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In a video by Sabine Hossenfelder, it is mentioned that in an expanding universe, energy is not conserved. The reason for this is that there is no time translation symmetry, an experiment performed at one time will not necessarily give the same result as an otherwise identical performed at another time, so Noether's theorem will not apply.

She then goes on to state, that strictly speaking, that energy cannot be defined in an expanding universe, but if we ask what the energy would be if the universe was not expanding, and then see what happens to if it starts expanding, we would see that this energy is not conserved.

The example given, is for light which is redshifted as the universe expands. She asks where does its energy go as it redshifts? Then answers her own question by saying that "it is just not conserved."

Is this true? Or are there some subtitles that I have missed in the explanation? To clarify, I am talking about the total energy of the universe, not some particular system within it.

I received a notification asking me to edit my question explaining why it is different from the linked one. So I am placing my comment on the matter here:

The linked question asks if the total energy is zero, and the answers reflect this. 3 of 6 equate the stress-energy tensor of the universe to zero, but this is not what I had in mind when I asked the question. 1 of the answers talks about a particular region of space within the universe. 1 of the answers just restates what I wrote in my question, about the difficulty defining energy, and the last one simply states that the total energy in the universe is not zero, but doesn't talk about other constant values.

Furthermore, the single answer this question received is in contradiction with the linked questions answers. It claims that time translation symmetry holds for the whole universe in the context of GR, while answers on the linked question claim it does not.

user400188
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    Does this answer your question? Is the total energy of the universe zero? – Young Kindaichi Nov 01 '20 at 04:05
  • @YoungKindaichi unfortunately not. The linked question asks if the total energy is zero, and the answers reflect this. 3 of 6 equate the stress-energy tensor of the universe to zero, but this is not what I had in mind when I asked the question. 1 of the answers talks about a particular region of space within the universe. 1 of the answers just restates what I wrote in my question, about the difficulty defining energy, and the last one simply states that the total energy in the universe is not zero, but doesn't talk about other constant values. – user400188 Nov 01 '20 at 04:29

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in an expanding universe, energy is not conserved. The reason for this is that there is no time translation symmetry, an experiment performed at one time will not necessarily give the same result as an otherwise identical performed at another time, so Noether's theorem will not apply.

This is incorrect. Time translation symmetry is what leads to conservation of energy. That is, conservation of energy is a consequence of Noether's Theorem where the equations describing the dynamics do not change with respect to a time translation. Noether's theorem clearly applies. The equations from general relativity and cosmology give solutions on large scales and from considering different conditions and points in the expanding universe. I think the confusion arises when you assume that just because the equations give you different results from one place/time/conditions to another, then the equations themselves are incorrect and therefore inapplicable. This is not the case.

energy cannot be defined in an expanding universe, but if we ask what the energy would be if the universe was not expanding, and then see what happens to if it starts expanding, we would see that this energy is not conserved. The example given, is for light which is redshifted as the universe expands. She asks where does its energy go as it redshifts? Then answers her own question by saying that "it is just not conserved."

This is again not correct. While it is true that while photons travel throughout the expanding Universe, they experience stretched wavelengths and longer wavelengths imply a decreased energy. But this decrease in energy does not imply that the energy is lost. The energy must go somewhere else.

From general relativity, this energy (in the form of work) goes toward causing the Universe's expansion. That is, if we could reverse the expansion of the Universe and the universe contracts, that work will be done in reverse and energy will go right back into the photons.

joseph h
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  • I don't think time translation symmetry applies. Consider a man falling off a building. In a non-expanding universe, he will hit the ground at the same time regardless of whether he jumps today or tomorrow. However if the space between the top and bottom of the building expands with time, then he will take longer to hit the bottom if he jumps tomorrow. So for the same experiment, the equations of motion will be different. There are actually some explanations regarding this in the question Young Kinaichi linked. – user400188 Nov 01 '20 at 04:55
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    My point is that time translation symmetry results in conservation of energy. The set of all time translations on a given system form a Lie group. – joseph h Nov 01 '20 at 05:04
  • What does it add to the question though? Time translation symmetry isn't applicable here. – user400188 Nov 01 '20 at 05:09
  • Yes it would. I still don't see what this adds though. The time translation you are considering in your example is $t+0$. – user400188 Nov 01 '20 at 05:20
  • If we could measure the energy of the entire universe now, then let it expand for 1000 years. Would the total energy then be the same as now? And then suppose we rewound this expansion back to now, would the total energy be the same as the original measurement? – joseph h Nov 01 '20 at 05:25
  • No it's t + 1000. Then t - 1000. t, t + 1000, t - 1000 would yield the same total energy. – joseph h Nov 01 '20 at 05:27
  • You re-wrote your question. Originally you considered letting it expand for 1000 years, then rewound it 1000 years, then asked if the energy would be the same. To answer your new question however, if your answer is correct, then the energy would be the same because some of the constituent particles of the universe (lets say photons) did work on the rest of the particles in the universe (system as a whole) by making it expand. – user400188 Nov 01 '20 at 05:36
  • What makes me skeptical however is that, what makes your answer correct and the claims made on the page Young Kinaichi linked wrong? Each explanation is self consistent, but they are not mutually consistent. One states that time translation symmetry does not hold, and yours states that it does hold. – user400188 Nov 01 '20 at 05:40