The discussion centers on the concept of a global energy conservation law in general relativity, specifically addressing the energy-momentum tensor ##T^{\mu\nu}## and its components. It is established that while the integral of ##T^0_{\,0} \sqrt{-g}## over a large 4D spacetime region may appear to suggest a conservation law, this is misleading as it only rediscover the Komar energy, which is valid solely in stationary spacetimes. The conversation highlights the distinction between tensor densities and tensors, emphasizing that ##T^{00} \sqrt{-g}## and ##T^0_{\,0} \sqrt{-g}## are not equivalent representations of energy density. Furthermore, the integration of tensor densities over finite volumes raises concerns about the validity of such operations in curved spacetime.
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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.
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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!