The Deng Algorithm in Higher Dimensions

We extend an algorithm of Deng in spherically symmetric spacetimes to higher dimensions. We show that it is possible to integrate the generalised condition of pressure isotropy and generate exact solutions to the Einstein field equations for a shear-free cosmological model with heat flow in higher dimensions. Three new metrics are identified which contain results of four dimensions as special cases. We show graphically that the matter variables are well behaved and the speed of sound is causal. 1. Introduction Spherically symmetric gravitating models with heat flow, in the absence of shear, are important in the study of various cosmological processes and the evolution of relativistic astrophysical bodies. For a variety of applications in the presence of inhomogeneity, see Krasinski [1]. Heat flow models are also important in analysing gravitational collapse and relativistic stellar processes. Astrophysical studies in which heat flow is important include the shear-free models of Wagh et al. [2], Maharaj and Govender [3], Misthry et al. [4], and Herrera et al. [5]. By studying shear-free models, we avail ourselves with a rather simpler avenue where we only need to provide solutions to the generalised condition of pressure isotropy containing two metric functions. A complete study of shear-free heat conducting fluids with charge was completed by Nyonyi et al. [6] using Lie’s group theoretic approach applied to differential equations. Shearing models where heat flow is significant have been recently studied by Thirukkanesh et al. [7] for radiating spherically symmetric spheres. It turns out that the resulting nonlinear equations with shear are much more difficult to analyse. A generic method of obtaining new solutions to the Einstein field equations with heat flow was provided by Deng [8]. Using this general method, we can regain existing results and obtain new classes of solutions. Nyonyi et al. [6], Ivanov [9], and Msomi et al. [10] have obtained new solutions using the Lie group theoretic approach and other methods, by solving the underlying pressure isotropy condition. These investigations are applicable to four dimensions. Extensions to higher dimensions have also been considered by many authors because of physical requirements; for example, Bhui et al. [11] showed the absence of horizons in nonadiabatic gravitational collapse. Studies of this type motivated the Lie symmetry analysis of heat conducting fluids by Msomi et al. [12] in dimensions greater than four. In the present treatment, we extend the Deng [8] algorithm to higher dimensions and show