Mussa, A; (2014) Spherical symmetry and hydrostatic equilibrium in theories of gravity. Doctoral thesis , UCL (University College London).

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Abstract

Static, spherically symmetric solutions of the Einstein-Maxwell equations in the presence of a cosmological constant are studied, and new classes of solutions are derived. Namely the charged Einstein static universe and the interior and exterior charged Nariai spacetimes, these solutions form a subclass of the RNdS solution with distinct properties. The charged Nariai solutions are then matched at a common boundary. When constructing solutions to gravitational theories it is important that these matter distributions remain in hydrostatic equilibrium. If this equilibrium is lost, with internal gravitational forces dominating internal stresses, the solution will collapse under its gravitational field. An upper bound on the mass-radius ratio Mg/R for charged solutions in de Sitter space is derived, this bound implies hydrostatic equilibrium. The result is achieved by assuming the radial pressure p≥0 and energy density ρ≥0, plus p+2p⊥≤ρ where the tangential pressure p⊥≠ p. The bound provides a generalisation of Buchdahl's inequality, 2M/R ≤8/9, valid for Schwarzschild's solution. In the limit Q→0, Λ→0, the bound reduces to Buchdahl's inequality. Solutions in hydrostatic equilibrium are also considered in modified f(T) gravity. It is shown that the tetrads eⁱμ impact the structure of the field equations, and certain tetrads impose unnecessary constraints. Two particular tetrads are studied in more detail, solutions are then found for both tetrads, and a conservation equation is obtained using an analogous method to obtaining the Tolman-Oppenheimer-Volkoff equation. Although both tetrad fields locally give rise to the spherically symmetric metric, the tetrad fields are not globally well-defined and hence cannot be described as spherically symmetric. We then derive an upper bound on M/R which also implies hydrostatic equilibrium, this yields some constraints on the form of f(T) given a particular tetrad that locally gives rise to the line element ds²=exp(a)dt²-exp(b)dr²-r²dΩ².

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