So we have an understanding of the special theory of relativity, and we
understand that the universe is Lorentz invariant, and we are now in the position
that Albert Einstein finds himself in, in 1905, 1906.
He has produced the special theory, and he now is trying to understand how to
write relativistic physics. And he runs into the problem that gravity
is not as we understand it, a good relativistic theory,
and we can understand what is wrong with it.
Look at what the fate familiar form of Newton's gravitational force law and
everything about it is wrong. For example there is a denominator there,
r squared. Who's measuring r?
Remember if someone is moving, they would measure a different distance because of
Lorentz contraction. Whose r do you use?
Worse than that, what do you, when do you measure this distance?
If I take the say the sum, at this point, and I somehow manage to move it, two
astronomical units to the left. what happens to the force on Earth?
Does Earth immediately respond? Because when do you measure this R?
Earth is not allowed to immediately respond, because Earth is not allowed to
know about the fact that the sun has moved until s- enough time, seven minutes
or whatever it is, has passed for light to have arrived from the sun to the
Earth, because no information is allowed to tr- to be transferred faster than the
speed of light. Now, we know the resolution of this.
Maxwell's equations, how do they do it? remember, Maxwell's equations,
essentially, start with Coulomb's force law between charges, remarkably similar
to Matt, Newton's force law between masses.
And you know how this is resolved in Maxwell's theory.
The way it's resolved in Maxwell's theory is,
if you take a proton and an electron, and you move the proton suddenly,
the electron does not respond instantaneously,
because the electric field where the electron is has not changed.
The motion of the proton generates an accelerating charge, generates an
electromagnetic wave. That wave is a ripple in the field that
changes it from the value associated to the first position to the value
associated to the final position, and when this ripple gets to the
electron, then the electron responds. And everything is nice and relativistic
and Lowrance invariant. So perhaps we can take Maxwell's
equations, apply them to gravity, and have a
relativistic version of gravity. And this is, indeed, Einstein's first
choice. The problem is a subtle one.
what we need is a field theory. We need something that will carry at the
speed of light the information from one place to the other.
Well, Einstein's aware of this. he attempts to use Maxwell's equations,
but he immediately knows that, that will not work.
The difference between this equation and this equation is an important one.
The quantity appearing here is, the electric charge.
The electric charge is a conserved quantity,
and Maxwell's equations imply in fact charge conservation,
whereas the object appearing here is mass.
And in Einstein's theory, mass is not conserved.
Okay. We'll modify things.
What is conserved, instead of mass. What is conserved as energy momentum?
But energy momentum is a completely different beast than electric charge.
Electric charge, remember, is Lorentz invariant.
Every inertial observer measures the same amount of electric charge in a given
region in space. On the other hand, mass is, or energy
momentum is not Lorentz invariant. Different observers measure different
energies and different momentum related, like T and X, by Lorentz transformations.
This complicates the theory significantly.
There's also philosophical question that a actually turns out to be a very
important for the answer, which is, what is a gravitational force?
So how do you measure a gravitational force?
Well let's start. How do you measure the electric force?
I want to measure the electric field at this point and space at this time.
Well I bring along something charged and I measure the force on it.
Divide the force by the charge. The answer is the electric field.
You see none, perhaps you were mistaken. Perhaps what you were measuring was just
the gravitational force on this object. I said, how, okay, I'll bring another
object with the same mass as the first but with the opposite charge.
Now, the gravitational force will be in the same direction, the electric force in
the opposite direction. I'll look at the difference between the
two, and the change is the elec, the one that changed between the positive and the
negative particle. That's the electric force.
Good. So now do the same for the elec, for the
gravitational force. So you bring along some object.
You measure the force that gravity applies to it.
You divide that by its mass. You get the gravitational acceleration.
Very good. Now bring another object upon oh wait,
there is no object upon which gravity acts in the opposite direction.
