A Lorentz invariant velocity distribution for a relativistic gas

Abstract

We derive a Lorentz invariant distribution of velocities for a relativistic gas. Our derivation is based on three pillars: the special theory of relativity, the central limit theorem and the Lobachevskyian structure of the velocity space of the theory. The rapidity variable plays a crucial role in our results. For $v^2/c^2\ll 1$ and $1/\beta =kT/2m_0{c}^2\ll 1$ the distribution tends to the Maxwell-Boltzmann distribution. The mean $⟨v^2⟩$ evaluated with the Lorentz invariant distribution is always smaller than the Maxwell-Boltzmann mean and is bounded by $⟨v^2⟩/c^2=1$. This implies that for a given $⟨v^2⟩$ the temperature is larger than the temperature estimated using the Maxwell-Boltzmann distribution. For temperatures of the order of $T\sim 10^{12}K$ and $T\sim 10^{8}K$ the difference is of the order of $10\%$, respectively for particles with the hydrogen and the electron rest masses.

Pdftex, 12 pages, 3 figures