# Relativistic heat capacity

1. Jan 28, 2014

### Sunfire

Hello All,

would anyone know an expression for the heat capacity of an ideal gas, moving with relativistic speeds. For non-relativistic gas, the heat capacity is a constant, proportional to R (the universal gas const)

Many thanks.

2. Jan 28, 2014

### Staff: Mentor

3. Jan 28, 2014

### Sunfire

I saw a homework guide on the page of Rochester I.T. I believe, they derived the heat capacity to be = d x Nk where d is the dimension of space. That will make it twice the value for nonrelativistic gases.

I was wondering, is this mainstream, because it seemed to me that relativistic thermodynamics is under debate (e.g. I am unsure of the mainstream transformation expression for the temperature)

4. Jan 28, 2014

### Sunfire

Then, is this true:

$c'_v$ = $\frac{c_v}{\sqrt{1-\frac{3}{4}\frac{v^2}{c^2}}}$

I mean, it would be lovely to have $c'_v$ somehow expressed through $c_v$ and $\gamma$. Is such thing possible?

5. Jan 28, 2014

### Staff: Mentor

What is v, and where does the formula come from?

6. Jan 28, 2014

### Sunfire

$c_v$ is the heat capacity at constant volume. Sorry for the mix-up, "v" under the root is the velocity.

The formula is a fib, if v approaches c, then the observed heat capacity approaches twice the value of the heat capacity of non-relativistic gas.

What I wanted to ask is, is there an expression for the heat capacity which bridges the non-relativistic and the relativistic limits. An example for an observable a is

a' = γa

Thus, when v << c, a'=a
when v --> c , a' --> ∞

Is there a similar expression for the heat capacity of ideal gas?

7. Jan 28, 2014

### pervect

Staff Emeritus
MTW's textbook approach (which I believe is standard) is to only use the temperature in the local rest frame of the gas. I've seen it in other texts too, though I couldn't name them.

If you pretend temperature is only defined in the rest frame, you don't need to know how it transforms :-). As far as heat capacity goes, if you make this assumption about temperature, heat capacity would also only be defined in the rest frame.

8. Jan 28, 2014

### Sunfire

I understand, it is just that in this particular case I need to express both the temperature and the heat capacity of an ideal gas in the stationary frame, while the gas moves with (relativistic) speed with respect to the stationary frame. For this reason I was trying to see if there is such an expression for T and c (heat capacity) which will contain their rest values and quite possibly,

$\gamma$=$\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

I would need the expressions of T and heat capacity as they appear to the stationary observer.

9. Jan 28, 2014

### Staff: Mentor

Ah sorry, I misunderstood your initial question. My reply was for a gas where the particles have relativistic speed, not a "cold" gas that is moving as an object.

The temperature of moving objects has multiple possible definitions, and they all lead to various issues. See this paper, for example.

10. Jan 28, 2014

### pervect

Staff Emeritus
The approach the textbooks I have take is to use as a parameter the the entropy / per particle. Entropy per particle is computed using the methods of statistical mechanics, a function of the possible state distributions of the particle. Thus, entropy per particle is just a number.

Then you use the well known (read up on it if you aren't familiar with it ) number-flux four vector to describe how the density and flux of particles transform.

If you have some interaction between two parts of a system moving with respect to each other, you compute the effect on the entropy/unit particle in the rest frame of each sub-system, and use this information to reassemble the total entropy in the system from whatever view point you desire.

So you basically avoid dealing with the issue of how the usual thermodynamic quantites transform by considering them only in the rest frame. Then the only thing you need to transform is the entropy / particle wich is a pure number. (Well, you also need to transform particle density, which is part of a 4-vector, the number-flux 4-vector).

11. Jan 30, 2014

### Sunfire

Just to repeat: If $c_v$ is the heat capacity of a volume of an ideal gas at rest, then how does one express the heat capacity $c'_v$ of a volume of an ideal gas moving with relativistic velocity $v$?
I would expect the dependence to be $c'_v = f(c_v, \gamma)$ where
$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, c is the speed of light.