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In Is the total energy of the universe zero? , the top answer states that the conservation of energy no longer holds as noether's theorem doesn't hold as the universe isn't space/time invariant as spacetime bends. Firstly, is this true? This seems quite surprising to me.

Because: space invariance still holds, even though the world isn't really space invariant. There is matter, so the universe and physics changes depending how close to planets and things one is. So, is there a similar chain of logic that allows for time and space invariance within general relativity?

Secondly, this is kind of unrelated, but does the 2nd law of themodynamics hold in general relativity? If not, is there an equivalent law, and why does it not hold?

Thank you for answering my question!

OdinOblivion
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General relativity has nothing to say about thermodynamics. It is a theory about how spacetime interacts with matter fields, but it does not really care about their entropy.

That said, there are some issues. One is that thermodynamics insists that these fields increase (or stay the same) entropy as time goes by, but GR does not necessarily have a well-defined time direction. For example, a wormhole solution could be used to set up a time loop (a CTC, closed timelike curve) and this might undermine thermodynamics. Or not: it has been argued that thermodynamics somehow prevents objects from traversing or forming CTCs. The issue is not resolved to my knowledge, but pure GR predicts that certain spacetimes give thermodynamics a headache.

The second issue is that there might be a kind of entropy of the gravitational field itself, expressed in terms of the Weyl tensor. This Weyl curvature hypothesis is so far unproven.

As for the energy conservation, locally energy and momentum are well-behaved since locally Noether's theorem applies in most reasonable spacetimes. It is just that energy and momentum in a general spacetime are not conserved if a particle moves about enough.

Anders Sandberg
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