Hooray. 
So, we understand something about how 
gravity affects light, how gravity 
affects time. 
Because we're dealing in a relativistic 
theory. 
It affects distances as well. 
And Einstein thinks when he figures this 
out, that's he's well on his way to 
resolving the theory. 
It turns out that the mathematics 
involved is difficult. 
luckily for him, unlike Newton after a 
few years of worrying about it, he 
discovers that mathematicians had already 
pretty much solved the problem he was 
trying to solve, albeit in a different 
context. 
And making the connection in 
conversations with Riemann, he very 
quickly is led to what becomes the Theory 
of General Relativity. 
And it, he comes up with this 
understanding that gravity is in a very 
technical sense, 
a description of geometry. 
And as I said, the mathematics was hairy. 
And we're not going to do it here. 
So, this is the hard part of the class 
where I get to do a lot of vigorous hand 
flapping and tell you the answers. 
But I'll try to motivate it. 
So, what does, what do we know about 
gravity? 
What are the characteristics of a 
relativistic theory of gravity? 
Well, in small regions of space-time, if 
you observe, 
there are a collection of inertial freely 
falling observers. 
And if they measure, then in their local 
region, the don't, 
that they claim that there is no gravity. 
These are the freely falling observers. 
Of course, there are many of them. 
They, they can have 
all different velocities, and their 
different experiences locally. 
Because they have rel, constant relative 
velocities relative to each other are 
related by the Lorentz transformations 
that we know about. 
And then, the reason that gravity is for 
real, 
what's really going on is the tidal 
effect. 
The fact that the acceleration of gravity 
at different positions in space and at 
different times, is not the same. 
This means that what 
but as an inertial observer here is not 
the same as what an inertial observer 
there. 
So, if two people are freely falling 
towards Earth say, a near to each other, 
they jump off the same spaceship and 
they're freely falling towards Earth. 
They will notice that they are slowly 
accelerating towards each other and each 
will blame the other for being that 
inertia, I'm in free fall. 
You can't be because you're accelerating 
towards me. 
Of course, what's really going on is 
they're both accelerating towards the 
center of the Earth. 
and it's the tidal effect that is the 
real gravity. 
on the other hand, if I give an object a 
velocity, any initial velocity at any 
given point and then just let it go then 
gravitation ensures that it knows where 
it's going to go. 
At that point, gravity is the only force 
acting. 
So, given any initial velocity, there is 
a unique inertial world line that starts 
with that initial velocity, starts moving 
at that slop relative to the vertical 
axis. 
And then, from there on let gravity do 
it. 
Gravity will tell this thing where to go. 
Why am I telling you all these stories? 
These are all things we know. 
Because they all have an analogy in these 
theories of geometry that were being 
developed in the 19th and early 20th 
century. 
And what do we know about? 
What did, what did these people teach us 
about geometry? 
You're talking about the geometry of 
curved spaces. 
And we'll give an example and try to make 
it 
more clear. 
But, small regions of occurred flay, 
space always appear like flat space and 
can be described at the normal x, y, z. 
Whatever number of coordinates your 
dimension is we all live on a curved 
space. 
For example, the surface of the Earth. 
But we all use 
coordinates like street and avenue number 
in the context of a city, and we can 
easily describe Manhattan ignoring the 
curvature of the Earth and navigate 
around Manhattan to very good precision. 
Because the curvature of the Earth takes 
affects us only over large distances. 
A small chunk of the Earth, of the 
spherical Earth, is effectively flat and 
we can use maps. 
But, we can't use a map, as we all know 
who've, when you look at, at trying to 
draw projections. A map of the entire 
Earth needs to be either a globe or have 
large distortions on it. 
Now again, just as with gravity, when you 
pick, sit at any point on a curved 
surface, you can draw the, these little 
straight lines and pretending that these 
things are flat, pick any direction. 
You can draw the beginnings of a straight 
line in any direction in the little flat 
region, 
and those lines are related by rotations. 
You rotate a line through one point in 
one direction. 
You rotate it, you get a line in a 
different direction. 
on the other hand, the lines that are 
straight. 
