  We have investigated the proof of the $H$ theorem within a manifestly
covariant approach by considering the relativistic statistical theory developed
in [Phy. Rev. E {\bf 66}, 056125, 2002; {\it ibid.} {\bf 72}, 036108 2005]. In
our analysis, however, we have not considered the so-called deformed
mathematics as did in the above reference. As it happens in the nonrelativistic
limit, the molecular chaos hypothesis is slightly extended within the
$\kappa$-formalism, and the second law of thermodynamics implies that the
$\kappa$ parameter lies on the interval [-1,1]. It is shown that the
collisional equilibrium states (null entropy source term) are described by a
$\kappa$ power law generalization of the exponential Juttner distribution,
e.g., $f(x,p)\propto (\sqrt{1+
\kappa^2\theta^2}+\kappa\theta)^{1/\kappa}\equiv\exp_\kappa\theta$, with
$\theta=\alpha(x)+\beta_\mu p^\mu$, where $\alpha(x)$ is a scalar, $\beta_\mu$
is a four-vector, and $p^\mu$ is the four-momentum. As a simple example, we
calculate the relativistic $\kappa$ power law for a dilute charged gas under
the action of an electromagnetic field $F^{\mu\nu}$. All standard results are
readly recovered in the particular limit $\kappa\to 0$.
