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.