Singularity Free Inhomogeneous Models with Heat Flow

We present a class of singularity free exact cosmological solutions of Einstein’s equations describing a perfect fluid with heat flow. It is obtained as generalization of the Senovilla class [1] corresponding to incoherent radiation field. The spacetime is cylindrically symmetric and globally regular. PACS numbers : 04.20 Jb, 98.80 Dr † E-mail address : [email protected] 1 Senovilla [1] has opened up a new vista of singularity free cosmological models. He obtained a class of cyclindrically symmetric inhomogeneous exact solutions of Einstein’s equations describing a perfect fluid with the equation of state ρ = 3p. The spacetime was globally regular, smooth and satisfied the energy and causality conditions. All the physical and curvature invariants were finite everywhere, and all causal curves were complete [2]. There was no singularity of any kind. The singularity structure of a general class of inhomogeneous perfect fluid solutions was analysed [3] and it was shown that existence or non-existence of singularity as well as its kind (spacelike, timelike, Weyl, Ricci, etc) depended upon the particular choice of certain parameters and metric functions. There exists a general class of inhomogeneous perfect fluid solutions without singularity. Recently we have also found [4] a new class of solutions for stiff fluid (ρ = p) that have vacuum spacetime as the matter free (ρ = 0) limit. It is very curious that spacetime is empty as well as without singularity. One wonders about the source of its curvature. How robust is the singularity free character of spacetime ? In particular what happens when dissipative effects of viscosity, heat flow etc. are incorporated ? In this note we wish to obtain a generalization of the Senovilla [1] model that includes heat flow. Dissipative effects in cosmology were considered in the context of large entropy per baryon and isotropy of microwave background radiation[5,6] and heat flow was in particular investigated by several authors [7 16]. In very early times, matter is supposed to be highly dense and hot, and hence consideration of heat flow for very 2 early evaluation of the Universe is quite appropriate. The energy momentum tensor incorporating heat flow with perfect fluid reads as Tik = (ρ+ p)uiuk − pgik + 2u(iqk) (1) where uiu i = 1, qiu i = 0, and qi is the heat flow vector. The expansion θ, shear scalar σ and the acceleration vector u̇i for the fluid are defined by θ = u;i, σ 2 = σikσ ik σik = u(i;k) − u(iu̇k) − θ 3 (gik − uiuk) u̇i = ui;ku k . For Einstein’s equations Rik − 1 2 Rgik = −8πTik (2) we obtain the following class of solutions for perfect fluid with radial heat flow; ds = A(dt − dr)−Bdz −Ddφ (3)