General Relativity Tutorial - Short Course Outline

John Baez

January 26, 1996

If you click on some of the capitalized concepts, you will jump to the corresponding place in the longer outline. In some cases, if you click on them again from there, you will jump to a still more thorough explanation.

1. A TANGENT VECTOR at the point p of spacetime may be visualized as an infinitesimal arrow with tail at the point p.

2. A TENSOR of "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 TENSOR of "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.

3. 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.

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 an important theorem saying that if we have a metric g, there is a unique way to do parallel translation which is:
   1. Linear: the output v' depends linearly on v.

   2. 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".

   3. Torsion-free: this is a way of making precise the notion of "no rotating". I don't think I want to go into the math of "TORSION" just yet. Let's see the overall picture first.

5. 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 εu and εv , where ε 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 = -ε²R(u,v,w) + terms of order ε³

   Thus the Riemann tensor keeps track of how much parallel translation around a wee parallelogram changes the vector w.

6. Introducing COORDINATES. Now say we choose coordinates on some patch of spacetime near the point p. Call these coordinates x^a (where a = 0,1,2,3). Then given any tangent vector v at p, we may speak of its components v^a in this basis. The inner product g(v,w) of two tangent vectors is given by

   g(v,w) = g_ab v^a w^b

   for some matrix of numbers g_ab, where as usual we sum over the repeated indices a,b, following the EINSTEIN SUMMATION CONVENTION. Another way to think of it is that our coordinates give us a basis of tangent vectors at p, and g_ab is the inner product of the basis vector pointing in the x^a direction and the basis vector pointing in the x^b direction.

   Similarly, the vector R(u,v,w) has components

   R(u,v,w)^a = R^a_bcd u^b v^c w^d

   where we sum over the indices b,c,d.

7. The EINSTEIN TENSOR. The matrix g_ab is invertible and we write its inverse as g^ab. 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. We will do this using coordinates to save time... though later we should do this over again without coordinates. This part is the only profound and mysterious part, at least to me.

   Okay: starting from the Riemann tensor, which has components R^a_bcd, we now define the RICCI TENSOR to have components

   R_bd = R^c_bcd

   where as usual we sum over the repeated index c. Then we "RAISE AN INDEX" and define

   R^a_d = g^ab R_bd,

   and then we define the RICCI SCALAR by

   R = R^a_a

   The Riemann tensor knows everything about spacetime curvature, but these gadgets distill certain aspects of that information which turn out to be important in physics. Finally, we define the Einstein tensor by

   G_ab = R_ab - (1/2)R g_ab.

   You still should not feel you understand why I am defining it this way!! Don't worry! That will take a bit longer to explain. But we are almost at Einstein's equation; all we need now is:

8. 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-v-direction is flowing through the point p in the u direction. Writing it out in terms of components in any coordinates, we have

   T(u,v) = T_ab u^a v^b

   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 T_ab:

   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" 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.

9. EINSTEIN'S EQUATION: This is what general relativity is based on. It says that

   G = T

   or if you like coordinates and more standard units,

   G_ab = 8 π k/c² T_ab

   where k is Newton's gravitational constant and c is the speed of light. This equation says how the flow of energy and momentum through any given point of spacetime affect the curvature of spacetime there.

   But what does Einstein's equation mean? Well, it turns out that that a = b = 0 component of this equation can be translated into plain English as follows:

   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.

   Even better, if this holds in every coordinate system, Einstein's equation holds. So, all of general relativity can be recovered from this one sentence if one really thinks hard. In practice, though, it's best to master tensors and learn how to calculate with them to extract the predictions of general relativity.

For more details, go to the longer course outline.

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