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.