Matter Gaining Energy from Expanding Spacetime?

[Suekdccia]

TL;DR Summary

Matter gaining energy from expanding spacetime?

Sean Carroll has an article (https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/) where he explains that matter can gain energy from spacetime expansion.

At the end of the article, he says: In general relativity spacetime can give energy to matter, or absorb it from matter, so that the total energy simply isn’t conserved.

So, there are natural processes where there is a loss of energy like photons being redshifted from spacetime expansion…

But what about gaining energy? Is there any example of matter or radiation gaining more and more energy as spacetime expands as Carroll seems to suggest?

 

[Ibix]

I don't think he actually specifies expansion. The obvious case is in a closed universe that collapses again, where the redshift becomes a blueshift in the collapse phase.

 

[vanhees71]

Of course you can also say that the gravitational field exchanges energy with matter although this sounds less sensational ;-)).

 

[Ibix]

Well, I think Carroll's point is that it isn't easy to say what "the energy of the gravitational field" is in order to be able to say whether it's exchanging energy with anything or not. I gather you can come up with Hamiltonians that look like an energy of the gravitational field, which is where the "total energy of the universe is zero" argument comes from, but Carroll does not seem convinced. I don't understand enough to have an opinion of my own on the topic.

 

[vanhees71]

That's of course true. There is no generally covariant energy-momentum tensor that obeys a local conservation law for the gravitational field in GR.

The best you can come up with, afaik, are socalled "pseudo-tensors" a la Landau and Lifhitz, which however is also difficult to properly interpret.

 

[PeterDonis]

Is there any example of matter or radiation gaining more and more energy as spacetime expands as Carroll seems to suggest?

No. As @Ibix has pointed out, using the heuristic viewpoint that Carroll gives, one would expect matter and radiation to lose energy to "gravity" in an expanding universe, and matter and radiation to gain energy from "gravity" in a collapsing universe.

 

[vanhees71]

The most simple example is the cosmic microwave background radiation. Today we observe it as a almost perfect Planck radiation at the temperature of about 2.725 K. Its origin is the radiation which was in thermal equilibrium with the charged medium in the early universe until around 380000 years after the big bang, when neutral atoms formed and the radiation decoupled, when the matter was at the Mott transition temperature of about 3000K. Since then the em. waves were free in the expanding universe and due to the expansion the wave-lengths stretched by the red-shift factor $(1+z) \simeq 1000$. The same scaling holds for the corresponding temperature, because the em. field is a massless field and thus the original Planck radiation spectrum always stays a Planck radiation spectrum with the correspondingly down-scaled temperature.

 

[PAllen]

That's of course true. There is no generally covariant energy-momentum tensor that obeys a local conservation law for the gravitational field in GR.

The best you can come up with, afaik, are socalled "pseudo-tensors" a la Landau and Lifhitz, which however is also difficult to properly interpret.

It has been proposed by Nakanishi that these pseudo-tensors have reasonable interpretation in special coordinates: harmonic coordinates (which they call de Donder).

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1006.9839&rep=rep1&type=pdf

[note: I disagree with a number of opinions in this article, but find the results on energy/momentum pseudo-tensor having plausible interpretation in special coordinates quite interesting and reasonable].