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

TL;DR

This paper analyzes the poles and dispersion spectra of the relativistic kinetic equation, revealing a gradient structure that is crucial for maintaining causality in relativistic hydrodynamics.

Contribution

It introduces a collision kernel conserving energy-momentum and particle current, and demonstrates the importance of non-local temporal derivatives for causality.

Findings

Dispersion relations show systematic gradient structure in the long wavelength limit.

Non-local temporal derivatives are essential for preserving causality.

Gradient expansion aligns spatial and temporal gradients in the relaxation operator.

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.