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Energy is a slippery concept in GR, but in an asymptotically flat spacetime there is a perfectly reasonable concept of energy (indeed, several) which goes by the name the ADM energy: $$ E_\mathrm{ADM} := \frac{1}{16 \pi} \lim_{r \to \infty} \int_{S^2_r} \mathrm{d}A \, n_i (\partial_j h_{ij} - \partial_i h_{jj} )$$ Where $h_{ij}$ is the 3-metric on a spatial hypersurface $\Sigma$. My question is simply: is the ADM energy time-independent, for an arbitrary (asymptotically flat) spacetime? Said another way, is the ADM energy independent of the chosen spatial hypersurface? In particular, for a spacetime with no timelike Killing vector field, is the ADM energy conserved? How about the ADM 3-momentum?

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The ADM energy is conserved at infinity, for asymptotically flat spacetime. It refers to spatial infinity, and the reason it is conserved is that it is asymptotically Minkowski, and so Noether's theorem says it is conserved asymptotically. See the energy section in the ADM wiki article, at https://en.m.wikipedia.org/wiki/ADM_formalism

It is not a totally trivial result. For instance in cosmological models spacetime doesn't have to be asymptotically flat.

Not sure what could be different spatial infinities in Minkowski spacetime, maybe just a mathematical construction. I do know that you get a different conserved entity at lightlike infinity from the construction of lightlike infinity. That's called the Bondi Metzner Sachs (BMS) formalism, and is more useful to determine the energy at infinity for isolated bodies that radiate gravitational waves. The BMS group at conformal lightlike infinity, defined rigorously, provides the conserved energy, and other conserved quantities that include those from the Poincare group but others in adidition. Hawking and his collaborators are using it to define conserved entities at black hole horizons, which are also lightlike infinities. See the BMS treatment at http://www.scholarpedia.org/article/The_Bondi-Sachs_Formalism.

For the Hawking et al work just google their arxiv posting early in 2016 on soft hair of black holes.

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  • $\begingroup$ Your answer seems to be missing some words at the end $\endgroup$ – Virgo Nov 18 '16 at 5:22
  • $\begingroup$ Yes, just fixed it, I hit post it too soon. $\endgroup$ – Bob Bee Nov 18 '16 at 5:31
  • $\begingroup$ Thanks for the answer. One question: what exactly do you mean by 'conserved at infinity'? Do you just mean that if I removed the limit from my definition, and considered 'the ADM energy at radius $r$', then such a quantity would only be conserved in the limit $r \to \infty$? (The only thing I don't understand is that the ADM energy as I've defined it is a single number, independent of space, and so I don't know what it means to talk about this quantity 'at infinity'). $\endgroup$ – gj255 Nov 18 '16 at 13:19
  • $\begingroup$ It means that the energy inside that does not change with time. And yes, do it at r and let r go to infinity. In BMS they actually define a covariant conformal infinity. In ADM a little different but still well defined $\endgroup$ – Bob Bee Nov 18 '16 at 14:20

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