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February 11, 1996

General Relativity Tutorial - Long Course Outline

John Baez

This is a longer version of the course outline. If you click on some of the capitalized concepts, you will seemore information on them.

A TANGENT VECTORor simply VECTOR at the point p of spacetime may be visualized as an

infinitesimal arrow with tail at the point p. The tangent vectors at p form a vector space called the

TANGENT SPACE; in other words, we can add them and multiply them by real numbers.

Suppose we work in a local COORDINATE SYSTEM with coordinates (x0,x1,x2,x3). (Since we are

working in 4d spacetime there are 4 coordinates; we may think of x0 as the time coordinate t and

the other 3 as x, y, and z, but we don't need to think of them this way, since we're using an utterly

arbitrary coordinate system.) Then we can describe a tangent vector v by listing its components

(v0,v1,v2,v3) in this coordinate system. For short we write these components as va, where the

superscript a, like all of our superscripts and subscripts, goes from 0 to 3.

1.

A COTANGENT VECTORor simply COVECTOR at the point p is a function f that eats a tangent

vector v and spits out a real number f(v) in a linear way. Cotangent vectors can be viewed as

ordered stacks of parallel planes in the tangent space at p. They don't "point" like tangent vectors

do; instead, they "copoint".

Working in local coordinates, we define the components of a covector f to be the numbers

(f0,f1,f2,f3) you get you get when you evaluate f on the basis vectors:

f0 = f(1,0,0,0)

f1 = f(0,1,0,0)

f2 = f(0,0,1,0)

f3 = f(0,0,0,1)

2.

A TENSORof "rank (0,k)" at a point p of spacetime is a function that takes as input a list of k

tangent vectors at the point p and returns as output a number. The output must depend linearly on

each input.

A TENSORof "rank (1,k)" at a point p of spacetime is a function that takes as input a list of k

tangent vectors at the point p and returns as output a tangent vector at the point p. The output must

depend linearly on each input.

More generally, a TENSORof "rank (j,k)" at a point p of spacetime is a function that takes as input

a list of j cotangent vectors and k tangent vectors and returns as output a number. The output must

depend linearly on each input. Note that this definition is compatible with the previous ones! This is

obvious for the rank (0,k) tensors, but for the rank (1,k) ones we need to check that a function that

eats k vectors and spits out a vector v can be reinterpreted as a function that eats k vectors and one

covector f and spits out a number. We just let the covector f eat the vector v and spit out f(v)!

Similarly, note that a vector can be reinterpreted as a tensor of rank (1,0), and a covector can be

reinterpreted as a tensor of rank (0,1).

3.

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In local coordinates we write the components of a tensor T of rank (j,k) as a monstrous array

Tab....cde....fwith j superscripts and k subscripts. Again, all superscripts and subscripts range from 0

to 3; each number Tab....cde....fis simply the number the tensor spits out when fed a suitable wad of

basis vectors and covectors. I will describe this in more detail in the following example:

The METRIC is the star of general relativity. It describes everything about the geometry of

spacetime, since it lets us measure angles and distances. Einstein's equation describes how the flow

of energy and momentum through spacetime affects the metric. What it affects is something aboutthe metric called the "curvature". The biggest job in learning general relativity is learning to

understand curvature!

Mathematically, the metric g is a tensor of rank (0,2). It eats two tangent vectors v,w and spits out a

number g(v,w), which we think of as the "dot product" or "inner product" of the vectors v and w.

This lets us compute the length of any tangent vector, or the angle between two tangent vectors.

Since we are talking about spacetime, the metric need not satisfy g(v,v) > 0 for all nonzero v. A

vector v is SPACELIKE if g(v,v) > 0, TIMELIKE if g(v,v) < 0, and LIGHTLIKE if g(v,v) = 0.

The inner product g(v,w) of two tangent vectors is given by

g(v,w) = gab va wb

for some matrix of numbers gab, where we sum over the repeated indices a,b (this being the

so-called EINSTEIN SUMMATION CONVENTION). Another way to think of it is that our

coordinates give us a basis of tangent vectors at p, and gab is the inner product of the basis vector

pointing in the xa direction and the basis vector pointing in the xb direction.

4.

PARALLEL TRANSPORT or parallel translation is an operation which, given a curve from p to q

and a tangent vector v at p, spits out a tangent vector v' at q. We think of this as the result of

dragging v from p to q while at each step of the way not rotating or stretching it. There's animportant theorem saying that if we have a metric g, there is a unique way to do parallel translation

which is:

Linear: the output v' depends linearly on v.1.

