Poles from the conserved kinetic equation : The emerging gradient structure and causality riddle of relativistic hydrodynamics

TL;DR

This work addresses causality and conservation issues in relativistic kinetic theory by introducing a collision kernel that conserves both the particle current $J^{\mu}$ and the energy-momentum tensor $T^{\mu\nu}$ by construction. In Fourier space, the linearized kinetic equation yields poles with a logarithmic dispersion structure that, in the long-wavelength limit, organize into an infinite gradient expansion where each higher spatial derivative is balanced by nonlocal temporal derivatives through the relaxation operator $(1+i\omega\tau_R)$. This all-order temporal nonlocality is shown to be essential for maintaining causality when truncating the gradient expansion, and the analysis connects the microscopic kernel to MIS-like hydrodynamic modes while revealing the role of non-hydrodynamic degrees of freedom. The results highlight how microscopic conservation laws and nonlocal relaxation shape the emergent hydrodynamics, with potential implications for constructing causal, frame-independent relativistic fluid theories from kinetic theory.

Abstract

In this work, the poles and the resulting dispersion spectra from the relativistic kinetic equation have been analyzed with the help of a proposed collision kernel that conserves both the energy-momentum tensor and particle current by construction. The dispersion relations, which originally come out in the form of logarithmic divergences, in the long wavelength limit exhibit the systematic gradient structure of the relativistic hydrodynamics. The key result is that, in the derivative expansion series, the spatial gradients appear in perfect unison with the temporal gradients in the non-local relaxation operator like forms. It is then shown that this dispersion structure, including non-local temporal derivatives, is essential for the preservation of causality of the theory truncated at any desired order.