In these local patches, in one patch, in 
one little piece of space are not and 
we'll see this, the same as the straight, 
do not look straight as viewed from a 
different position in a particular not 
related to the straight lines through 
this point by, the first point by 
rotation. 
And third, given a point in an original 
direction, there is a unique geodesic, 
the analog in a curved space of a 
straight line. 
And we'll see an example of how to do 
that. 
But in a, any curved manifold at any 
point, 
there is sort of this analog of the 
straightest thing there is. 
The closest there is to a straight line. 
So, we'll define geometric object. 
It's called the geodesic. 
And at any point if you pick a direction, 
there is a unique geodesic that starts at 
that point moving in that direction. 
Just as there was a unique inertial world 
line starting at any point with any 
velocity, you're beginning to make the 
analog that the shape of a space isn't 
coded exactly in the fact that what are 
straight or, or if it's curved. 
That what are straight lines at one point 
are not straight lines at another. 
Or it's encoded in the distance between 
points as some of course, the distances 
between points depends on how you label 
the points. 
But, some coordinate invariant 
information there and in geometric terms, 
this is called curvature. 
This is a lot of fancy words about 
geometry, but I haven't really told you 
anything. 
Let's do an example that might help us 
understand what we are talking about and 
the example is, let's think about Earth's 
surface. 
It's a two dimensional surface we live on 
a surface and it's a curved space. 
The surface of a sphere is not flat, try 
to wrap a piece of paper around a ball or 
make a, a, a planer, a map two 
dimensional map of the Earth on a flat 
piece of paper and you'll realize that 
the Earth is curved. 
So, get an approximation it's spherical, 
spherical is special and it makes some of 
the observations sort of trivial. 
On a sphere, every point is the same as 
every other. 
And so, if we think of the Earth as a 
sphere, then might as well start at the 
pole because we want a point. 
Every point is like any other. 
If you start off in any direction, you're 
going south. 
And now, we understand on Earth exactly 
what it means to go in a straight line. 
Sit at the pole, 
pick some direction and keep going 
straight. 
What are you going to do? Well, you'll 
start out getting south along some 
meridian, and following that meridian 
you'd be going in a straight line, 
turning east or west of that meridian 
would not be going in a straight line. 
We all have this understanding and 
eventually, of course, you'll come to the 
other, cross the other pole, keep going 
around the Earth and come back to the 
place you started. 
what you will have completed is what is 
called the Great Circle. 
These are those geodesics that I talked 
about. 
And you notice that starting at the pole, 
going off in any direction, there are 
meridians, there are these Great Circles. 
Nothing special about the pole, as I 
said, every point is like every other 
point. 
So, indeed, on Earth, 
there are precisely straight lines, great 
circles, geodesics. 
Unique one starting at any point, heading 
off in any direction. 
Now initially, as you mentioned, a bunch 
of penguins walking off from the north, 
exact North Pole in all directions, they 
won't notice they're on a curved planet. 
They will be walking on a straight sheet 
of ice. 
And everything will look completely flat. 
And, in particular, if they measure the 
distances between them, they will match 
what you'd expect from the angle, the 
small angle approximation if they're 
moving in small angles to each other and 
the distance they've traveled. 
As they get farther and farther away, 
they will find that each successive 
penguin is farther and farther apart. 
on the other hand, as they get really 
far, they'll notice that the distances 
are not increasing sufficiently. 
this is where the curvature of Earth 
comes in and perhaps a demonstration of 
this would be helpful. 
So, 
here's our demonstration. 
It's our old friend 
right, right assention declination 
demonstrator from UNL. 
But, 
we are going to re-purpose it for what 
are our own devious goals here. And so 
what we're going to do is we're going to 
imagine this collection of ants or 
penguins that we talked about. 
They start at the North Pole, and as they 
move off in all directions, initially 
they don't notice the curvature of the 
Earth. 
And what that means is that, one walks 
along the green meridian, one walks along 
the red meridian and 
they will notice that the distance 
between them is given by the distance 
they have traveled. and then, determined 
by the small angle for them any angle 
between them. But as they get farther and 
further apart, you notice that these 
lines that, that one of the penguins, as 
they get farther and farther down their 
lines, you'll notice that their lines are 
not getting separated by a sufficient 
distance. 
