metric
blends smoothly into the
ordinary flat space.
One is if the mass is zero,
that's not surprising.
If the mass is zero then
space-time isn't curved.
Or alternatively,
if you're really,
really far away from the mass.
If R is much,
much bigger than the
Schwarzschild radius,
you're way out there.
There's no gravitational effect.
Space-time remains flat.
So, these the--this is the
limiting case where you recover
special relativity.
Now, the other case is when
R gets close to the
Schwarzschild radius and
approaches it.
So then 1 -
R_S / R
approaches zero,
because these two are going to
get closer and closer together.
1 - 1 = 0.
What happens then?
So, this is now,
first of all--in physical terms
you're getting really close to
the Schwarzschild radius.
So now, what happens to the
metric if you do that?
The dR term gets very
big, because it's got a zero in
the denominator.
The dT term gets really
small, because it's got that
thing that's going to zero in
the numerator.
Fine.
What does that mean?
Well, remember,
this is the negative term.
This is the positive term.
So the positive term is getting
really, really big.
The negative term is getting
really, really small.
And that means that all
intervals are gradually becoming
space-like.
What do I mean by that?
Well, the negative term is
getting small.
One of the positive terms is
getting big.
So the sum of those tends to be
positive.
It's becoming more and more
positive.
Positive intervals are these
space-like intervals,
and you can't communicate or
travel across space-like
intervals.
When you get all the way to the
Schwarzschild radius,
this blows up completely,
becomes infinite.
This becomes zero,
and there are no time-like
intervals.
There are no time-like
intervals that cross the event
horizon.
That's why you can't get out.
This takes us back to the basic
principle of black holes.
So, cannot communicate or
travel over space-like
intervals.
And so, you can't cross
R equals the
Schwarzschild radius.
All right.
Let's see.
Let me write the thing down
again, here, for you.
Okay, so that's the metric
we're worrying about here.
And now, let's think about what
happens inside the Schwarzschild
radius.
R less than
R_Sch
warzschild.
That means the dR term
becomes negative,
because 1 -
R_S / R.
If R_S is
bigger than R then this
term is--this term is greater
than 1,
and this whole thing is less
than zero, and the signs change.
And the dT term becomes
positive.
So, that means this is the
time-like term,
where this one is the
space-like term,
because it's positive.
That's what I meant three,
four, five lectures ago,
when I said that inside the
Schwarzschild radius,
when you're inside the
Schwarzschild radius,
space and time reverse.
It's a sign change in the
metric.
That's what it means.
And you can only travel along
negative intervals.
That means you have to move in
R the same way outside
the Schwarzschild radius you
have to move in T.
But notice it's only the radial
term.
This term hasn't changed.
You could go around in circles,
but whatever you do,
you still have to move,
as it turns out,
toward the center of the thing
in radius, in order to have a
time-like interval.
And so, motion in R is
required for inside the
Schwarzschild radius,
whereas motion in T is
required outside.
So space and time reverse.
All of which is very nice,
but I've left out
something--I've left something
out,
which is the fact--inside the
event horizon,
how do you know that this is
still the metric?
One could invent some function
that looks just like this
outside the Schwarzschild
radius,
but then looks like something
else inside the Schwarzschild
radius.
And because no communication
across the Schwarzschild radius
is possible, you'd never be able
to test it.
And so, this is how one gets
away with doing non-testable
physics.
You say, well,
we're just going to assume that
the metric hasn't changed.
Why should it change?
After all, it's the same
equation.
But inside the Schwarzschild
radius you can't actually test
this.
Outside the Schwarzschild
radius you can test it,
because you see whether objects
behave as they ought to behave
in a space that's curved in this
particular way--in a space-time
that's curved in this particular
way.
And so, this is what I meant
by, five classes ago,
by saying space and time
reverse.
These two quantities reverse
their signs.
All right, that's as far as we
can go, because the next thing
that one would want to do is,
you find out what the equation
is for finding out how things
move in these curved
space-times.
Basically, you remember,
things go from one event to
another in the shortest possible
path, that's the equivalent of a
straight line.
That means if you integrate
over dT,
it has to be minimized.
So, you minimize this integral.
That tells you how things move.
We're not going to do that.
Sighs of relief?
And because--for obvious
reasons.
So, this is as far as we can
go, just to write down the
metric here.
