## 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.