The relativistic equilibrium velocity distribution plays a key role in describing several high-energy and astrophysical effects. Recently, computer simulations favored Juttners as the relativistic generalization of Maxwells 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 Juttners distribution and Equipartition Theorem. The standard procedure to get the equilibrium distribution as a solution of the relativistic Boltzmanns 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 Juttners 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 Juttners 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.
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