Spherically symmetric lightlike radiation and conformally flat space-times

Abstract

The problem of the coexistence of spherical symmetry of a 3-space, tracelessness of the energy-momentum tensor, conformally flat 4-metrics, and the validity of the Einstein equations is investigated. The assertion is proved that when spherical symmetry is present nonequilibrium lightlike radiation with the energymomentum tensor T_µv = ρl _µ l _v (l _α v ^α = 0) cannot serve as the source of the gravitational field corresponding to a conformally flat space-time (type 0 according to the algebraic classification). An exact spherically symmetric solution with a conformally flat metric is obtained which describes dust and equilibrium isotropic radiation without energy exchange between them. This solution is rewritten for a synchronous reference frame in which it is evident that it describes a homogeneous and isotropic universe. In the limit of the absence of radiation the solution changes into the well-known Friedmann solution for an open universe.