They're, in fact, failing to get to be 
far enough to satisfy the small angle 
formula. 
In fact, as they go farther, the distance 
stops increasing all together when they 
hit the pole and then it starts the 
equator, and then it starts decreasing 
eventually. 
These two lines that went off at 
different angles do something they would 
never do in a flat planer world, they 
collide again at the South Pole. 
So, the fact that these two geodesics are 
not going far away enough from each 
other, not going as far from each other 
as they would in the plane. 
And then, eventually, they meet as an 
indication of what we call the positive 
curvature of the surface of the Earth. 
Less we get too caught up in all curved 
surfaces are like the surfaces of the 
earth. 
we have here this nice demonstration of 
the surface, again two dimensional. 
We can only really imagine curved two 
dimensional spaces because we think of 
them as sitting inside our three 
dimensional space. But there are curved 
three and four and six dimensional 
spaces, at least mathematically. 
And so you can this surface, this 
so-called one-sheeted hyperboloid is a 
surface with negative curvature. 
And I'll make it a little more 
transparent so we can see and the 
geodesic that I'm this program very 
nicely draws geodesics through a chosen 
point and I can vary both the location of 
the point in the angle. 
What I want you to see is what happens 
when I make a small change in the initial 
angle. 
See, I'm rotating the initial angle at 
which this geodesic comes out, ever so 
slightly. 
And what I'm seeing is that geodesics 
that start very nearby, here are notice 
that the change far off is much larger 
than the change I am making nearby. 
So, geodesics that start very near each 
other will eventually grow very far. 
The distance between geodesics grows more 
than you would expect in on on the plane. 
These are getting farther apart more than 
the small angle formula specifies a 
negative curvature leads to diverging 
geodesics where as a positive curvature 
leads to converging geodesics as they do 
on the surface of the earth. 
So, we have here this analogy between 
geometry for which we have some 
intuition, and for which mathematicians 
have the formalism. 
And 
the theory of relativistic gravity the 
role of inertial world lines is played by 
geodesics. 
Logan's transformations correspond to 
rotations, rotations preserve the 
distance between two points. 
We know that making a rotation does not 
change the lengths of things. 
Lorentz transformation has also preserved 
something. 
They preserve the value of this invariant 
interval that we defined. 
the minus sign here, as opposed to the 
plus sign here, has an important role. 
It means that the geometry, technically 
we're going to describe is 
pseudo-maronian rather than maronian, and 
it makes many differences. Gravity is not 
the same as the description of a curve 
of, of [UNKNOWN] spaces. 
But, gravitational forces are therefore 
the changes in 
that the, the, the, the fact that the 
title forces that make what seems to be 
inertial at one point, not look inertial 
at another point, 
play the role of the curvature. 
So, the idea is the presence of a massive 
object deforms time and it's vicinity and 
the motion of inertial freely falling 
objects is a long geodesic, these 
straightest of lines that you can draw. 
Remember, inertial world lines were those 
straight lines and the flat case absent 
gravity and now they're going to be as 
straight as they can be in this curved 
space-time. 
And the picture that is often brought up 
as a visualization is this rubber sheet 
image. 
You put a bowling ball in the middle of a 
rubber sheet, it bends the rubber sheet 
down and then rolling a ping pong ball 
will cause the ping pong ball to move 
around it. 
This is misleading on a whole number of 
levels. 
But, I guess the general intuition in 
there in some sense the presence of a 
massive object in particular the main 
ingredient here that is completely absent 
from the real case is 
gravitational field of the Earth pulling 
the bowling ball down. 
This, it's preferred down direction is 
completely non-existent in the actual 
case, but I will give you a better 
visualization in a minute. 
a few caveats to taking this picture too 
seriously. 
First of all, the shape of that deformed 
surface, the rubber sheet is not related 
to curved space-time. 
I'll describe the curved space-time in a 
minute. 
In general relativity, it is not space 
that deforms, but space-time. 
This 4-dimensional object, or if you want 
the, in the 2-dimensional case that we've 
been drawing, 
the whole 2-dimensional sheet is curved, 
not just the x-axis. 
