What is the equation of state for a relativistic fluid/gas? Say we have a relativistic fluid/gas, as we have in some astrophyical systems.

Now let us write:


$e$ - energy density in the fluid's rest frame.
$P$ - pressure in the fluid's rest frame.
$n$ - number density in the fluid's rest frame.
$m$ - mass of the particles.


I know that for the non-relativistic case we have:

$$e=nmc^2+\frac{1}{\hat{\gamma}-1}P$$

where $\hat{\gamma}$ is the adiabatic index. $\hat{\gamma}=1+\frac{2}{f}$ for a gas with $f$ degrees of freedom.

For the ultra-relativstic case we have:

$$e=3P$$

My question is what is $e(P,n)$ for a relativstic case (which is the general case of the 2 limits shown above)? I would also like to know how to derive it.



Is the following way the correct way to do it ? :

The number density of particles is:
$$n=\int_{0}^{\infty} n_p(p) dp $$

The pressure is:
$$P=\int_{0}^{\infty} \frac{1}{3} p v(p) n_p(p) dp $$

The energy density is:
$$e=\int_{0}^{\infty} \epsilon(p) n_p(p) dp $$

where:

$$n_p(p)= (2s+1)\frac{1}{ e^{({\epsilon(p)-\mu})/{k_B T}}+(-1)^{2s+1} } \frac{4\pi p^2}{h^3}$$

Here $s$ is the spin of the particles, for electrons $s=\frac{1}{2}$.

$$ \epsilon(p)=(m^2c^4+p^2c^2)^{\frac{1}{2}} $$

$$ v(p)= \frac{d\epsilon}{dp}=\frac{p}{m}\left(1+\left(\frac{p}{mc}\right)^2\right)^{-\frac{1}{2}} $$

From calculating the three integrals above we can finally obtain $e(P,n)$.


Can anyone confirm this is the proper way to do it, or am I missing something here?
It seems as if those integrals cannot be solved analytically - is this
true?
Perhaps in this case there is no explicit formula for $e(P,n)$?
Your equations seem correct. Note, you can also get $v$ without having to differentiate, from $E=\gamma mc^2$ and $p=\gamma mv$. 


Here are some notes on relativistic fluids as related to stellar interiors, including derivation of these equations and their solutions in the non- and ultra-relavistic limits http://www.jb.man.ac.uk/~smao/starHtml/equationState.pdf


Correct, the integrals cannot be solved analytically (only in the two limits) so there is no general analytic formula