Temperature in relativistic fluids

For static matter in a gravitational field, different conventions for equilibrium local temperature exist in the classic physics literature. We illustrate the difference between two popular conventions using black-body radiation in a spherically symmetric gravitational potential. Equilibrium temperatures defined by the “Landau frame” or “Eckart frame” prescriptions most commonly used in relativistic fluid dynamics do not satisfy the statistical-mechanical relation $1\mathrm{/}T=dS\mathrm{/}dE$.