It is not space that is curving, 
it is space-time. 
In some circumstances particular cases 
called static space-time where nothing 
changes, essentially in a static 
space-time, you can define something 
called space. 
And think of space at any given time as 
we could think of the entire universe at 
any given time. 
Those horizontal lines in our space-time 
diagrams. 
Those make sense in, in the sense that 
all of these horizontal lines are 
essentially the same in the static 
space-time case, 
but not always. 
most of the space-time we'll need to 
consider are going to be static. 
inertial orbits are, when you look at the 
space, the inertial orbits, the ellipses 
with the sun at one focus, are not 
geodesics of the curved space that the 
sun creates, 
they're geodesics in space-time. 
It's like helical motion that I drew. 
That is the straightest line in the 
curved space, that is a geodesic in the 
curved space time, about a massive 
object. 
So remember, we drew that helix. 
That helix is a representation of the 
geodesic motion in the presence of a 
massive object. 
That's the straightest line you can draw. 
And, in general 
a set of coordinates that allows you to 
describe all of space-time does not 
exist. 
And we'll see some times when that is a 
problem, and we'll actually have to pay 
attention to this. 
A better visualization of, in fact, the 
gravitational effect the curved, curved 
space around space-time. The curved space 
around a stationary gravitating object 
like a star is this so called Flams 
Parabloid and, this actually is related 
to the curvature of space. 
In the following sense 
the space around a star is spherically 
symmectric, so you can, break it up into 
balls. 
And you can assign each ball a size based 
on its area. 
And what we do here is to each, instead 
of this, we draw circles and the 
circumference of a circle is two pi R 
where the surface of that ball is four pi 
R squared. And if the space is flat then 
the distances between the balls are the 
differences in their Rs that you get from 
their areas. 
But in this curved space, the distances 
between Jason falls are not given by the 
differences in their Rs. 
In fact, there is variation in the 
distance between them. 
And as you get closer and closer to the 
gravitation object, in this case the 
star, 
then the distance between adjacent 
spheres, or in this picture circles, gets 
large. 
And the way we, 
telative to the change in their areas. 
And the way we do this here is we pull 
down the bottom of this sheet so that 
these two circles, if this were flat, 
these two circles which have radii that 
are circumferences that are very close to 
each other would actually be right next 
to each other and were extending it so 
the distance between adjacent circles is 
different. 
And this actually does represent, in a 
somewhat reliable way, the space around a 
gravitating object. 
So, I've been very hand wavy about how 
gravity is curvature. 
I can write you the equation. 
So, here is what Einstein tells us. 
Here is Einstein's Theory of General 
Relativity and the discovery that gravity 
is Geometry is only the first step. 
the next step is writing the equation 
which tells you that the way in which 
mass energy generates a source of 
curvature, its a source of space-time 
curvature in the same way that electric 
charge is a source of electric field. 
And when you change the distribution of 
energy and momentum, it is the curvature 
of space-time that is this field the 
ripples in space time are the waves in 
this field that carry this information 
off at the speed of light. 
And this is the equation, 
this is Maxwell's equation for gravity. 
It looks simpler but only because all 
these symbols are con, confusing. 
We will not do anything with this 
equation except write it down, but I just 
want to write it down for you, 
this is Einstein's equation. 
these are looking objects are of the 
Riemann Tensor. 
They express the curvature of space at a 
particular point in space and time. 
The curvature of space-time, G, we know 
that is Newton's constant, C is the speed 
of light. And this T object is the energy 
moment of stress tensor. 
This is 
our characterization of the flow and 
amount of energy and momentum at a given 
point in space and time or at a given 
event. 
So, at a given event, you have this 
acting like the electric charge and this 
tells this is a differential equation 
that tells the gravitational field how to 
respond. 
You can solve this equation. One of the 
consequences of this equation is that 
inertial motion, the motion of objects 
under the influence of nothing but 
gravity is along geodesics of this curved 
space-time that you obtained by solving 
the equation. 
