Spherically Symmetric Static Solutions of the Einstein Field Equations using Segre Classification

Abstract

Obtaining solutions to the Einstein field equations remains an interesting research area within general relativity. In four dimensions, these field equations are a system of ten non-linear partial differential equations. Finding general solution of this system is al most impossible. Therefore, one deals with this system under different physical and geometrical restrictions, like spherical or cylindrical symmetry etc. In literature, many researchers have made arbitrary assumptions/ansatz about various unknown variables, such as the electric field intensity, the gravitational potentials, and the equation of state. This thesis aims to develop, using the Segre classification, a systematic scheme to reduce the arbitrariness in taking different ansatz. The developed scheme offers a structured method for obtaining solutions of the field equations. It is shown that spher ically symmetric static solutions can be categorized into four Segre types: [(1,111)], [1, (111)], [(1,1) (11)], and [1,1(11)]. These types correspond to the degeneracy of eigen values, which in turn helps to determine the type of matter distribution in space based on the correlation to timelike and spacelike eigenvectors. It is to be mentioned that the Segre type [(1,111)] admits only Schwarzschild de Sit ter and anti-de Sitter solutions. The Segre type [1, (111)] describes an ideal fluid with isotropic pressure. The corresponding solution represents a stable stellar con f iguration, where the uniform pressure distribution counteracts gravitational collapse. The type [(1,1)(11)] is associated with a non-null electromagnetic field and signifies an anisotropic pressure distribution. This type admits solutions that exhibit the presence of dark energy. The Segre type [1,1(11)] also corresponds to an anisotropic pressure distribution. This thesis presents solutions of each type along with a detailed physical analysis. The numerical tests for the solutions, against Segre types [1,(111)] and [1,1(1,1)], are conducted by comparing them with 4U 1538 − 52, PSR J1614 − 2230, 4U 1608 − 52, and EXO 1785−248. The density, pressure, and compactness factor in each case are observed, showing consistency.