There's not even an object upon which gravity does not act that you can bring
and replace it with. In fact all objects fall with the same
acceleration there's this universality that suggests that thinking of gravity as
a force is kind of difficult, because you can't even really operationally define
how to measure it. Thinking along these lines, Einstein is
working through it, and in 1907, he has what he considers and in writing, his
happiest thought ever. And his happiest thought basically
involved jumping off the roof. If you jump off the roof he realizes,
there is no gravity until you hit the ground.
This, the, the idea that Einstein is discovering Is and discovering the
importance of is the principal of equivalence.
We made a big deal, a few weeks ago, about the fact that when you are in free
fall there is no gravity, and I emphasize that this is not some random idea.
In space everything is in free fall. Stars, galaxies, planets, they're all in
gravitational free fall, so they're not experiencing any gravitational force.
The converse is, in fact, also true. I can play with the apparent
gravitational force. By, accelerating, measuring, making
measurements in an accelerating frame. Not in an inertial frame.
Being an observer who is accelerating. For example, we've all had this
experience. As the elevator starts going up, you feel
as though your weight had increased. Why does your weight increase?
Because now, the elevator floor has not only to hold you up against the pull of
Earth's gravity, but also to supply the extra force to
accelerate you up. And so the floor is pushing on you
harder. That's the experience of weighing more,
precisely. And so the gravitational acceleration at
a given point, at any given time, depends on your
acceleration, and by picking the suitable acceleration,
ie letting go of everything and falling down,
I can make the gravitational acceleration relative to me at any space or point in
time be zero. How do I do that?
Just let go, and when I free fall I don't experience gravity.
The gravitational acceleration I then measure when I drop something like when I
jumped off that chair. When I was jumping and I dropped the
ball, I measured the gravitational acceleration of exactly zero.
So Einstein realizes this, and then maybe the whole idea of gravity needs to be
abolished. You don't need to think of gravity you
just need to understand that we are working in an accelerating frame.
So everywhere the surface of the earth should be thought of as accelerating up.
I mean of course it's not getting anywhere, but accelerating up and
relative to some rest, inertial frame, which is in free fall.
And because the floor is accelerating up, that is why we experience weight.
This would work well were it not for the tricky fact that Earth is a sphere.
You can't accelerate the entire surface of Earth up, you can accelerate it out.
But if it did accelerate out, then it would not only be pushing us up but also
we'd be accelerating away from each other.
We know how that happens. That has to do with tidal forces.
What makes gravity, gravity, what distinguishes gravity from, just the
result of measuring an accelerating frame, what, the actual gravitational
force lies in, is in those tidal forces of which I made a big deal a couple of
weeks ago. The fact that the acceleration of gravity
is not the same everywhere in space and time,
that is what the result of the fact that, that there is actually a gravitating body
called Earth. If the gravitational acceleration were
equal everywhere, then, you're actual in an accelerating elevator.
Get out of the elevator, and you'll measure a gravitational acceleration of
zero. So this idea proves to be key in
Einstein's solution of the problem of finding a relativistic theory of gravity,
but right at the top, we can use it, to good effect. So let's learn one important
thing about gravity that will serve us well.
So here's a aartoon version of the principle of equivalence.
Here we have a space ship. It's out in space.
There's no gravity. It's in the middle of nowhere in space.
It's accelerating up at an acceleration that I choose to call G, and what that
means is that if, if you sit here and throw a little rubber ball, well the
rubber ball, of course, will, there's no, no gravity, there's nothing touching it.
It will proceed to move in a straight line at a constant speed across the,
across space. However, because the space ship is
accelerating up, relative to the space ship, the ball is not accelerating.
It's moving in a straight line. Relative to the space ship.
The ball will appear to be accelerating down with acceleration g because the ball
is not accelerating the spaceship is of course.
Bring that same situation over to the right hand side, here, I have the same
spaceship. It's sitting on the ground, it's not
going anywhere but there's a gravitational force.
A gravitational acceleration of g, as there is on Earth.
And if I throw the rubber ball, well then it will fall on the ground with
acceleration g. From the point of view of looking inside
the spaceship, you cannot tell the difference between being on the ground
and experiencing gravitational acceleration, or being in the middle of
nowhere and accelerating up. Good, so that's the principle of
equivalence. We know about that.
