Conservation of Energy in General Relativity

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].

Answer

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.