Spherically Symmetric Static Solutions in General Relativity

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This thesis studies spherically symmetric static solutions in general relativity. The most general form of matter in general relativity compatible with staticity and spherical symmetry is anisotropic fluid. We study all possible algorithms that can generate all solutions of the anisotropic fluid system via quadrature using all possible pairs of the four basic functions of the system as input functions. We also study sub-algorithms that generate all solutions that are regular at the center and, for this, we revisit the conditions for central regularity for both isotropic and anisotropic systems and obtain all possible sets of equivalent initial conditions for regularity by combining the Einstein equations with the previouslyknown geometric conditions of regularity. Our study provides a reformulation of an existing algorithm for the system and provides its first regularity analysis. A surprisingly simple new algorithm for the anisotropic system follows from our study that aligns itself with the regularity conditions. This concordance enables us to find solutions that satisfy all the other hard-to-achieve conditions of physical acceptability. Anisotropy has increasingly been shown to be physically relevant in recent times. We keep the well-studied isotropic system as a special case and use it as a frame of reference for measuring the success of our study of the anisotropic system. We then study the hydrostatic equilibrium of static (an)isotropic fluid spheres. From the condition of hydrostatic equilibrium, we explore maps between (an)isotropic solutions with the same density profiles and develop solution-generating techniques to find new solutions from existing ones. We compare and give physical interpretations of several equilibrium configurations in terms of fluid variables and provide several examples where the solutiongenerating theorems can be utilized to find physically acceptable anisotropic solutions. This include a new exact solution that satisfies all physically desirable conditions. Finally, we study light propagation in Kottler, i.e., Schwarzschild-(anti-)de Sitter, spacetime. The metric of this spacetime is known in canonical coordinates and, unlike its Λ = 0 version (i.e, Schwarzschild metric), this metric was not known in isotropic coordinates (in which the constant-time hypersurfaces are flat). We obtain the Kottler metric in isotropic coordinates. This further enables us to plot the refractive indices of Kottler spacetime and show that the invariance of Snell’s law in ordinary geometric optics is analogous to projective equivalence in isotropic static coordinates. We conclude with a summary and some future directions.

Physics, Theory