General relativistic heat flow from first order hydrodynamics

Following the recently proposed stable and causal first order relativistic hydrodynamics by Bemfica, Disconzi, and Noronha, we find the heat flow equation in the presence of gravity for a nonviscous fluid, which suffers heat dissipation. The derivation is confined to static and stationary backgrounds. We find that in the presence of gravity, the heat flux times a redshift factor is conserved. Then for radial heat flow, the temperature profiles are obtained from the heat equation when the gravity is sourced by Schwarzschild, Schwarzschild-de Sitter, Kerr, and Kerr-de Sitter, respectively. Consequently, the chemical potential profile is also discussed.