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 non-viscous 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-dS, Kerr and Kerr-dS black holes, respectively. Consequently, the chemical potential profile is also discussed.