So, let me know pause for
questions, and then we're going
to go back and talk about
astronomy--about things in the
sky that actually exhibit these
relativistic behaviors.
Yes?
Student: You were
talking before about intervals,
and how all the intervals are
negative.
What exactly is one interval?
[Inaudible]
Professor Charles
Bailyn: Sorry?
Student: What exactly is
one interval?
[Inaudible]
Professor Charles
Bailyn: Oh,
an interval.
So, what I'm doing is I'm
taking two events,
each of which is one of these
points in space-time,
and I'm asking,
"What is the interval between
them?"
What is--you could measure the
distance between them,
you could measure the time from
one event to another.
But as it turns out,
those aren't invariants.
And so, there's this other
thing, the metric,
which is invariant.
And so, that's a measure--an
invariant measure of how
separated these two events are.
So, you take two events and you
ask yourself,
"Are they separated by zero,
a positive quantity or a
negative quantity?"
Where by separation,
I mean, this curious
combination of space and time.
Student: So the
intervals are before the metric,
before the interval?
Professor Charles
Bailyn: Yeah,
it's an interval--think of--let
me go back to the analogy I
started with.
Here's--in two spatial
dimensions, x and
x, here are two points.
And depending on how I set the
coordinate system up,
the x-distance--the
x difference between them
and the y difference
between them can change,
but the distance is always the
same.
So now, I've got two points
with--each with and x,
y, Z and a T. And
depending on how I change my
velocity or my coordinates the
particular values of x,
y, Z,
and T can change,
but this (Δ T^(2)) defined
by--I'm in flat space now,
right?
This separation,
this interval between those two
points, this is the
invariant--in the same way that
the distance between two points
doesn't change if you change the
coordinate system,
even though the x and
y separations do.
Student: [Inaudible]
Professor Charles
Bailyn: It gives you--no.
Well, it combines these four
things into one thing that
doesn't change.
That's the point.
Yes?
Student: Is there a way
that you can--like as an
interval from zero basically
describes two events that appear
simultaneous?
Professor Charles
Bailyn: As the--yeah
exactly.
Student: So,
is that then collapsed into a
Newtonian theory,
things with two--things appear
simultaneously if they happen at
the same time -- Professor
Charles Bailyn: Well okay.
So, there's two different ways
things can appear,
quote, simultaneously.
One is if they are two
different events in space-time
and light travels from one,
and they're exactly separated
by--the amount of time between
them is the same as the distance
between them if you multiply by
c^(2).
They can also appear
simultaneously if they are the
same point as each other.
And then everything goes to
zero.
And it's only in that second
case that it--that the Newtonian
concept of simultaneous kicks
in.
Simultaneous is usually taken
to mean that the time separation
is zero.
Two things happen
simultaneously when they happen
at the same time.
Student: [Inaudible]
Professor Charles
Bailyn: On Earth the--the
distances and the velocities and
the gravitational fields are
never so strong that you have
any trouble--that the Δ T
changes significantly depending
on what your point of view is.
So, in our everyday life,
we have a strong concept of
simultaneity.
It's two things that happen at
the same time.
Turns out, though,
that if you move at close to
the speed--if you observe two
events to be at the same time
and I'm moving at close to the
speed of light,
I don't observe those two
events at the same time,
even though you do.
And so, at that point,
you have to abandon the
Newtonian concept that Δ T = 0
tells you that two events are
simultaneous.
And the whole concept of
simultaneity takes on a
different task.
Other questions, yes?
Student: [Inaudible]
Professor Charles
Bailyn: Okay,
so these units can be any units
of length you like provided
that--any units of length you
like,
provided the time units are
related to it by c.
That is to say,
if your distance units are
light-years, your time units
have to be years.
If your distance units are
meters, then your time units are
some kind of meter,
light-meter-second thing.
And so, the only restriction on
the units you use is that the
time and the space units have to
be convertible into each other
through c^(2).
Or, alternatively,
another way of saying it is,
you use any units you like,
and as long as you express the
speed of light in those units.
If you have a time unit and a
space unit, if you're in--if
you're measuring your space in
furlongs and your time in
fortnights,
as long as your c is in
furlongs per fortnight,
it's going to come out okay.
So, as long as it's convertible.
Other questions?
Okay, if you don't get all the
details and nuance of what I've
said this period,
don't worry too much about it.
