Oscillating non-singular relativistic spherical model

A particular choice of the time function in the recently presented spherical solution by Dadhich [1] leads to a singularity free cosmological model which oscillates between two regular states. The energy-stress tensor involves anisotropic pressure and a heat flux term but is consistent with the usual energy conditions (strong, weak and dominant). By choosing the parameters suitably one can make the model consistent with observational data. An interesting feature of the model is that it involves blue shifts as in the quasi steady state model [2] but without violating general relativity. 04.20, 04.60, 98.80Hw ∗E-mail: [email protected] 1 Following the discovery of non-singular cylindrically symmetric perfect fluid exact cosmological solution of the Einstein equation by Senovilla [3], some spherically symmetric nonsingular models have been presented by Dadhich et.al [1,4]. These models have an energystress tensor with anisotropic pressure and heat flux but obeying the strong, weak and dominant energy conditions. The metric has a time function which can be arbitrarily chosen subject to the constraint of non-singularity and the energy conditions. It turns out that there exist different such choices which will be discussed in a detailed paper separately. In this letter we shall confine to the choice that gives an oscillatory behaviour of the universe without any singularity. The authors are not aware of any oscillatory singularity free model in classical general relativity (GR) while there are some oscillatory behaviour models proposed in the recent formulation of of quasi steady state cosmology (QSSC) [2]. As the QSSC models predict the possibility of blue shift, our model would also admit that possibility. Thus should observations in future reveal blue shifts, it may simply indicate that the matter in the uinverse is not perfect fluid and one need not bring in the ideas of non-conservation as in QSSC contradicting GR. Our oscillatory model is described by the metric [1], ds = (r + P )dt − 2r 2 + P r2 + P dr − r(dθ + sinθdφ) (1) where P = P (t) which can be chosen arbitrarily. The choice P (t) = a + bcosωt with a > b will render oscillatory behaviour to the model without encountering divergence of any kinematical and physical parameters. In Ref. [1] the choice made was P (t) = a + bt, which of course did not give oscillatory behaviour. The energy-stress tensor for imperfect fluid is given by [5], Tik = (ρ + p)uiuk − pgik +4p[cick + 1 3 (gik − uiuk)] + 2qc(ick) (2)