Compatible with the metric: if we parallel translate two vectors v and w from p to q, and get

two vectors v' and w', then g(v',w') = g(v,w). This means that parallel translation preserves

lengths and angles. This is what we mean by "no stretching".

2.

Torsion-free: this is a way of making precise the notion of "no rotating". We can define the

TORSION tensor, with components tab, as follows. Take a little vector of size epsilon pointingin the a direction, and a little vector of size epsilon pointing in the b direction. Parallel

translate the vector pointing in the a direction by an amount epsilon in the b direction.

Similarly, parallel translate the vector pointing in the b direction by an amount epsilon in the a

direction. (Draw the resulting two vectors.) If the tips touch, up to terms of epsilon3, there's

no torsion! Otherwise take the difference of the tips and divide by epsilon2. Taking the limit

as epsilon -> 0 we get the torsion tab. We say that parallel translation is "torsion-free" if tab =

0.

3.

5.

A GEODESIC is a curve whose tangent vector is parallel transported along itself. I.e., to follow a

geodesic is to follow ones nose while never turning ones nose... to follow a completelyunaccelerated path. A particle in free fall follows a geodesic in spacetime. In this sense, in general

relativity gravity is not a force!

6.

The CONNECTION is a mathematical gadget that describes "parallel translation along an7.


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infinitesimal curve in a given direction". In local coordinates the connection may be described using

the components of the CHRISTOFFEL SYMBOL Gammaabc. There is an explicit formula for these

components in terms of components gab of the metric, which may be derived from the assumptions

1-3 above. However, this formula is very frightening.

The RIEMANN CURVATURE TENSOR is a tensor of rank (1,3) at each point of spacetime. Thus it

takes three tangent vectors, say u, v, and w as inputs, and outputs one tangent vector, say R(u,v,w).

The Riemann tensor is defined like this:

Take the vector w, and parallel transport it around a wee parallelogram whose two edges point in

the directions epsilon u and epsilon v , where epsilon is a small number. The vector w comes back a

bit changed by its journey; it is now a new vector w'. We then have

w' - w = -epsilon2 R(u,v,w) + terms of order epsilon3

Thus the Riemann tensor keeps track of how much parallel transport around a wee parallelogram

changes the vector w. When we say spacetime is curved, we mean that parallel transport around a

loop can change a vector. As it turns out, all the information about the curvature of spacetime is

contained in the Riemann tensor!

In addition to this simple coordinate-free definition of the Riemann tensor, we may describe its

components Rabcd using coordinates. Namely, the vector R(u,v,w) has components

R(u,v,w)a = Rabcd ub vc wd

where we sum over the indices b,c,d. Another way to think of this is that if we feed the Riemann

tensor 3 basis vectors in the xb, xc, xd directions, respectively, it spits out a vector whose

component in the xa direction is Rabcd.

There is an explicit formula for the components Rabcd in terms of the Christoffel symbols. Together

with the aforementioned formula for the Christoffel symbols in terms of the metric, this lets us

compute the Riemann tensor of any metric! Thus to do computations in general relativity, these

formulas are quite important. However, they are not for the faint of heart, so I will only describe

them to readers who have passed certain tests of courage and valor.

See the adventures ofOz and the Wizard for an example of one such daring reader!

8.

The RICCI TENSOR. The matrix gab is invertible and we write its inverse as gab. We use this to

cook up some tensors starting from the Riemann curvature tensor and leading to the Einstein tensor,which appears on the left side of Einstein's marvelous equation for general relativity.

Okay, starting from the Riemann tensor, which has components Rabcd, we now define the Ricci

tensor to have components

Rbd = Rcbcd

where as usual we sum over the repeated index c.

The physical significance of the Ricci tensor is best explained by an example. So, suppose an

astronaut taking a space walk accidentally spills a can of ground coffee.

Consider one coffee ground. Say that a given moment it's at the point P of spacetime, and its

velocity vector is the tangent vector v. Note: since we are doing relativity, its velocity is defined to

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be the tangent vector to its path in *spacetime*, so if we used coordinates v would have 4

components, not 3.

The path the coffee ground traces out in spacetime is called its "worldline". Let's draw a little bit of

its worldline near P:

|

v^

|

P

|

|

|

The vector v is an arrow with tail P, pointing straight up. I've tried to draw it in, using crappy ASCII

graphics.

