Is energy conserved during the formation of local systems?

[Suekdccia]

TL;DR Summary

Is energy conserved during the formation of local systems?

I found an old article (https://journals.aps.org/pr/abstract/10.1103/PhysRev.137.B1379) which talks about conservation of energy in an expanding space. Apparently, the author found that energy is conserved at local scales (like the motion of planets in our solar system) as one would expect, but he also concluded that at cosmological scales, during the formation of local systems (like galaxy clusters), there are deviations from the law of conservation of energy.

The paper also cites this one from David Layzer (https://adsabs.harvard.edu/full/1963ApJ...138..174L) which apparently reached a similar conclusion (see section IV).

My questions are: Is energy not conserved during the formation of such structures due to spacetime expansion? And if there are indeed deviations from the laws of conservation of energy, can energy be "created" in such situations? And if it can, in what form (thermal, electromagnetic, kinetic...)?

 

[PAllen]

I think the issue is more fundamental. To discuss conservation of energy at some scale of system, you first need an accepted definition of energy. There simply is none at large scales in general GR solutions, and, in particular there is no accepted definition of energy at large scales in realistic cosmologies. This means you cannot even pose the question of conservation.

Some authors claiming explicit violation of conservation of energy (rather than the ill posed nature of the question) choose to use a definition of energy that is known to be wrong for the special cases in GR where conservation of energy is well posed. This makes no sense to me.

 

[PeterDonis]

The paper also cites this one from David Layzer (https://adsabs.harvard.edu/full/1963ApJ...138..174L) which apparently reached a similar conclusion (see section IV).

Section IV of this paper is using "velocity" to mean what cosmologists call "peculiar velocity", i.e., velocity relative to comoving observers. But in an expanding universe, comoving observers are not at rest relative to each other, so this "velocity" is not the same as "velocity relative to the center of mass frame of a particular isolated system"; yet the paper's conclusion that you describe rests on the (invalid) assumption that it is.

Also, the equations in Section IV of the paper assume a universe that is filled with matter of a uniform density everywhere (which is the standard assumption used to derive the Friedmann equations). But of course that is not actually the case; and when trying to model the development of an isolated system separated by vacuum from other isolated systems, obviously that assumption cannot be taken to be valid and you cannot use equations derived from it. Yet that is what this paper does.

Using correct equations for an isolated system surrounded by vacuum avoids these issues and gives the result that the usual conservation of energy applies just fine. The only underlying assumption that needs to be made in this case is that, to a good enough approximation, the matter in the rest of the universe, outside the isolated system, is distributed around the isolated system in a spherically symmetric fashion. If that is the case, then the shell theorem says that the spacetime geometry around the isolated system is unaffected by the matter in the rest of the universe, which again leads to the conclusion that the usual conservation of energy applies to the isolated system.