Thermodynamics in Friedman World Models

[cepheid]

Hi there,

I have a question regarding thermodynamics in the expanding universe. I'm specifically looking at section 7.1 of Galaxy Formation by Malcolm S. Longair. In this section, he sets out to show that the dynamical equations for the scale factor automatically incorporate the first law of thermodynamics in its "full, relativistic sense." He does so by using the first law to derive a relationship that expresses the rate of change of energy density (or inertial mass density) with scale factor, and then showing that this relation is part of the reason why the dynamical equations have the form that they do. He starts off with:

$$dU = -pdV$$​

This is a statement that the change in internal energy is equal to the work done on the surroundings. Longair then went on to state that the energy density, which he calls $\epsilon_{\textrm{tot}}$, is a sum of all possible contributions (rest energy of particles, thermal kinetic energy etc.). I.e. it can be thought of as $\epsilon_{\textrm{tot}} = \sum_i \epsilon_i$, a sum of all possible terms that can contribute to the energy density. He therefore states that $U = \epsilon_{\textrm{tot}}V$, and so, differentiating both sides with respect to a (the scale factor):

$$\frac{dU}{da} = -p\frac{dV}{da}$$

$$\frac{d}{da}(\epsilon_{\textrm{tot}}V) = -p\frac{dV}{da}$$

$$V\frac{d\epsilon_{\textrm{tot}}}{da} + \epsilon_{\textrm{tot}}\frac{dV}{da} = -p\frac{dV}{da}$$​

Longair then uses the fact that the volume is proportional to the cube of the scale factor to write this as:

$$a^3 \frac{d\epsilon_{\textrm{tot}}}{da} = -3a^2 (\epsilon_{\textrm{tot}} + p)$$

$$\frac{d\epsilon_{\textrm{tot}}}{da} = -3\frac{(\epsilon_{\textrm{tot}} + p)}{a}$$​

My question for this thread is simple: how am I supposed to interpret this derivation? When my prof was talking about it, he spoke of U as the total energy "of the universe" and V as the "volume of the universe." But the picture of a system with some internal energy expanding and doing work on its surroundings sort of breaks down if the system is the whole universe! There ARE no surroundings, and the universe may not even have a finite volume.

Then I got to thinking about it some more, and I realized: this is an equation in differential form. That means it is applicable at a specific point in space. Therefore, I reinterpreted V, thinking of it as any finite volume in the universe. If it is true that the energy density in any finite volume varies like that with a, then it must be true for the energy density of the universe as a whole, right? Is that the right way to interpret this derivation? If so, I have some further questions that stem from it. If not, then how should I interpret it?

 

[cepheid]

If it is true that the energy density in any finite volume varies like that with a, then it must be true for the energy density of the universe as a whole, right?

Upon further reflection, I'm wondering how that works with the uniform expansion. If every finite volume expands, and in so doing, does "work" on its surroundings, where does that work go? I mean the "surroundings" of a given finite volume are, in fact, the other finite volumes in question! I'm beginning to think that the answer to my question might be, "you can't really understand what's happening to the energy density in an expanding universe without invoking GR, and this classical derivation has been chosen because it happens to give the right answer and looks superficially plausible, even though it really doesn't make any sense." Is that the case? Longair did later go on to say that the term p/c² (which is missing if you derive the Friedman equation using Newtonian analogues) doesn't have a conventional interpretation, but can be thought of as a relativistic correction to the inertial mass density, and that together they make up the active gravitational mass density.

 

[cepheid]

Does anybody have any insights on this issue?