A global energy conservation law in general relativity

[Bosko]

...
Thus, I am tempted to conclude that there is a global conservation law for energy in general relativity.
..

I think that the answer is in the Noether's theorem.
That question appears in the early stage after Einstein published the GR theory.

https://en.wikipedia.org/wiki/Noether's_theorem​

 

[PeterDonis]

I think that the answer is in the Noether's theorem.

The fact that there is a global conserved energy in a stationary spacetime is a consequence of Noether's theorem, yes. The Komar energy is the Noether charge associated with the time translation symmetry.

 

[JimWhoKnew]

Notice that, first, there is no factor of $\sqrt{-g}$ anywhere,

Wald says that $dV$ in equation (11.2.10) represents the "natural volume element" on $\Sigma$ . His definition of the natural volume element includes $\sqrt{-g}$ . See note below equation (3.4.6) and appendix B.

 

[PeterDonis]

Wald says that $dV$ in equation (11.2.10) represents the "natural volume element" on $\Sigma$ . His definition of the natural volume element includes $\sqrt{-g}$ . See note below equation (3.4.6) and appendix B.

Yes. And that means he is treating $\sqrt{-g}$ as part of the volume element, not as part of the integrand. In other words, he's doing what I said in post #19 that most other textbooks do--which is different from what Dirac does.

 

[Demystifier]

Thus, I am tempted to conclude that there is a global conservation law for energy in general relativity.

Of course there is, it is well known that one can construct an energy-momentum pseudo-tensor that leads to a globally conserved notion of energy. However, the problem with this is that it is a pseudo-tensor, not a true tensor. This means that it is not general-covariant, i.e., not independent on the choice of coordinates.

 

[martinbn]

Of course there is, it is well known that one can construct an energy-momentum pseudo-tensor that leads to a globally conserved notion of energy. However, the problem with this is that it is a pseudo-tensor, not a true tensor. This means that it is not general-covariant, i.e., not independent on the choice of coordinates.

Hm, you start with "of course there is", but then you say it is not indipendent of coordinates. Isn't that self contradictoty? In general there are no global coordinates, so how do you even define this "energy" if it depends on coordinates!

 

[Demystifier]

Hm, you start with "of course there is", but then you say it is not indipendent of coordinates. Isn't that self contradictoty? In general there are no global coordinates, so how do you even define this "energy" if it depends on coordinates!

By first fixing coordinates. For example, if you need to define conserved energy-momentum of gravitational waves, then you work with linearized gravity and in this approximation GR is similar to a gauge theory like electrodynamics, so fixing coordinates becomes fixation of a gauge.