We study the gravitational vacuum star (gravastar) configuration as proposed by [1] in a model where the interior de Sitter spacetime segment is continuously extended to the exterior Schwarzschild spacetime.

Using the improved quantization technique to the minisuperspace approximation of loop quantum gravity, we study the evolution of black holes supported by a cosmological constant. The addition of a cosmological constant allows for classical solutions with planar, cylindrical, toroidal, and higher-genus black holes. Here we study the quantum analog of these space-times. In all scenarios studied, the singularity present in the classical counterpart is avoided in the quantized version and is replaced by a bounce, and in the late evolution, a series of less severe bounces. Interestingly, although there are differences during the evolution between the various symmetries and topologies, the evolution on the other side of the bounce asymptotes to space-times of Nariai-type, with the exception of the planar black hole analyzed here, whose T-R ¼ constant subspaces seem to continue expanding in the long-term evolution. For the other cases, Nariai-type universes are attractors in the quantum evolution, albeit with different parameters. We study here the quantum evolution of each symmetry in detail.

In the context of loop quantum gravity, we construct the phase-space of isolated horizons with genus greater than 0. Within the loop quantum gravity framework, these horizons are described by genus g surfaces with N punctures and the dimension of the corresponding phase-space is calculated including the genus cycles as degrees of freedom. From this, the black hole entropy can be calculated by counting the microstates which correspond to a black hole of fixed area. We find that the leading term agrees with the A/4 law and that the sub-leading contribution is modified by the genus cycles.

The stationary gravitational equations in vacuum are expressed in five different forms. A necessary integral condition on the twist potential φ is derived. The Papapetrou-Ehlers class of stationary solutions is rederived in a different way. In the study of the complex potential theory it is proved from the field equations that a body admitting an arbitrary symmetry must satisfy an integral condition analogous to the equilibrium criterion. It is proved that the vanishing of the scalar curvature of the associated space implies the flatness of the space-time metric. A proof is given for the fact that the only analytic functions of the complex potential F which preserve the field equations form a four-parameter Möbius group. It is also shown that any differentiable function of F and F̄ which preserves the field equations must either be an analytic function of F or the conjugate of such a function. Next the conformastationary vacuum metrics are classified. In the study of the axially symmetric stationary fields a class of metrics (outside the Papapetrou-Ehlers class) is found depending on Euclidean harmonic functions.

The system consisting of a self gravitating perfect fluid and scalar field is considered in detail. The scalar fields considered are the quintessence and "tachyonic" forms which have important application in cosmology. Mathematical properties of the general system of equations are studied including the algebraic and differential identities as well as the eigenvalue structure. The Cauchy problem for both quintessence and the tachyon is presented. We discuss the initial constraint equations which must be satisfied by the initial data. A Cauchy evolution scheme is presented in the form of a Taylor series about the Cauchy surface. Finally, a simple numerical example is provided to illustrate this scheme.