# On the Manifestly Covariant Juttner Distribution and Equipartition Theorem

## TL;DR

This paper derives a manifestly covariant form of the Juttner distribution and equipartition theorem, establishing invariant temperature in relativistic kinetic theory using a new approach with Cartesian coordinates in momentum space.

## Contribution

It presents a new derivation of the covariant Juttner distribution and equipartition theorem, emphasizing invariant temperature from manifest covariance and using Cartesian coordinates in momentum space.

## Key findings

- Derived a covariant form of Juttner distribution.
- Established invariant temperature from covariance principles.
- Proposed a new covariant equipartition theorem.

## Abstract

The relativistic equilibrium velocity distribution plays a key role in describing several high-energy and astrophysical effects. Recently, computer simulations favored Juttner's as the relativistic generalization of Maxwell's distribution for d=1,2,3 spatial dimensions and pointed to an invariant temperature. In this work we argue an invariant temperature naturally follows from manifest covariance. We present a new derivation of the manifestly covariant Juttner's distribution and Equipartition Theorem. The standard procedure to get the equilibrium distribution as a solution of the relativistic Boltzmann's equation is here adopted. However, contrary to previous analysis, we use cartesian coordinates in d+1 momentum space, with d spatial components. The use of the multiplication theorem of Bessel functions turns crucial to regain the known invariant form of Juttner's distribution. Since equilibrium kinetic theory results should agree with thermodynamics in the comoving frame to the gas the covariant pseudo-norm of a vector entering the distribution can be identified with the reciprocal of temperature in such comoving frame. Then by combining the covariant statistical moments of Juttner's distribution a novel form of the Equipartition Theorem is advanced which also accommodates the invariant comoving temperature and it contains, as a particular case, a previous not manifestly covariant form.