Why is energy density divided by c² equal to mass density?

[Leonardo Machado]

Hi everyone!

I'm currently strudying some astrophysical equation of states, some stuff about Fermi's gas and I'm kinda confused about the relation between the energy density and the mass density,

$$
\frac{\epsilon}{c^2}=\rho.
$$

I don't get why they do not use whole

$$
\epsilon=\frac{E}{V}=\frac{\sqrt{p^2c^2+m^2c^4}}{V}
$$

could someoen explain me this ?

Thanks!

 

[Ibix]

I would guess they are working in coordinates where the gas is at rest on average, where $p=0$ and the kinetic energy of the atoms (i.e., the heat) contributes to $\epsilon$.

That's just a guess, though. Do you have a reference or link?

 

[Herman Trivilino]

I agree with @Ibix . For a gas it's pretty common to study it in its center-of-momentum frame, where by definition the sum of the particles' momenta is zero.

From there it's easy to see that the mass of the system has to change in direct proportion to the rest energy.