New solutions for heat conducting fluids with a normal shear-free flow

Abstract

The integration of Einstein's equations is considered for the most general spacetime with a non-rotating, non-static, shear-free fluid source with bulk viscosity and heat conduction. The constraints of the field equations are shown to reduce to a linear algebraic system on the derivatives of the metric functions. An examination of this system leads to the derivation of the following three large classes of exact solutions characterized by a generating function J: (1) time-dependent and non-zero J yields new Petrov type D solutions admitting no Killing vectors in general, although plane and spherically symmetric subcases are readily identified; (2) time-independent and non-zero J leads to another class of Petrov type D new solutions, all of which are either spherically or plane symmetric; (3) the case J=0 corresponds to conformally flat, Petrov type O solutions reducing to the 'Stephani Universe' as heat conduction vanishes. All Petrov type D solutions (regardless of the form of J not=0) lack a non-static perfect fluid limit. In the three classes of solutions one finds particular cases admitting a conformal Killing vector parallel to the 4-velocity or a one-parameter self-similar motion. Since all previously known solutions with a heat conducting, shear-free fluid source are either conformally flat and/or spherically symmetric, all of them are shown to be particular cases of the solutions derived here.