Superposed_Cat said:
I guess my more accurate question was whether everything was conserved in GR when kinetic energy was changed to gravitational potential
It is in a special class of spacetimes, stationary spacetimes (heuristically these are spacetimes that you can view as describing a system, like a central mass, that doesn't change with time). In a stationary spacetime there are two conserved "energy" quantities of interest:
(1) Energy at infinity: this is a constant of geodesic (free-fall) motion, and it is what is conserved when, as you describe, kinetic energy and gravitational potential energy are converted back and forth as an object follows a free-fall orbit. The name can be somewhat confusing since, for example, an object in a bound orbit is never
at infinity; but you can think of it as the sum of kinetic and gravitational potential energy that works more or less the same way as it does in Newtonian physics (the only complication in GR is that you have to work in the rest frame of the central mass, since that is the proper frame for defining the kinetic and potential energy in the right way).
(2) Komar energy: Also called Komar mass, this is the GR version of the intuitive notion of obtaining the total mass of the central body by adding up the contributions of the stress-energy tensor of all the parts of the body. Note, however, that this is a
different quantity from energy at infinity above; energy at infinity has nothing to do with the stress-energy tensor at all, and is almost always made use of in the vacuum region outside the central mass, since that's where objects will be in free-fall orbits--and in the vacuum region the stress-energy tensor is zero. The Komar mass is obtained by an integral over the
interior of the central mass, and doesn't involve kinetic or potential energy; the key complications in GR are that the geometry of space inside the mass is not Euclidean, that the stress-energy tensor inside the mass includes more than just its energy density (for a static object the key additional contribution is from pressure), and that gravitational binding energy makes a negative contribution to the total mass. The Komar mass integral incorporates all of this.
Superposed_Cat said:
, as with electric potential, elastic tension, and all the other ways to store it, there was a clear way to write it in the SE tensor, whereas when energy is stored gravitationally, I don't see where it goes in a recoverable way in the field equation
See the discussion of Komar mass above.