A consistent first-order model for relativistic heat flow

Abstract

This paper revisits the problem of heat conduction in relativistic fluids, associated with issues concerning both stability and causality. It has long been known that the problem requires information involving second order deviations from thermal equilibrium. Basically, any consistent first-order theory needs to remain cognizant of its higher-order origins. We demonstrate this by carrying out the required first-order reduction of a recent variational model. We provide an analysis of the dynamics of the system, obtaining the conditions that must be satisfied in order to avoid instabilities and acausal signal propagation. The results demonstrate, beyond any reasonable doubt, that the model has all the features one would expect of a real physical system. In particular, we highlight the presence of a second sound for heat in the appropriate limit. We also make contact with previous work on the problem by showing how the various constraints on our system agree with previously established results. I. CONTEXTRelativistic thermodynamics continues to provide interesting challenges, in particular in the context of dissipative and nonlinear phenomena. The issues involved range from direct applications in various areas of physics to fundamental problems like the nature of time (visavi the second law of thermodynamics) and the formation of structures at nonlinear deviations from thermal equilibrium. Much recent work has been motivated by the modelling of complex astrophysical systems, like neutron stars [1], and cosmology [2]. There has also been a resurgence of interest in dissipative systems in the context of colliders like RHIC at Brookhaven and the LHC at CERN [3-5]. These latter developments, which have to a large extent been driven by the need to understand the dynamics of a hot quark-gluon plasma, are often linked with underlying principles like the AdS/CFT conjecture and holography [6]. Even though the problem dates back to the origins of relativity theory, it remains (in a slightly different guise) at the forefront of modern thinking.According to the established consensus view, one must account for second-order deviations from thermal equilibrium in order to achieve causality and stability. This is certainly the lesson from the celebrated work of Israel and Stewart [7,8], see [9][10][11] for recent work on the problem. We have recently revisited the key points in the context of heat conduction [12], taking a multi-fluid prescription based on Carter's convective variational formulation for relativistic fluids [13] as our starting point. This is a mathematically elegant approach that has the flexibility required to account for the physics that we need to consider. A particularly appealing feature of the variational approach is that, once an "equation of state" for matter is provided, the theory provides the relation between the various currents and their conjugate momenta [1]. The variational analysis leads to a second-order model which has the key elements required for causality and stability, in particular, it clarif...

Quantised vortices and mutual friction in relativistic superfluids

“…In a multi-component system the choice of frame is less obvious. This is clearly illustrated by the classic problem of relativistic heat-flow [27,28].…”

Electron thermodynamics in GRMHD simulations of low-luminosity black hole accretion

“…Israel & Stewart (1979) derived a second order solution for q µ e which was later shown to be conditionally stable (Hiscock & Lindblom 1985;. Here we use a first order reduction of that second order model that has been shown to be both stable and self-consistent (Andersson & Lopez-Monsalvo 2011). We refer the reader to for more details.…”

A variational approach to resistive relativistic plasmas

“…issues arising when components become superfluid, when heat flows and when the electromagnetic charge current is treated as a dynamical variable [1, [9][10][11]. These advances allow us to consider a wide range of relevant phenomena, but the general theory is incomplete in two important respects.…”