Abstract: The goal of this course is to derive from kinetic theory a relativistic dissipative fluid theory on an arbitrary curved background spacetime. To this purpose, we start with a brief introduction to the relativistic Boltzmann equation for a simple gas, which is an integro-differential equation for the one-particle distribution function on relativistic phase space. Next, the relativistic version of the H-theorem will be discussed, which establishes the existence of an entropy flux whose divergence is non-negative. This singles out equilibrium configurations which are characterized by having zero entropy production. The difference between local and global equilibrium will be explained. Then, we introduce formal series expansions to describe close-to-equilibrium configurations, and we derive the fluid theory resulting from considering the first-order approximation in this scheme. Finally, we show that this theory provides a physically sound description of dissipative fluids which may be considered to be a relativistic generalization of the Navier-Stokes equations. In particular, it will be shown that this theory is governed by hyperbolic evolution equations with causal propagation for which global equilibrium configurations in Minkowski spacetime are stable.

Comments: The material of this course is mainly based on the following articles (it is not necessary to understand them before taking the course, only basic knowledge from differential geometry and classical mechanics will be assumed)

An introduction to the relativistic kinetic theory on curved space times

R.O. Acuña-Cárdenas, C. Gabarrete, OS, Gen. Rel. Grav. 54, 3, 23 (2022), arXiv:2106.09235

The Chapman-Enskog approximation for a relativistic charged gas in the trace-fixed particle frame

C. Gabarrete, A.L. García-Perciance, OS, arXiv:2508,14251

Relativistic dissipative fluids in the trace-fixed particle frame

J.F. Salazar, A.L. García-Perciante, OS,

Phys.Rev.D 111 (2025) 8, L081501, arXiv:2412.03712

Phys.Rev.D 111 (2025) 8, 084024, arXiv: 2412.03713