Energy conservation and universe expansion

[nomadreid]

I am sure this question comes up frequently in the forum, and I have read a number of answers to it in different places; namely, that since the expansion of the universe stretches the wavelengths and thereby lowers the energy of light, what happens to the principle of the conservation of Energy? From the answers I have read, it seems that the answer is that Energy is not conserved, as it doesn't obey Noether's Theorem, but that a combination of Energy and stress tensors is what ends up being conserved. However, I doubtful whether this is a good statement of the proper answer. Could someone give me, in relatively straightforward terms (no pun intended), the answer to this question?

 

[Dale]

I would start with this page:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

 

[nicksauce]

GR Answer: Since the FRW metric does not possesses a timelike Killing vector, there is no conserved quantity that we can call energy. However, GR automatically enforces local energy conservation, since $\nabla_{\mu}T^{\mu\nu}=0$, where T is the stress tensor, is always true.

Newtonian Answer: Photons lose energy as the universe expands, however, PdV work must be done to expand the universe. These two factors exactly cancel out.

 

[nomadreid]

Thank you, DaleSpam and nicksauce. I went to the link recommended by DaleSpam, and found the response of nicksauce quite helpful; I shall now pursue these leads. (After posting my original question, I also came across nicksauce's reply in previous posts in the Forum. I must admit that I should have looked through the Forum better before posting.)

This brings me to a related question. On many sites I read the explanation of the possibility of spacetime being flat locally and curved globally by the analogy of "the Earth appears flat when one is small and close to its surface, whereas it appears curved when one is further away or large enough." I find this analogy extremely suspect: after all, when one is close to its surface it will appear, with sufficient measuring instruments, as curved, even though only very lightly curved. From what I understand of curvature, the difference of curvature is not of quantity, but rather of quality. Does someone have a better explanation of the difference?

 

[Dale]

On many sites I read the explanation of the possibility of spacetime being flat locally and curved globally by the analogy of "the Earth appears flat when one is small and close to its surface, whereas it appears curved when one is further away or large enough." I find this analogy extremely suspect: after all, when one is close to its surface it will appear, with sufficient measuring instruments, as curved, even though only very lightly curved. From what I understand of curvature, the difference of curvature is not of quantity, but rather of quality. Does someone have a better explanation of the difference?

You are exactly correct and have instinctively identified the very heart of the matter. When you are talking about space being "locally flat" how "local" you have to go depends on the accuracy of your measuring instruments.

Experimentally, local flatness means that for any given measuring device with a finite precision, e, you can always find a sufficiently small region of spacetime around any given event such that the error introduced by the curvature will be less than e.

Mathematically, local flatness means that at any given event you can always transform to a coordinate system which approximates a flat coordinate system to first order and that any deviations are second order or higher.

 

[nomadreid]

Thank you, DaleSpam. These definitions are very useful.