Conformal symmetry and applications to spherically symmetric spacetimes.

Abstract

In this thesis we study static spherically symmetric spacetimes with a spherical conformal symmetry and a nonstatic conformal factor. We analyse the general solution of the conformal Killing vector equation subject to integrability conditions which impose restrictions on the metric functions. The Weyl tensor is used to characterise the conformal geometry. An explicit relationship between the gravitational potentials for both conformally and nonconformally at cases is obtained. The Einstein equations can then be written in terms of a single gravitational potential. Previous results of conformally invariant static spheres are special cases of our solutions. For isotropic pressure we can find all metrics explicitly and show that the models always admit a barotropic equation of state. We show that this treatment contains well known metrics such Schwarzschild (interior), Tolman, Kuchowicz, Korkina and Orlyanskii, Patwardhan and Vaidya, and Buchdahl and Land. For anisotropic pressures the solution of the fluid equations is found in general. We then consider an astrophysical application of conformal symmetries. We investigate spherical exact models for compact stars with anisotropic pressures and a conformal symmetry. We generate a new anisotropic solution to the Einstein field equations. We demonstrate that this exact solution produces a relativistic model of a compact star. The model generates stellar radii and masses consistent with PSR J1614-2230, Vela X1, PSR J1903+327 and Cen X-3. A detailed physical examination shows that the model is regular, well behaved and stable. The mass-radius limit and the surface red shift are consistent with observational constraints.