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Astronomy and Cosmology
Cosmology
"Energy is not conserved" vs. energy is conserved: Friedmann Equations
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[QUOTE="PeterDonis, post: 6869256, member: 197831"] Because it would be wrong. Global conservation is not the same thing as local conservation. Local conservation is a differential law, obtained by taking the covariant divergence of the Einstein Field Equation. It holds at every event--which is consistent with the cosmological principle--but it's still local. Global conservation is an integral law: you have to have some invariant way of picking out an integral that describes the "total energy". The only spacetimes in which such an invariant exists are asymptotically flat spacetimes (but even then there are [I]two[/I] such integrals, not one--the ADM energy and the Bondi energy) and stationary spacetimes (for which the Komar energy is the invariant). For other spacetimes, no such invariant integral exists. The cosmological principle says that the universe should be the same [I]locally[/I] everywhere. It makes no sense to say the universe is globally "the same everywhere", since a global property is not a property that could vary or not vary "from place to place"; it's an integral property. If "don't care about" means "don't have", yes, this is correct. FRW spacetimes are not stationary, and they are not asymptotically flat. So they don't fall into either of the two classes of spacetimes above that have invariant global "total energy" integrals. So they don't have any. Noether's theorem for energy requires a timelike Killing vector field, i.e., it requires the spacetime to be stationary. FRW spacetimes are not stationary, so Noether's theorem for energy does not apply to them. If you mean that energy in a spacetime that isn't stationary or asymptotically flat can be conserved locally but not globally, yes, this is correct. [/QUOTE]
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Astronomy and Cosmology
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"Energy is not conserved" vs. energy is conserved: Friedmann Equations
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