Last time around, I described the Einstein equation in terms of the Einstein tensor and the stress energy tensor, two rather abstract entities. Now, I'd like to describe the content of the equation in more physical terms to give you more of an insight into what the equation actually means. To do this I'll first have to talk about the idea of relative accelerations in general relativistic spacetimes and then consider the motion of initially comoving particles arranged on spherical shells (both in vacuum and in the presence of matter). In the next two parts, I'll go on to talk about the solutions of the field equations for homogeneous isotropic universes and for non-rotating, uncharged black holes. After that I'll describe variational formulations of general relativity. Finally, we'll move into the murky depths of quantum gravity.

Recall that in special relativity there was no absolute meaning for "at rest". Instead, we had families of inertial frames related by Lorentz transformations and any of these frames was as good as any other. However, we did still have an absolute sense of "at rest relative to each other" - we could grab a little piece of one particle's worldline and carry across to a point on another particle's worldline and see if they were parallel. If they were parallel for all times then the two particles were at rest relative to each other. As previously described, however, in curved spacetimes the result of carrying a piece of worldline around ("parallel transporting a tangent vector") in general depends on the path taken. That means that in curved spacetimes we can no longer say whether two particles are at rest relative to one another. There's no longer a special family of inertial frames that cover all of spacetime. In a really, really small region, though, we can say that two particles are at rest relative to each other, because if you look ever more closely at a curved spacetime it starts to look just like a flat one. A pair of particles at rest relative to each other in this sense are said to be "comoving". The curvature of the spacetime may make initially comoving particles become closer or more distant with time. For example, imagine that Jo and I stand on the equator ten metres apart and both start moving north at the same speed. Initially we're comoving because we think we're moving in the same direction at the same speed but as we head towards the North Pole we'll find that we get closer and closer to each other and finally at the Pole itself we'll collide. We might think that there's a force acting between us and pulling us towards each other, but really we can see that it's an effect of the curvature of the surface of the Earth. Gravitation is really just like this - it looks like it's a force but it's really the curvature of spacetime. Relative accelerations of particles moving under the absence of all forces are caused by geometry.

Now, imagine arranging a set of comoving test particles on the surface of a little spherical region and watching them as they move along geodesics in the spacetime. We can't see the geodesics, of course, but we can see what happens to the shape of the shell. Einstein's equation can be entirely reconstucted from the equation describing the behaviour of such initially comoving spheres. Here's how it works. First measure the volume of the region inside the shell for all proper times as you travel along with the centre of the shell. Then measure the energy density and pressures in the three orthogonal directions at the sphere's centre. The equation describing the behaviour of the shell then says that if you work out the "acceleration" of the volume (that is, the second derivative of the volume by time) and divide this by the current volume then this is proportional to minus the sum of the energy density and three components of the pressure at the centre of the sphere. The particles on the shell are tracing the geometry of the spacetime and the energy density and pressure are parts of the stress-energy tensor (remember that this describes how energy and momentum are flowing around in spacetime; energy density is the flow of energy in the "time" direction and pressure is related to the flow of momentum perpendicular to a surface across a surface). Requiring that the equation that I described above be true in all coordinate systems requires that the full Einstein equation that I spoke about last time be true. Now, though, we're talking about things that can be much more easily visualised and which are related to much more elementary physical quantities than stress-energy tensors.

Let's first see what this new formulation has to say about the motion of particles in free space. In this case the energy density and pressure are both zero, so the Einstein equation says that the volume of the shell remains constant. This isn't to say that the shape of the shell remains the same and in general it won't. For example, if we just drop a spherical shell of particles somewhere above the Earth then it will become distorted into an ellipsoid, because each particle will move along a line (in space) directed towards the centre of the Earth so that those on the sides of the sphere will become closer together; and the top and bottom particles will become further apart (in terms of forces, the "gravitational force" on the lower particle is stronger than that on the upper particle, but in terms of spacetime geodesics it's just the observation that the curvature in the vicinity of the Earth is such that the geodesics of the two initially comoving particles diverge from each other). We call the changing of shape but not volume of the sphere a "tidal effect", because this is the behaviour associated with particles under "tidal forces" in Newtonian gravitation. But remember that in this case there aren't really any forces, just curvature of spacetime.

Now let's look at a simple case in which there's matter present: a spherical shell of test particles initially comoving with a little blob of mass at the centre of the shell. In this case the equation and the symmetry of the situation predict that the shell will begin to fall towards the central blob. In terms of spacetime, the worldlines of the test particles on the shell and the central blob intially look parallel, but the mass of the blob curves spacetime so that the worldlines of the test particles converge on its worldline. Blobs of mass thus attract other particles by curving spacetime in their vicinity. It's very simple really - in some ways it's simpler than the Newtonian picture of fields of force and inertial and gravitational masses and all of that stuff. And, of course, this simple picture forms the basis of the Einstein equations and hence the whole theory of gravitation embodied in general relativity. In the next part, I'll use this simple new formulation to talk about the behaviour of the entire universe (at least in the approximation that the universe is filled with "dust" and is homogeneous and isotropic; but those are very good approximations to the properties of the real universe [at least on sufficiently large scales]).