From Noether's Theorem, is it true that the law of conservation of energy can be proved?

Question

So what I understand is that the law of conservation of energy, likes Newton's law of motion, can't be proved. However, by Noether's Theorem, if there is a time symmetry, the energy is conserved. It seems to me that we have actually "proven" the law of conservation of energy, is it true?

Answer 1

Sure, you've "proven" the conservation of energy - under the even stronger assumptions that the universe follows suitable Lagrangian dynamics and that those dynamics have a continuous time-translation symmetry. If you really feel those assumptions are easier to swallow, then yes, you've gone forward.

However, there are situations where there is no guarantee that there is time-translation invariance, such as general settings in GR, and there are situations where it is explicitly broken, like several standard cosmological models, so all you've really done is shift the weight of truth from one solid-but-not-quite-solid-enough pillar to another solid-but-not-quite-solid-enough pillar.