As I previously described, the equations for various fields all look pretty much the same. On the left side of the equation is some property of the field, and on the right side is the source of the field. For general relativity we'll want to put something describing curvature on the left of the equations and something describing matter on the right. Now, in relativity mass and energy are more or less the same thing, and the energy and momentum are related to each other in the same way as time and space are (as described in the discussion of symmetries and conservation laws). This means that not just energy but momentum too will have to act as the source for gravity or else the equations would fail to be invariant under a change of coordinate frames. Furthermore, the thing on the right of the field equations will have to be a density of energy and momentum, and this idea is also not relativistically invariant, because a density is the amount in a small region of space, and different observers will slice up spacetime into space in different ways. The mathematical object that encodes the energy-momentum density in a relativistically invariant way is the "stress-energy tensor".

The stress-energy tensor is a gadget with two slots that can be cooked up at each event from the particles and (non-gravitational) fields at that event. Remember that the metric tensor is also a gadget with two slots and that if you took a little piece of worldline and put it into both slots it tells you how much proper time elapses as you travel along that piece of worldline. Similarly, if you put a little piece of worldline into both slots of the stress-energy tensor then it tells you how much energy density would be measured by somebody travelling along that bit of worldline. There are other useful you can do with the stress-energy tensor. If you put a piece of worldline into just one slot then it tells you the energy-momentum density that somebody travelling along that piece of worldline would see flowing past them. Finally, the stress-energy tensor lets you work out things like the stress and pressure in materials at the event, but that won't be very important for a while.

If the stress-energy tensor is the right side of the field equations, we need to figure out what goes on the left side. Whatever goes there has to be some measure of how spacetime is curved, so it should be something constructed from the metric and/or the Riemann curvature tensor (and must have two slots, and there must be no way to tell apart the slots; in maths jargon this means it must be a "symmetric second rank tensor"). Furthermore, if spacetime is flat then it should be zero, and if you double the value of the Riemann tensor then the left side should also double (which means that it will be a true measure of curvature). Unfortunately, these requirements are not enough to uniquely determine the left side of the field equations.

Now, it's been known for a long time that energy and momentum are conserved, and, more importantly, are conserved locally. If you're thinking non-relativistically then this is the same as saying that the change in the amount of energy or momentum in a region of space is given by the amount of energy or momentum flowing into or out of the region - energy and momentum cannot be created or destroyed, just moved from one place to another. Relativistically, this has to be true not just for any region but also when viewed in any frame. If energy and momentum are to be locally conserved then there must be a restriction placed on the possible forms of stress-energy tensor, because that tensor determines the way that energy and momentum are flowing around in spacetime; in the jargon we say that the divergence of the stress-energy tensor must be zero. The key insight in building the Einstein field equations is that it's possible to "wire up" the geometry and the stress energy in such a way that local conservation isn't something that is imposed as an extra principle, but is a geometric necessity. This additional requirement uniquely determines the left side of the field equations; the tensor that appears there is called the Einstein tensor.

The field equations thus say that the Einstein tensor is proportional to the stress-energy tensor. To find out what the metric looks like for a given distribution of matter and energy in the spacetime, what we have to do is take a general metric that reflects the symmetry of the spacetime, make the connection and hence the Riemann curvature tensor from that, use all of these things to cook up the Einstein tensor and finally set that equal to the stress-energy tensor for all the fields and particles we've put in the spacetime. If we do this then we get equations for each of the components of the metric tensor, and we can solve those to find what the metric actually looks like. In later parts I'll describe the solutions to the field equations for various situations with high degrees of symmetry and particularly simple matter distributions. In the next part, however, I'll discuss variational approaches to general relativity.