I just wanted to get the
concept of the metric out there
and show you how,
if you look at that equation
these concepts of space and time
reversing,
and so forth,
have a kind of mathematical
consequence, as well as just
spouting words.
And if you get,
sort of, the basic outline of
the argument,
that's fine.
Okay, back to actual things in
actual--that actually exist.
So, what I want to talk about
now is evidence for general
relativity from astronomical
objects--real black holes,
stuff like that.
Now, one of the curious things
about this is that when Einstein
thought all this stuff up,
he thought it up from basically
these philosophical concerns
about mass – that the inertial
mass turned out to always be
equal to the gravitational mass.
Why would that be?
And there wasn't,
when he thought it up,
a great body of evidence for
his theory in the real world.
This is in contrast to special
relativity.
Special relativity,
there were all these
experiments that needed to be
explained.
General relativity--very,
very little.
In fact, when Einstein first
put forward the theory in 1917,
there was only one thing that
had ever been observed that
actually showed an effect of
general relativity,
and that was the orbit of
Mercury, which you're reading
about for this week's problem
set.
So, just going back a little
bit, in the nineteenth century,
people had observed the orbits
of planets in great detail.
And they found out that two of
the planets were moving in ways
they couldn't quite explain.
There were very small
deviations from the predictions
orbit.
In particular,
the orbit of Uranus was a
little weird.
And that was quickly explained
by the presence of an
unknown--hitherto unknown
planet,
which was also exerting a
gravitational force on Uranus
and pulling it out of the orbit
that it should have been,
by a very small amount,
because the gravitational force
of another planet is very small
compared to that of the Sun.
But by the middle of the
nineteenth century they could
measure such things.
And they therefore predicted
the presence of this other
planet, of Neptune,
and they calculated where it
should be.
And some guy went off and
observed in that spot and found
it--predicted presence of
Neptune and discovered it in the
predicted place.
Big triumph!
Everybody--if they had had
Nobel Prizes back then,
they would have won it for
this, for sure.
And then, there was a whole big
kerfuffle because they couldn't
decide whether the French guy
had done it before the English
guy, or vice versa.
And they argued with each other
for decades about who gets the
credit.
But in scientific terms,
there was a prediction,
and the prediction was
verified.
Excellent news.
Now, there was also a problem
with the orbit of Mercury--also
perturbed, from what you would
expect.
And having had this big triumph
in the Outer Solar System,
they figured,
well, we know how to deal with
this.
There's got to be another
planet in there.
So, they predict the presence
of a planet called Vulcan,
which then disappears from the
scientific literature until it's
resurrected by Star Trek.
But Vulcan--the concept of
Vulcan was, this was going to be
a planet that's closer to the
Sun than Mercury.
That's why they haven't been
able to find it,
because it's too near the Sun
to be easily observed.
And it's going to pull on
Mercury in such a way that it's
going to explain the problems
with the orbit of Mercury.
And so, they then look for
Vulcan in the predicted place,
and they find it.
And then somebody else finds it.
And they find it many times and
each time it's different--all
different.
And it gradually becomes clear
that everybody's fooling
themselves.
That there's no--this is a
really hard observation to make,
right?
Because the thing is right near
to the Sun.
And so, it turns out that all
of this is wrong.
None of these observations are
really any good.
It's not repeatable--so,
not really.
And so, after some attempts to
find Vulcan--and then,
they rule out the presence of
Vulcan in various places.
So, then the people calculating
the orbits have to go back and
say, well, if Vulcan isn't
there,
maybe there are two or three
planets combining together to do
the thing that we originally
wanted Vulcan to do.
This gets sort out of control
after a while.
And at a certain point,
people just kind of give up,
and they say,
well, it's a great big mystery
about Mercury.
And after a while,
after that, people kind of even
stopped caring.
Because, you know,
we know Newton's laws worked.
This is just some weirdness
about Mercury that we don't
understand.
And then, when Einstein creates
his new theory of gravity,
he then computes in the new
theory of Mercury's orbit.
And he now gets something that
agrees with the observations,
without the need for a new
planet.
And so, what happened was,
Mercury's orbit is a little
different from the Newtonian
prediction.
The general relativity
prediction is a little bit
different in just the same way
to explain this problem that
people had been trying to solve
for fifty years unsuccessfully.
And so, this was the first
verification,
empirical verification,
of general relativity.
