- #1

Qwet

- 13

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Hello.

I have a question about the law of energy conservation in GR.

As time is inhmogeneous, we don't have energy-momentum 4-vector which would be preserved during system's dynamical change. It is only possible to define 4-vector locally. And next, the problem regarding how to sum this vectors arises.

Do I understand correctly that this problem can be solved only if there is Killing vector field? As the space has to be isometrical according to Noether's theorem (conservation of energy-momentum is given in relation to space-time movement which means we have to consider isometry group). Or it is sufficient to have any kind of symmetry (not just isometry)?

I considered the case where there is no Killing field. And as I understand, it is still possible to construct values satisfying a differential continuity equation. And using Gauss's theorem we can get an integral equation to correspond to the law of conservation. But, again, it is only possible if we have Killing field because the derivatives are covariant in Gauss's theorem (for energy-momentum tensor:

I have a question about the law of energy conservation in GR.

As time is inhmogeneous, we don't have energy-momentum 4-vector which would be preserved during system's dynamical change. It is only possible to define 4-vector locally. And next, the problem regarding how to sum this vectors arises.

Do I understand correctly that this problem can be solved only if there is Killing vector field? As the space has to be isometrical according to Noether's theorem (conservation of energy-momentum is given in relation to space-time movement which means we have to consider isometry group). Or it is sufficient to have any kind of symmetry (not just isometry)?

I considered the case where there is no Killing field. And as I understand, it is still possible to construct values satisfying a differential continuity equation. And using Gauss's theorem we can get an integral equation to correspond to the law of conservation. But, again, it is only possible if we have Killing field because the derivatives are covariant in Gauss's theorem (for energy-momentum tensor:

**D**, where_{i}T^{i}k = 0**D**is covariant). And the equation for covariant divergence doesn't correspond to any conservation law._{i}
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