Gravito-thermal transports, Onsager reciprocal relation and gravitational Wiedemann-Franz law

Using the near-detailed-balance distribution function obtained in our recent work, we present a set of covariant gravito-thermal transport equations for neutral relativistic gases in a generic stationary spacetime. All relevant tensorial transport coefficients are worked out and are presented using some particular integration functions in $(\alpha,\zeta)$, where $\alpha = -\beta\mu$ and $\zeta =\beta m$ is the relativistic coldness, with $\beta$ being the inverse temperature and $\mu$ being the chemical potential. It is shown that the Onsager reciprocal relation holds in the gravito-thermal transport phenomena, and that the heat conductivity and the gravito-conductivity tensors are proportional to each other, with the coefficient of proportionality given by the product of the so-called Lorenz number with the temperature, thus proving a gravitational variant of the Wiedemann-Franz law. It is remarkable that, for strongly degenerate Fermi gases, the Lorenz number takes a universal constant value $L=\pi^2/3$, which extends the Wiedemann-Franz law into the Wiedemann-Franz-Lorenz law.