Relativistic heat conduction

Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. In special (and general) relativity, the usual heat equation for non-relativistic heat conduction must be modified, as it leads to faster-than-light signal propagation. Relativistic heat conduction, therefore, encompasses a set of models for heat propagation in continuous media (solids, fluids, gases) that are consistent with relativistic causality, namely the principle that an effect must be within the light-cone associated to its cause. Any reasonable relativistic model for heat conduction must also be stable, in the sense that differences in temperature propagate both slower than light and are damped over time (this stability property is intimately intertwined with relativistic causality). Heat conduction in a Newtonian context is modelled by the Fourier equation, namely a parabolic partial differential equation of the kind: where θ is temperature, t is time, α = k/(ρ c) is thermal diffusivity, k is thermal conductivity, ρ is density, and c is specific heat capacity. The Laplace operator,, is defined in Cartesian coordinates as This Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector, q, as a function of temperature gradient, into the first law of thermodynamics where the del operator, ∇, is defined in 3D as It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics, where s is specific entropy and σ is entropy production. This mathematical model is inconsistent with special relativity: the Green function associated to the heat equation (also known as heat kernel) has support that extends outside the light-cone, leading to faster-than-light propagation of information. For example, consider a pulse of heat at the origin; then according to Fourier equation, it is felt (i.e. temperature changes) at any distant point, instantaneously.