Why do newer degrees of freedom appear in higher-order truncated hydrodynamic theory?

TL;DR

This paper explains why higher-order truncated relativistic hydrodynamic theories introduce new degrees of freedom, emphasizing the importance of all-order temporal derivatives for causality and the role of non-local relaxation operators.

Contribution

It provides an exact derivation showing how higher-order truncations in relativistic hydrodynamics necessitate additional degrees of freedom due to all-order temporal derivatives.

Findings

Higher-order spatial truncations require new degrees of freedom.

All-order temporal derivatives are essential for causality.

Non-local relaxation operators resum temporal derivatives.

Abstract

An exact derivation of relativistic hydrodynamics from an underlying microscopic theory has been shown to be an all-order theory. From the relativistic transport equation of kinetic theory, the full expressions of hydrodynamic viscous fluxes have been derived which turn out to include all orders of out-of-equilibrium derivative corrections. It has been shown, that for maintaining causality, it is imperative that the temporal derivatives must include all orders, which can be resummed in non-local, relaxation operator-like forms and finally `integrated in' introducing newer degrees of freedom. The theory can of course be truncated at any higher spatial orders, but the power over the the infinite temporal sum increases correspondingly such that the causality is respected. As a result, the theory truncated at any higher order of spatial gradient, requires newer degrees of freedom for each increasing order.