Now imagine a bunch of coffee grounds right near our original one, that are initially at rest relative

to it --- or "comoving". What does this mean? Well, it means that for any tangent vector w at P

which is orthogonal to v, if we follow a geodesic along w for a certain while, we find ourselves at a

point Q where there's another coffee ground. Let me draw the worldline of this other coffee ground.

| |

v^ ^v'

| w |

P->-----Q

| |

| |

| |

I've drawn w so you can see how it is orthogonal to the worldline of our first coffee ground. The

horizontal path is a geodesic from P to Q, which has tangent vector w at Q. I have also drawn theworldline of the coffee ground which goes through the point Q of spacetime, and I've also drawn the

velocity vector v' of this other coffee ground.

What does it mean to say the coffee grounds are initially comoving? It means simply that if we take

v and parallel translate it over to Q along the horizontal path, we get v'.

This may seem like a lot of work to say that two coffee grounds are moving in the same direction at

the same speed, but when spacetime is curved we gotta be very careful. Note that everything I've

done is based on parallel translation! (I defined geodesics using parallel translation.)

Now consider, not just two coffee grounds, but a whole swarm of comoving coffee grounds near P.If spacetime were flat, these coffee grounds would *stay* comoving as time passed. But if there is a

gravitational field around (and there is, even in space), spacetime is not flat. So what happens?

Well, basically the coffee grounds will tend to be deflected, relative to one another. It's not hard to

figure out exactly how much they will be deflected. We just use the definition of the Riemann

curvature! We get an equation called the GEODESIC DEVIATION EQUATION.

But let me not do that just yet. Instead, let me just say what this has to do with the Ricci tensor.

Imagine a bunch of coffee grounds near the coffee ground that went through the point P. Consider,

for example, all the coffee grounds that were within a given distance at time zero (in the local rest

frame of the coffee ground that went through P). And suppose that at time zero all the coffee

grounds are comoving. A little round ball of coffee grounds in free fall through outer space! As time

passes this ball will change shape and size depending on how the paths of the coffee grounds are

deflected by the spacetime curvature. Since everything in the universe is linear to first order, we can


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imagine shrinking or expanding, and also getting deformed to an ellipsoid. There is a lot of

information about spacetime curvature encoded in the rate at which this ball changes shape and

size. But let's only keep track of the rate of change of its volume! This rate is basically the Ricci

tensor.

More precisely, the second time derivative of the volume of this little ball is approximately

-Rab va vb

times the original volume of the ball. This approximation becomes better and better in the limit as

the ball gets smaller and smaller. The first time derivative of the volume is zero, since the coffee

grounds started out comoving.

In 4-dimensional spacetime, the Riemann tensor has 20 independent components. 10 of these are

captured by the Ricci tensor, while the remaining 10 are captured by the WEYL TENSOR.

The RICCI SCALAR. Starting from the Ricci tensor, we define

R

a

d = g

ab

Rbd.

As always, we follow the Einstein summation convention and sum over repeated indices when one

is up and the other is down. This process, which turned one subscript on the Ricci tensor into a

superscript, is called RAISING AN INDEX. Similarly we can LOWER AN INDEX, turning any

superscript into a subscript, using gab.

Then we define the Ricci scalar by

R = Raa.

This process, whereby we get rid of a superscript and a subscript in a tensor by summing over thema la Einstein, is called CONTRACTING.

10.

The EINSTEIN TENSOR. Finally, we define the Einstein tensor by

Gab = Rab - (1/2)R gab.

You should not feel you understand why I am defining it this way!! Don't worry! That will take

quite a bit longer to explain; the point is that with this definition, local conservation of energy and

momentum will be an automatic consequence of Einstein's equation. To understand this, we need to

know Einstein's equation, so we need to know about:

11.

The STRESS-ENERGY TENSOR. The stress-energy is what appears on the right side of Einstein's

equation. It is a tensor of rank (0,2), and it defined as follows: given any two tangent vectors u and

v at a point p, the number T(u,v) says how much momentum in the u direction is flowing through

the point p in the v direction. Writing it out in terms of components in any coordinates, we have

T(u,v) = Tab ua vb

In coordinates where x0 is the time direction t while x1, x2, x3 are the space directions (x,y,z), and

the metric looks like the usual Minkowski metric (at the point in question) we have the following

physical interpretation of the components Tab:

The top row of this 4x4 matrix, keeps track of the density of energy --- that's T00 --- and the density

of momentum in the x,y, and z directions --- those are T01, T02, and T03 respectively. This should

make sense if you remember that "density" is the same as "flow in the time direction" and "energy"

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is the same as "momentum in the time direction". The other components of the stress-energy tensor

keep track of the flow of energy and momentum in various spatial directions.