And if you think about it,
you would expect that Mercury
would be the place you would
find this out.
For Mercury,
R_S /
R, this is the
Schwarzschild radius of the Sun
because that's what's doing the
gravitating,
is the biggest in the Solar
System.
Because the R,
the distance from Mercury to
the Sun, is the smallest of any
of the planets in the Solar
System.
And so, the relativistic
effects, the general relativity
effects, are relatively large.
But, you know,
this is still a really small
number.
This is 3 kilometers,
the Schwarzschild radius of the
Sun.
Mercury is way out there
somewhere.
So, even though this is the
most--this is the biggest
relativistic effect in the Solar
System, it still isn't that
huge.
Let me just remind you what
this effect is.
Here's the Sun.
Mercury's going around the Sun.
And it's going around in a
slightly elliptical orbit.
I'm going to draw a very
elliptical orbit,
here, but it's really not that
big.
And there's a point in the
orbit where it is closest to the
Sun.
That point is called the
perihelion.
"Peri" for close,
"helios" for Sun--of Mercury.
And in the Newtonian theory,
you should have exactly the
same orbit every time.
You should come back and the
perihelion should be in the same
place in each successive orbit.
The orbit doesn't move or
doesn't change.
But, in general relativity,
the perihelion moves.
So, after a while the
perihelion will be here.
The whole orbit will kind of
tip this way,
and it'll look like this.
So, this is the perihelion
later.
And it looks like that.
And the angle which the
perihelion makes with the Sun
has changed.
This angle is called the angle
of the perihelion.
And this precesses.
So this is called the
precession of the perihelion.
And it's measured in some angle
per time.
Because the question is,
"How long does it take for the
perihelion to precess across
some angle?"
And the key number for Mercury
is 43 arc seconds.
Remember arc seconds?
Those are small angles.
Per century – a really small
movement, but something that can
be measured, and had been
measured.
And it's not surprising that
this is small,
because the relativistic
effects are going to be small,
because the Schwarzschild
radius of the Sun is really
small compared to the size of
the orbit of Mercury.
But this was observed before
Einstein made his theory.
Nobody understood it.
Einstein came up with his
theory.
It turned out it predicted a
precession of the perihelion in
a way that Newton didn't,
and it turned out to work out
precisely.
So, that was good.
And at the time Einstein
published the theory,
this was the only piece of
evidence that it was correct.
Pretty small empirical
verification.
And so, let's just write down
the fable, here.
This is Einstein and the
precession of the perihelion.
And there are two versions of
the moral.
Sometimes in textbooks,
you know, they make a big deal
out of this.
They say, oh,
there was this terrible problem
with Mercury,
and then Einstein came along
with this great new theory,
solved that problem.
In the same way that they say,
there was this terrible problem
with the speed of light being
constant from all frames,
and Einstein came along with
special relativity and solved
that problem.
That's a misreading of history.
This was a by-product of
Einstein.
It wasn't that there was a
problem with the data and he
went out to try and fix the
theory to conform with the data.
There was very little data.
So, the moral here is aesthetic
considerations,
aesthetic--perhaps you want to
call this philosophical,
considerations can lead to a
good new theory because he
didn't really do it to explain
the data.
This is, however,
the only time I can think of
where this actually happened
this way.
Every other major advance in
science came about because the
observers or the experimenters
had a problem--but not G.R.
Only for general relativity.
Now, subsequent to that,
between 1917,
when this theory was
promulgated, and now,
there have been a variety of
tests of general relativity
using astronomical objects.
You always have to use
astronomical--or almost always
have to use astronomical objects
to test this,
because you need really strong
gravitational fields,
and it's hard to produce a
really strong gravitational
field in the laboratory.
You're kind of limited to what
the Earth provides you with,
and that isn't such a strong
gravitational field.
We computed,
at some point,
the Schwarzschild radius of the
Earth, R /
R_Sch
warzschild is the relevant
quantity,
and it's really small from the
Earth's gravitational field.
So, it's hard to see these
things.
So, all these tests tend to be
astronomical in nature.
Since then, a variety of tests
of G.R., and the punch line is
that it passes all of them.
G.R. is still a good theory.
There's no contradictory data.
But the tests still aren't as
strong as you might have hoped
they would be.
And we'll talk about some of
those on Thursday.
 
