# Thermodynamics in Friedman World Models

• cepheid
In summary, the conversation discusses the first law of thermodynamics in the context of the expanding universe. Specifically, it examines a derivation by Malcolm S. Longair in section 7.1 of Galaxy Formation, where he shows that the dynamical equations for the scale factor incorporate the first law in its full, relativistic sense. This is done by using the first law to derive a relationship between energy density and scale factor, and then showing that this relation is a part of the dynamical equations. The conversation also explores the interpretation of this derivation and its implications, including how it works with the uniform expansion and the role of the term p/c^2. It is concluded that the derivation, although it may not make complete sense, provides a
cepheid
Staff Emeritus
Gold Member
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 said:
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.

Does anybody have any insights on this issue?

## What is a Friedman World Model?

A Friedman World Model is a mathematical model used to describe the expansion of the universe in terms of the cosmological constant and the curvature of space. It was proposed by physicist Alexander Friedman in the early 1920s and has been used to study the evolution of the universe in the context of thermodynamics.

## What is the cosmological constant in Friedman World Models?

The cosmological constant is a term in the equations of general relativity that represents the energy density of the vacuum in the universe. It is denoted by the Greek letter lambda (Λ) and plays a key role in Friedman World Models by determining the rate of expansion of the universe.

## How does thermodynamics apply to Friedman World Models?

Thermodynamics is the study of energy and its transformations, and it plays a crucial role in understanding the behavior of the universe in Friedman World Models. These models use thermodynamic principles to describe the evolution of the universe and its energy distribution over time.

## What is the first law of thermodynamics in Friedman World Models?

The first law of thermodynamics, also known as the law of conservation of energy, states that energy cannot be created or destroyed, only transformed from one form to another. In the context of Friedman World Models, this law helps to explain how the energy in the universe is distributed and how it changes over time.

## What is the second law of thermodynamics in Friedman World Models?

The second law of thermodynamics states that in any spontaneous process, the total entropy (measure of disorder) of a closed system always increases. In Friedman World Models, this law helps to explain how the universe tends towards a state of maximum entropy, also known as the heat death of the universe.

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