Originally Posted by

**bangstrom**
But he did not go so far as to say that it does not apply over large scales

What I was trying to say is that there is

**no** global law of energy conservation across regions of space-time where curvature cannot be neglected - it is simply not definable, and it does not make any physical sense. That's because the energy-momentum tensor is the conserved Noether current of space-time translation invariance, but in a curved space-time notions of "space" and "time" and purely local ( so in general there is no such invariance ), so this tensor is quite simply not defined for an extended region, hence it makes no sense to talk about its conservation either. The entire concept is not physically meaningful. Energy-momentum is rigorously defined and conserved everywhere locally, but not defined globally.

And, since he mentioned nothing except how difficult it is measure the conservation of energy, he made it sound as if we don't know if energy is conserved or not- anywhere.
While the notion of "conservation of energy-momentum" makes no physical sense globally in a curved space-time, one can define another object which is a linear combination of the energy-momentum tensor and a mathematical entity called the Landau-Lifschitz pseudotensor. This latter entity can intuitively be understood as gravitational self-energy, i.e. it is a measure of how much energy gravity itself "carries" with it. As being a pseudotensor, it is an observer-dependent quantity ( as it must be of course ); however, the divergence of its combination with the energy-momentum is again is fully covariant tensor.

To cut a long ( and mathematically complicated ) story short - conservation of energy-momentum is not defined globally, however, conservation of a combination of energy-momentum

**and** gravitational self-energy is in fact globally conserved. In other words, to generalise the Newtonian energy conservation law to curved space-times, you need to account for both sources of energy-momentum

**and** gravitational self-energy; neither one in isolation is conserved, but the combination of the two is, and forms a conserved current. See here for more details :

Stress