EINSTEIN'S EQUATION: This is what general relativity is all about. It says that

G = T

or if you like coordinates and more standard units,

Gab = 8 pi k/c2

Tab

where k is Newton's gravitational constant and c is the speed of light. So it says how the flow of

energy and momentum through a given point of spacetime affect the curvature of spacetime there.

But what does it mean? To see this, let's do some "index gymnastics". Stand with your feet slightly

apart and hands loosely at your sides. Now, assume the Einstein equation!

Gab = Tab

Substitute the definition of Einstein tensor!

Rab - (1/2)R gab = Tab

Raise an index!

Rab - (1/2)R gab = T

ab

Contract!

Raa - (1/2)R gaa = T

aa

Remember the definition of Ricci scalar, and note that gaa = 4 in 4d!

R - 2R = Taa

Solve!

R = - Taa

Okay. That's already a bit interesting. It says that when Einstein's equation is true, the Ricci scalar R

is the sum of the diagonal terms of Taa. What are those terms, anyway? Well, they involve energy

density and pressure. But let's wait a bit on that... let's put this formula for R back into Einstein's

equation:

Rab + (1/2) Tcc gab = Tab

or

Rab = Tab - (1/2) Tcc gab.

This equation is equivalent to Einstein's equation. What does it mean? Well, first of all, it's nicebecause we have a simple geometrical way of understanding the Ricci tensor Rab in terms of

convergence of geodesics. Remember, if v is the velocity vector of the particle in the middle of a

little ball of initially comoving test particles in free fall, and the ball starts out having volume V, the

second time derivative of the volume of the ball is

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-Rab va vb

times V. If we know the above quantity for all velocities v (or just all timelike velocities, which are

the physically achievable ones), we can reconstruct the Ricci tensor Rab. But we might as well work

in the local rest frame of the particle in the middle of the little ball, and use coordinates that make

things look just like Minkowski spacetime right near that point. Then gab looks like

-1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

and va is just

1

0

0

0

So then --- here's a good little computation for you budding tensor jocks --- we get

Rab va

vb

= R00

So in this coordinate system we can say the 2nd time derivative of the volume of the little ball of

test particles is just -R00.

On the other hand, check out the right side of the equation:

Rab = Tab - (1/2) Tcc gab

Take a = b = 0 and get

R00 = T00 + (1/2) Tcc

Note: demanding this to be true at every point of spacetime, in every local rest frame, is the same as

demanding that the whole Einstein equation be true! So we just need to figure out what it MEANS!

What's T00? It's just the energy density at the center of our little ball. How about Tcc? Well,

remember this is just gca Tac, where we sum over a and c. So --- have a go at it, tensor jocks and

jockettes! --- it equals -T00 + T11 + T22 + T33. So we get

R00 = (1/2) [T00 + T11 + T22 + T33]

What about T11, T22, and T33? In general these are the flow of x-momentum in the x direction, and

so on. In a typical fluid at rest, these are all equal to the pressure.

So the "simple geometrical essence of Einstein's equation" is this:

Take any small ball of initially comoving test particles in free fall. Work in the local rest frame of

this ball. As time passes the ball changes volume; calculate its second derivative at time zero and

divide by the original volume. The negative of this equals 1/2 the energy density at the center of the

ball, plus the flow of x-momentum in the x direction there, plus the flow of y-momentum in the y

direction, plus the flow of z-momentum in the z direction.

Or, if you want a less precise but more catchy version:


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Take any small ball of initially comoving test particles in free fall. As time passes, the rate at which

the ball begins to shrink in volume is proportional to the energy density at the center of the ball plus

the flow of x-momentum in the x direction there plus the flow of y-momentum in the y direction

plus the flow of z-momentum in the z direction.

Note: all of general relativity can in principle be recovered from the above paragraph! Also note

that the minus sign in that paragraph is good, since it says if you have POSITIVE energy density,

the ball of test particles SHRINKS. I.e., gravity is attractive.


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