How do we use, do something useful with it?
Well, here's what Einstein says. Imagine putting a little blue light bulb
at the tail end of our spaceship and let this little blue light bulb make a pulse
of light. A little short wavelength pulse of blue
light. Now when the pulse is emitted, the
spaceship is moving maybe at some velocity v.
What v is, is completely irrelevant. I can always pick an observer that is
moving at a speed v and then for him the initial speed of this spaceship is zero.
And were the spaceship moving at a constant speed, indeed then this pulse of
light will travel at the exactly at the speed of light relative to the spaceship
up to the top of the spaceship and if I put a detector at the top to absorb it I
would observe that there was no Doppler shift, because detector and a,
emitter are moving at zero relative velocity, like moves at the speed of
light anyway. there would be zero redshift.
However, in this case the spaceship is accelerating if the height of the
spaceship is denoted by H, then it takes a time of the order of H divided by the
speed of light, for this pulse of light to climb to the top of the space ship.
If the space ship is accelerating with acceleration G, the, the change in its
speed during the, this time, while the photon is moving up the space ship is of
the order of its acceleration times that time.
And so, when the photon is absorbed, the space ship is no longer moving at speed
V. It's moving faster.
The result will be of course, that, you can imagine V to be zero by picking an
appropriate frame, you have something emitted, the observe-, the, reciever is
not moving at the same speed as the emitter.
The reciever is moving faster by this amount, and, we can use, are
understanding of the Doppler formula to compute, the, degree by which, the,
photon will reddened, because the receiver is, receiving from the observer,
at this relative speed. And so dividing V by C, we find this is
the non-relativistic expression for the Doppler shift were working with small
age, and so in that approximation we can use the non-relativistic version we're
not going to be doing anything superlativistic here.
What's good about this is that now the principal of equivalence tells me that
the exact same thing happen on the right if I put on the bottom floor of a
building or the bottom of this space ship sitting on the ground, a blue flashlight
and it makes a blue wave, that wave when absorbed at the top floor of the
building. Will be red-shifted.
And what this means is any light. Light is effected by gravity it has to be
because of the principle of equivalence and, what we understand about special
relativaty and, as you shine a light from the bottom floor to the top floor the
light is reddened of course, the reddening is not significant, GH over C
squared is not a large fraction, unless your building is exor, exorbitantly tall.
But if you make a precision measurement you could do it and, this is not just
about light remember, because right here is an atom, that is oscillating or
whatever, a clock that is oscillating, with some Characteristic period as
observed by someone at the top of the building clocks on the bottom floor runs
slow the red shift remember is accompanied just as it was in the a
relativistic red shift case it just reflects the change in.
the way clocks work. So if you observe from the top floor a
clock on the bottom floor, you will see that the clock runs slow.
If you want to live a very long time, go live near the Dead Sea.
You will age slower, albeit by a very small factor.
Is this for real? Yes.
This is definitely for real and in fact it was measured in a beautiful
experimental tour de force by Pound Rebka in 1959 in this tower, over here on the
left hand size of this building, the Jefferson Lab at Harvard University.
The tower was designed initially to do some magnetism measurements in the
nineteenth century. Pound Rebka re-purposed it. they put
their emitter down on the bottom floor of this tower and the receiver at top floor
of the tower and over this height. They found a very minuscule, but
completely consistent with a Einstein's argument, redshift and then when they of
course conversely went the admitted light at the top, it was glue shifted by the
time it reached the bottom floor Everything is symmetric in that regard.
But they needed, the, the, the, redshift was
miniscule. They used a technique called Mossbauer
spectroscopy to measure the wavelength of light with exquisite precision.
These days, this gravitational red shift is all over the place the little GPS
reciever in you cellphone knows about the gravitational red shift so it can
translate the time of travel of the signals that it gets from various GPS
satellites to their distance in a correct way taking into account that the receiver
is here on the ground and the satellites are up in space.