The discussion revolves around the concept of energy conservation in general relativity, particularly focusing on the implications of the energy-momentum tensor and its components in the context of stationary spacetimes. Participants explore the validity of a proposed global conservation law for energy, the nature of tensor densities, and the interpretations of various texts, including Dirac's work.
Participants express multiple competing views regarding the existence of a global conservation law for energy in general relativity, with no consensus reached on the interpretations of the energy-momentum tensor components or the validity of Dirac's arguments.
Limitations include the dependence on specific definitions of energy density and the coordinate dependence of tensor components, which remains unresolved in the discussion.
I think that the answer is in the Noether's theorem.Kostik said:...
Thus, I am tempted to conclude that there is a global conservation law for energy in general relativity.
..
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.Bosko said:I think that the answer is in the Noether's theorem.
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 said:Notice that, first, there is no factor of ##\sqrt{-g}## anywhere,
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.JimWhoKnew said: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.
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.Kostik said:Thus, I am tempted to conclude that there is a global conservation law for energy in general relativity.
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 said: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.
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.martinbn said: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!