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In the comment in What are the reasons to expect that gravity should be quantized? by Ron Maimon, it is mentioned that taking analogy from classical electromagnetic wave to classical grvational wave, one can notice that conservation of energy is violated.

However, general relativity does not really have conservation of momentum as fundamental concept, and thus it is unclear what this would mean. Can anyone explain this?

Can Bohr-Kramers-Slater (BKS) theory really serve as an example refuting possible validity of classical gravity?

Qmechanic
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Matsu Takama
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  • Energy is the Noether charge of the time translation symmetry. The conservation of this charge in only one specific case of the theorem. The charge is conserved only if the time translation symmetry exists. In the general case, in GR, this symmetry does not exist. This simply means that the corresponding Noether charge (energy) is variable, but it does not at all invalidate the concept. "Not well defined" means a technical difficulty in writing the equation, but not that the concept is invalid. So energy is variable, why is this strange? – safesphere Jul 28 '18 at 16:13
  • @safesphere: Energy is the Noether charge of the time translation symmetry. The conservation of this charge in only one specific case of the theorem. The charge is conserved only if the time translation symmetry exists. In the general case, in GR, this symmetry does not exist. This explanation keeps propagating on the internet, and it's just plain wrong. Time-translation symmetry of a spacetime is neither necessary nor sufficient for conservation of energy. If it holds, it only guarantees energy conservation for test particles, not the spacetime as a whole. If it fails, it doesn't[...] –  Jul 28 '18 at 18:04
  • [...] imply nonconservation of energy, even in the global sense. For example, an asymptotically flat spacetime always has certain globally conserved measures of its energy, regardless of whether it's static. –  Jul 28 '18 at 18:05
  • @BenCrowell Thanks Ben for your deeper insight! I understand and agree with your point. The point of my comment was that energy is a Noether charge that may or may not conserve depending on the circumstances. And if it does not conserve in certain cases, this doesn't make energy meaningless, but only variable. Also, while I have your expert attention, is there any chance you could please post an answer or comment to my earlier question? Thank you! https://physics.stackexchange.com/questions/415484/does-an-expanding-event-horizon-swallow-nearby-objects – safesphere Jul 28 '18 at 18:19

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However, general relativity does not really have conservation of momentum as a fundamental concept, and thus it is unclear what this would mean. Can anyone explain this?

GR does have conservation of momentum as a fundamental concept. Specifically, the structure of GR requires that the stress-energy tensor have zero divergence, which is a statement of local conservation of the energy-momentum four-vector. What GR doesn't have is a generic global conservation law for energy-momentum, but I don't think that has any logical consequences for the argument you refer to, because we do have such conservation laws for special cases like asymptotically flat spacetimes, and one can in principle detect gravitons, and falsify a classical theory of gravity, in an asymptotically flat spacetime.

In any case, the argument about the nonconservation of energy in BKS is really more about the nonconservation of probability, i.e., it's about unitarity. It's just that in 1927, people described it in terms of having only statistical conservation of energy and momentum.

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