So, just as Maxwell's equa, equations 
imply charge conservation, these 
equations imply energy momentum 
conservation, but also the fact that 
objects upon which only gravity acts move 
along geodesics of the space-time that 
has been created. 
there is of course, as always, there has 
to be a limit in which you reproduce 
Newton's theory, because Newton's theory 
has worked so well for us in describing, 
say, the motion of Earth around the sun, 
and binary stars, and so on. 
What is the limit? 
The limit is that if the curvatures are 
small, if you don't get too close to 
objects that are too massive, the 
gravitational fields are not extreme. 
And if those speeds at which objects are 
moving are slow, 
then you reproduce Newton's formula. 
You can express, for example, the motion 
of a planet about a stationary star in 
terms of, sort of, some effective 
potential energy function which will 
appear somewhat familiar. 
It's a function of R. 
It starts off with our favorite familiar 
term, -GmM/R, 
this is just the Newtonian potential 
energy. 
It's got this weird looking. 
Think others the angular momentum, we'll 
see in a minute what that is. 
And then, the first correction, there's a 
whole series of corrections. 
But the leading correction in Newtonian 
physics is this L^2/c^2r^3 correction. 
This becomes a little more sensible if 
you make the somewhat unjustified 
statement that the angular momentum is 
mvr. 
This is true for motion in a circular 
orbit. 
Plugging L = mvr into this equation 
simplifies things a great deal. 
For example, L^2/2mr^2 becomes. 
Let's write it out so we know what we are 
doing so you don't accuse me of cheating. 
L^2 m^2v^2r^2/2mr^2. 
And now, I do all my handy dandy 
cancellations. 
The r^2 is cancel, one of the m is 
cancel, and this is a half mv^2. 
So, this term in the case of circular 
orbits is a fancy way of saying Kinetic 
energy, this term is Newtonian potential 
energy. 
This term, when you plug everything in, 
turns just into v^2/c^2 times this term. 
And so this is the leading correction to 
Newtonian physics, and of course, there 
are higher order corrections. 
But when v is small compared to c and the 
curvatures are not too extreme, we see 
that we reproduce precisely Newtonian 
physics. 
So, good. We expect Einstein's theory, as 
we said, to reduce to Newtonian theory in 
suitable limits, and this is the suitable 
limit in which that works. 
And then, 
the last thing I want to point out is 
that there's a complication in 
gravitational theory that makes the 
mathematics of solving Einstein's 
equation and understanding the structure 
of its solutions vibrant, and lively, and 
interesting field of mathematical 
research as well as physics. 
Which is a complication that is not 
present in in Maxwellian electrodynamics. 
And the idea is this. Take the mass of 
say the Earth and the Sun, The Earth-Sun 
system. 
The mass of the Earth-Sun system I claim 
is a little bit less than the mass of the 
Earth plus the mass of the sun. 
And the reason is that Earth is bound to 
the sun, there's some negative 
gravitational potential binding energy 
between the two. 
Remember, if I took the Earth and the Sun 
very far away, as they fell, they would 
release energy. 
Having lost energy, that reduces their 
mass because mass and energy are 
equivalent. 
So, the mass of the earth and the sun 
together is a little bit less than the 
mass of the earth and the sun separately 
just as the mass of a helium atom nucleus 
is less than the mass of two neutrons and 
two protons. 
But, that means that the gravitational 
field that the Earth-Sun system together 
creates is different from the 
gravitational field that would be created 
by the earth and the sun. 
There are some contribution due to the 
gravitational binding energy. 
In other words, since gravitational 
fields carry energy, gravitational fields 
gravitate. 
Which is, almost as if you had the 
photon, 
the particle that carries light, being a 
charged particle so creating its own 
electric field. 
This would be like an electric field 
creating an electric field. 
This makes Einstein's equation, unlike 
Maxwell's equation, 
non-linear. 
A superposition principle for waves does 
not hold. 
These equations are very complicated. 
And as I said, studying their solutions 
is an ongoing enterprise. 
We won't get to do a lot more of the meat 
of general relativity. 
As I said, the mathematics there is more 
than we can afford to do. 
But, we'll see some of the consequences 
and hopefully that will be worth the 
effort to try to understand something.