Is Energy Conserved in General Relativity?

[pervect]

An inertial frame is a freely falling coordinate system. In such a frame of reference particles do not suffer accelerations unless there are specific non-gravitational forces acting on them. Such a frame can only be defined for a sufficiently small region around its origin, otherwise tidal forces will be experienced.

The tidal forces aren't a problem, if they approach the Newtonian tidal forces in the limit as you go to infinity. The standard definition of energy and energy conservation in GR can deal with tidal forces that approach Newtonian tidal forces as one goes to infinity.

It does appear to me that the expansion of the universe is not something that can be dealt with in this (standard) manner, however. This problem can only be dealt with by dealing with sections of the universe small enough that the cosmological expansion isn't important over the timescale studied.

The overall insight is that GR does not in general conserve energy, it is an improper energy theorem, it conserves energy-momentum instead.
The principle of the conservation of energy-momentum is not a concatenation of the principle of the conservation of energy and principle of the conservation of momentum; energy-momentum is a geometric concept in its own right, invariant under Lorentz transformations.

Energy and momentum are frame dependent concepts; therefore it is necessary to define a frame of reference, a preferred frame. in order to restore the principle of the conservation of energy.

Garth

The notion of a preferred frame of course requires a rather fundamental re-write of GR - one which a certain author just happens to have done :-).

We'll see how this new theory works out when the Gravity probe B results get back.

Meanwhile, I have to say that it is quite possible that the universe is screwy enough that the standard GR notion of energy conservation is the correct one - something that works over human time and distance scales, but something that doesn't work over cosmological time and distance scales.