I Energy conservation in an expanding universe

[elcaro]

TL;DR Summary

In an universe with positive and constant energy density the total amount of energy seems not to be conserved. However, if one also considers the negative energy contained in the gravitaional potential, which can become arbitrarily negative, then total energy in the universe IS conserved.

The total amount of energy is still a conserved quantity, even in an expanding universe based on a positive and constant energy density, and even under the rapid exponential expansion during inflation, total amount of energy is conserved. For how this works, see this lecture by Alan Guth, the father of inflationary cosmology.

 

[PeterDonis]

The total amount of energy is still a conserved quantity

Only if you define "the total amount of energy" to be a pseudotensor. But a pseudotensor is not an invariant; it depends on a particular choice of coordinates. The viewpoint of most physicists working in General Relativity is that all of the physics is contained in invariants, and coordinate-dependent quantities have no physical meaning.

For a different viewpoint than Guth's, see, for example, this article by Sean Carroll (which you will find referenced fairly frequently here on PF when this subject comes up):

https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[PeterDonis]

The total amount of energy is still a conserved quantity

Note, btw, that there is another possible interpretation of this statement besides the pseudotensor one, which might be being implicitly used (at least in part) in Slide 6 of the lecture slides corresponding to the video you gave:

https://ocw.mit.edu/courses/physics...all-2013/lecture-slides/MIT8_286F13_lec01.pdf

At the bottom of this slide, it says: "The TOTAL ENERGY of the universe may very well be zero." There is a quantity called the "Hamiltonian constraint" (IIRC) in GR, which is in fact identically zero for any spacetime that is a solution of the Einstein Field Equation. If we interpret this Hamiltonian as "the total energy" of the solution, then this means the total energy is zero.

However, in this formulation, there is no split in this "total energy" between "positive energy of material" and "negative potential energy in the gravitational field"; the total energy is just zero. To get the split described in the lecture between the "positive" and "negative" parts of the total energy, you need to pick a pseudotensor, which has the problem I described in my last post.