Where is the work done of adiabatically expanding universe going into?

Question

According to this Wikipedia article, the universe is expanding adiabatically.

However, any system that is expanding adiabatically must be losing energy $\left(\mathrm{d}E = -p\,\mathrm{d}V\right) .$

Does not this violate conservation of energy, as the internal energy of the universe is decreasing?

For any finite system expanding adiabatically, the conservation of energy is not violated as the system transfers its internal energy into the rest of the universe through the form of work.

However, when the whole universe is taken as the system, assuming there is nothing outside it (I am not very sure about this statement), where is the work done going into?

Note: A constant energy free expansion model will imply energy density times the volume is constant, so $T^4 V = \text{constant}$, or $T \propto a^{-\frac{3}{4}}$.

But the adiabatic expansion model implies $TV^{\frac{1}{3}} = \text{constant}$, or $T \propto \frac{1}{a}$, where $a$ is the scale factor.

Please note that I am looking for an explanation that can be understood without much knowledge in General Relativity.

Answer 1

Well, there are a few ways to answer this question.

Suppose you draw an imaginary (comoving) box around some region of spacetime. The volume of this box expands over time, so you can think of the contents of this box as doing work on the rest of the universe, outside of the box.

Now, you might say this isn't satisfying if we want to consider the universe as a whole. In that case, you can think of the energy as all going into gravitational potential energy. This is a perfectly good picture in the Newtonian limit.

However, it turns out that in general relativity, it's very hard to make the notation of "gravitational potential energy" precise. For example, you can't talk about the gravitational potential energy density at a point, because you can always go into a freely falling frame there, where the observed gravitational field is zero. For this reason, relativity textbooks generally say that the gravitational potential energy is not defined at all; instead energy (defined as not including this extra ill-defined piece) simply isn't conserved in general relativity. The energy doesn't "go" anywhere, it just vanishes.

If you think this is unacceptable, remember that the only reason we elevated conservation of energy to an important principle in the 19th century was that it was observed to work in everyday situations. We never tested it in exotic situations like those with curved spacetime, so there's no reason to expect the principle to continue to hold up. At a deeper level, Noether's theorem tells us that energy conservation is related to time translation invariance, and we don't have that in an expanding universe.

Answer 2

Just a small addition. It is crucial here that the density of the vacuum energy is constant over time. This means that if a given volume of vacuum energy expands, its energy density remains constant, in contrast to what happens if gas expands adiabatically instead. This means that the work done to adiabatically expand the universe is just sufficient to keep the vacuum energy density constant.

Answer 3

The relevant form of adiabatic expansion comprises two components. One is the expanding gas and the second is the system that opposes the expansion, without which the expansion would not perform $pdV$ work. The relation $dE=-pdV$, describes the transfer of energy to the second system. In the case of the universe, gravity opposes the expansion of the photon gas. The expanding gas of photons loses energy while the universe gains the work energy.

The previous answer describes an expanding box of photons doing work on the rest of the universe which suggests that the pdV energy might be transferred out of the box. However, this is forbidden because the universe is isotropic and homogeneous. If there is any net energy transfer out of an expanding volume, by the homogeneity assumption there must be equal and opposite energy transfer out of an immediately adjacent volume, therefore there cannot be any net energy transfer out of any volume of the universe. The pdV energy must remain within the expanding volume.

The universe is governed by the Friedmann equations. The $pdV$ energy must be found in one of the parameters of the Friedmann equations, which are the energy density and pressure of the components, the curvature parameter, and the cosmological constant. Since experiments show that the curvature and the cosmological constant are negligible at the time of photon decoupling, the only possible manifestation of the pdV energy is as an energy density.

The relation $dE=-pdV$ can be derived directly from the Friedmann equations. Hence neglecting the pdV energy is in contradiction to the Friedmann equations. The conclusion is that the lost photon energy must be included as an additional energy density in any model of the universe.