Generalising certain cosmological solutions sourced by a stiff perfect fluid
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ORIGINAL PAPER
Generalising certain cosmological solutions sourced by a stiff perfect fluid B B Hazarika* Department of Physics, Gauhati University, Guwahati 781014, India Received: 20 August 2019 / Accepted: 02 March 2020
Abstract: In this article, we show that a number of well-known time-dependent, spherically symmetric solutions of the Einstein’s field equations sourced by a stiff, perfect fluid with a cosmological constant can be generalised into a solution with arbitrary metric functions. This metric can be applied to construct stiff, perfect fluid metrics with (or without) a cosmological constant. We also explore the possibility that this metric may allow us to generate a singularity-free, spherically symmetric cosmological model. Keywords: Spherically symmetric metric; Stiff perfect fluid; Cosmological constant; Singularity-free
1. Introduction The study of spherically symmetric solutions is an important sub-field of the general theory of relativity. We note that although it is commonly believed that a large number of spherically symmetric solutions of the Einstein’s field equations are known, the actual situation is quite the opposite. In fact, as stated in [1], ‘‘Most known solutions are static or shearfree, and only very few of them satisfy the physical demands of having a plausible equation of state or being free from singularities’’. There are a number of time-dependent, stiff fluid, spherically symmetric solutions in the literature. In this article, our aim is to write down a metric for time-dependent perfect fluid solutions, with a stiff equation of state and a cosmological constant, which generalises some of these solutions and other solutions which are slight variations of some other solutions of this type. Here, the metric functions are products of arbitrary functions of the radial coordinate and time. Substituting appropriate known functions allows us to reproduce a number of well-known metrics. We also show that the spacetime is sourced by a stiff fluid that has non-zero expansion, shear and acceleration. The metric is algebraically special belonging to the type D in the Petrov classification scheme. Finally, we use this general metric to show that it could be possible to generate a spherically
symmetric, singularity-free cosmological model sourced by a stiff perfect fluid. A perfect fluid has the energy-momentum tensor of the form Tlm ¼ ðq þ pÞul um þ pglm
ð1Þ
where q is the energy density, p, the isotropic pressure and ul its four velocity. For fluids with a linear equation of state, the relation between pressure and energy density is usually written as p ¼ ðc 1Þq
ð2Þ
where 1\c\2 is a constant. The case c ¼ 2, which corresponds to p ¼ q, is known as a stiff fluid. A physical characteristic of such a fluid is that the speed of sound equals that of light in the medium. There have been several prior attempts to generalise perfect fluid solutions. Two notable papers in this connection are the ones by Lake [2] and by Maharaj et al. [3]. In [2], the functions l, m and k in the spherically s
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