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I recently heard about energy "not being conserved" in General Relativity and i had doubts. Is this true, cause the following questions deeply worry me if that is the case?:

  • Wasn't the whole point of introducing the concept of energy in Classical Physics that it was conserved?
  • And isn't it true that the theory of Quantum Mechanics, despite everything else, has a form of energy "that is conserved"? Why wasn't the expression for energy "tweaked a bit" in GR so that a conserved quantity is reached at? [Isn't this how a conserved energy was successfully introduced in QM].
  • So, what about Noether's theorem? If a physical theory of dynamics stays the same for all time, there must be a quantity called energy conserved in that theory, ryt? [Or, did i get it wrong]. Then, does GR predict its own in-applicability at different times? [I don't even know how that works].

Note: I have no prior knowledge of the GR theory, except rumors that a violation of the principle of energy conservation is inbuilt in it.

Edit to prove my question is not a duplicate: This question addresses the same problem essentially, but asks only if energy can be lost or gained from the universe as a whole. But I am asking that and the various implications of such a violation and therefore the answers there do not answer my question [at least adequately].

PhyEnthusiast
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    Possible duplicate of Is the law of conservation of energy still valid? – Rumplestillskin May 16 '17 at 10:32
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    Note that the issue is not just lack of energy-conservation; the deeper issue is that there is no satisfactory definition of gravitational energy in the first place. Possible duplicates: http://physics.stackexchange.com/q/2838/2451 , http://physics.stackexchange.com/q/10309/2451 , https://physics.stackexchange.com/q/2597/2451 , https://physics.stackexchange.com/q/7060/2451 , https://physics.stackexchange.com/q/35431/2451 , https://physics.stackexchange.com/q/109532/2451 , https://physics.stackexchange.com/q/127397/2451 , https://physics.stackexchange.com/q/175186/2451 and links therein. – Qmechanic May 16 '17 at 10:53
  • GR violates Noether's Theorem, yes, there's no time translation symmetry. But that's on very large scales only. Example: the energy of the CMWB photons decreases over time. – Jens May 16 '17 at 10:59
  • "Wasn't the whole point of introducing the concept of energy in Classical Physics that it was conserved? " where does that come from? Energy may or may not be conserved: it needs not to, in general. – gented May 16 '17 at 11:35
  • Energy is a concept created by man its utility majorly in the fact that it is conserved. – PhyEnthusiast May 16 '17 at 16:54
  • Still not sure how the question is a duplicate of any of the mentioned links – PhyEnthusiast May 16 '17 at 16:56
  • Related: https://physics.meta.stackexchange.com/q/9863/2451 – Qmechanic May 17 '17 at 15:00

1 Answers1

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Noether's theorem states that:

every differentiable symmetry of the action of a physical system has a corresponding conservation law.

The action in GR is the Einstein Hilbert action plus any contributions from matter:

$$ S = \frac{c^4}{16\pi G}\int R \sqrt{-g} \, d^4x + L_\text{matter} $$

Conservation of energy requires that the action be unchanged by a translation in time, but in general if the metric $g$ is a function of time then the action will be changed by a translation in time and therefore energy won't be conserved.

In Newtonian mechanics, or indeed special relativity, the metric is constant so this problem doesn't occur. That's why energy is conserved in classical mechanics and quantum field theory.

For completeness we should note that energy can be a slippery concept in GR and by no means everyone agrees that energy is violated in e.g. an expanding universe. It depends on what you count when calculating the total energy. Phil Gibbs is the main dissenter.

John Rennie
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  • what would be the "tiniest" change to the action integrand $R\sqrt{-g}$ that would make it time invariant and thus keep energy conserved formally? – hyportnex May 16 '17 at 15:20
  • @hyportnex no tiny change could archive that. In general, energy is not conserved - that's something we must learn to live with. – AccidentalFourierTransform May 16 '17 at 15:33
  • "but in general if the metric g is a function of time then..." Is there a distinction btw implicit time dependent and explicit time dependent here? In classical mechanics, potential energy can have implicit time dependency (position of particles determine the potential energy) and still have conserved energy. The metric g in GR comes from distribution of matter and energy, no? Is it similar to an implicit time dependent that doesn't break energy conversation? – Bohan Xu Dec 01 '23 at 14:27