Nonlocal extension of relativistic causal thermodynamics and general relativistic Burgers equation

A phenomenological approach to the construction of a relativistic model of thermo-visco-elasticity is elaborated. The main element of presented approach is the generalized Burgers equation. As a key step, we have constructed a non-local generalization of the Israel-Stewart model for relativistic causal thermodynamics of a homogeneous, isotropic cosmic fluid, in which the coefficient at the integral operator is responsible for the elastic properties of the medium. Based on the second low of thermodynamics we derive the integro-differential equation for the evolution of non-equilibrium pressure scalar and we show that the differential version of this equation is a relativistic analog of the Burgers equation describing visco-elastic processes in classical media. Based on the obtained equations for a medium model with bulk visco-elasticity, we present a hypothesis that the equations for a model with shear visco-elasticity can also be represented using the corresponding relativistic generalization of the Burgers equation; in other words, the equation for the traceless shear part of the non-equilibrium pressure tensor was obtained phenomenologically using an analogy with the exact equation for the bulk part of this tensor.