The concept of work in classical mechanics is useful in dealing with conservation of energy, since it tells you how you can add or subtract energy from a system, especially a system in a potential. So my question is this: does conservation of energy hold in GR? And if it does or doesn't, could someone explain why or why not? (Just from the fact that gravity isn't treated as a force and that it deals with non-inertial reference frames, I would think you'd run into problems...)

JP that's a lovely question

And one I'm wondering over too as it connects to my question of the plate jiggling.

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In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

In flat spacetime (the backdrop for special relativity) you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity) this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any infinitesimal piece of spacetime. The integral form says the same for a finite-sized piece. (This may remind you of the "divergence" and "flux" forms of Gauss's law in electrostatics, or the equation of continuity in fluid dynamics. Hold on to that thought!)

An infinitesimal piece of spacetime "looks flat", while the effects of curvature become evident in a finite piece. (The same holds for curved surfaces in space, of course). GR relates curvature to gravity. Now, even in newtonian physics, you must include gravitational potential energy to get energy conservation. And GR introduces the new phenomenon of gravitational waves; perhaps these carry energy as well? Perhaps we need to include gravitational energy in some fashion, to arrive at a law of energy conservation for finite pieces of spacetime?

Casting about for a mathematical expression of these ideas, physicists came up with something called an energy pseudo-tensor. (In fact, several of 'em!) Now, GR takes pride in treating all coordinate systems equally. Mathematicians invented tensors precisely to meet this sort of demand -- if a tensor equation holds in one coordinate system, it holds in all. Pseudo-tensors are not tensors (surprise!), and this alone raises eyebrows in some circles. In GR, one must always guard against mistaking artifacts of a particular coordinate system for real physical effects. (See the FAQ entry on black holes for some examples.)

These pseudo-tensors have some rather strange properties. If you choose the "wrong" coordinates, they are non-zero even in flat empty spacetime. By another choice of coordinates, they can be made zero at any chosen point, even in a spacetime full of gravitational radiation. For these reasons, most physicists who work in general relativity do not believe the pseudo-tensors give a good local definition of energy density, although their integrals are sometimes useful as a measure of total energy.

One other complaint about the pseudo-tensors deserves mention. Einstein argued that all energy has mass, and all mass acts gravitationally. Does "gravitational energy" itself act as a source of gravity? Now, the Einstein field equations are

Gmu,nu = 8pi Tmu,nu

Here Gmu,nu is the Einstein curvature tensor, which encodes information about the curvature of spacetime, and Tmu,nu is the so-called stress-energy tensor, which we will meet again below. Tmu,nu represents the energy due to matter and electromagnetic fields, but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory). So one can argue that "gravitational energy" IS a source of gravity.

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Is Energy Conserved in General RelativityAnd this one too.

"The Newtonian concept of "potential energy" appears at first glance to work in GR as well when describing the motion of a test mass in the field of a central body, in that the effective rest energy change due to time dilation in a static field matches the potential energy, and the potential energy plus (relativistic) kinetic energy remain constant for motion in a static field.

However, I was surprised to find that it doesn't appear to work if you also consider the energy change of the source mass, or consider two similar masses orbiting around one another, even in a linearized weak field approximation. In that model, each of the objects apparently experiences an equal decrease in rest energy as it approaches the other (because of time dilation due to the other object) matching the Newtonian potential energy, so the rest energy of the system changes by twice the amount in the Newtonian model (where the potential energy is a property of the configuration of the system, not of the individual objects). However, the kinetic energy of the system only increases by the Newtonian kinetic energy, equal to the Newtonian potential energy, so this doesn't seem to add up.

In gravitational Quantum Field Theory, some people apparently assume that the energy of the field increases by the same amount, preserving the total energy, and that works very nicely from a mathematical point of view, giving a result very much like the Maxwell energy density of the field in electromagnetism. However, GR says that there is no energy in a vacuum regardless of the strength of the field, so that appears to mean either that QFT isn't compatible with GR or that the definition of "energy" is not the same in these two cases.

Given that the Newtonian concept of conservation of energy allows us to calculate complex gravitational interactions of many bodies to very high accuracy in non-relativistic situations, it seems surprising to me that there should be a problem with the concept in such a simple two-body situation in a weak GR approximation." By Jonathan Scott