Mathematical Formulation of Causal Propagation in Relativistic Ideal Fluids

We establish a rigorous kinetic-theoretical framework to analyze causal propagation in thermal transport phenomena within relativistic ideal fluids, building a more rigorous framework based on the kinetic theory of gases. Specifically, we provide a refined derivation of the wave equation governing thermal and density fluctuations, clarifying its hyperbolic nature and the associated characteristic propagation speeds. The analysis confirms that thermal fluctuations in a simple non-degenerate relativistic fluid satisfy a causal wave equation in the Euler regime, and it recovers the classical expression for the speed of sound in the non-relativistic limit. This work offers enhanced mathematical and physical insights, reinforcing the validity of the hyperbolic description and suggesting a foundation for future studies in dissipative relativistic hydrodynamics.