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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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  • It is better to ask one question at a time, if they are unrelated. – Anders Sandberg Jun 19 '23 at 00:15
  • Some people think that general relativity has a lot to say about thermodynamics. The laws governing black hole behavior can apparently be cast in a form in which direct analogies can be drawn between those laws and the three fundamental laws of thermodynamics.

    I can't provide the narrative in this case, and must instead rely on the experts here to do so.

     – niels nielsen Jun 19 '23 at 03:16
  • @AndersSandberg Sometimes, people think that two questions are unrelated, but they may not be the best judges. In my opinion, this question is about the single issue of the relation between Thermodynamics and GR. – GiorgioP-DoomsdayClockIsAt-90 Jun 19 '23 at 08:03

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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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  • Is there a known realistic energy-momentum tensor field which implies a CTC? Something that can form from normal matter distribution moving around. – Ján Lalinský Jun 19 '23 at 00:31
  • @JánLalinský - Not as far as I know. There are a few "realistic" fields in some wormhole papers, but even if they can be formed how to get the topology change necessary to move from a singly connected to a multiply connected spacetime is unclear. – Anders Sandberg Jun 19 '23 at 15:47