Sean Carroll writes that in GR “spacetime can give energy to matter, or absorb it from matter, so that the total energy simply isn’t conserved”.
He hopefully means “space” not “space-time”, because space-time includes time and has no further time to do anything, let alone absorb stuff. Let us nevertheless do a sympathetic reading and firstly show what he may mean with a simple example:
Imagine electromagnetic radiation (light) inside a box of volume V = L3. Now remove one of the walls and let the radiation spread out into a larger volume, say V’ = 8V. The energy U of the radiation is conserved: U’ = U. However, if you let the radiation pressure move a plunger instead so that at some point the volume is eight times larger, half the energy goes into the plunger: U’ = U/2.
Just remember: plunger = taking out half the energy when expanding volume 8 fold.
Straw-man: “Why half the energy?”
Sock-puppet: “Consider a column with volume V = L2 x with a plunger of surface L2 starting at x = L with energy U(L). Like in the picture, but note that Sascha was too lazy to draw it three dimensional. Moving the plunger slowly reduces the energy via dU = - F dx = - P L2 dx = - (ρ/3) L2 dx = - [U/(3 L2x)] L2 dx = - [U/(3x)] dx. Thus U = Constant * x-1/3 and hence U = U(L) (L/x)1/3.
At V’ = 8V holds U(8L) = U(L) (1/8)1/3 = U(L)/2.”
Sock, you scare away readers! Now to cosmology: Consider a cube of volume V = R3 expanding to a cube twice the length, resulting in the volume being again eight times more: V’ = (2R)3 = 8V. Radiation energy density ρ = U/V is proportional to 1/R4, where R-3 is for the volume and the further factor R-1 takes care of the redshift, the stretching of the wavelength of the light.
The energy of the radiation is initially U = ρ R3, but after expanding the universe, it is U’ = ρ’ (2R)3. However, because of the redshift, it also holds that ρ’ = ρ/24.
And so we get U’ = ρ (2R)3/24 = U/2, i.e. the expansion takes out half the energy again.
In cosmology, space growing is the plunger, the “space-plunger”.
I will rip this argument above (and Sean's "solution") apart the next time. Here a more recent post on that energy is not a substance and how to most easily argue that fact.
