Gonçalo A S Dias

The action for a class of three-dimensional dilaton-gravity theories, with an electromagnetic Maxwell field and a cosmological constant, can be recast in a Brans-Dicke-Maxwell type action, with its free $\omega$ parameter. For a negative cosmological constant, these theories have static, electrically charged, and spherically symmetric black hole solutions. Those theories with well formulated asymptotics are studied through a Hamiltonian formalism, and their thermodynamical properties are found. The theories studied are general relativity, a dimensionally reduced cylindrical four-dimensional general relativity theory, and a theory representing a class of theories, all with a Maxwell term. The Hamiltonian formalism is setup in three dimensions through foliations on the right region of the Carter-Penrose diagram, with the bifurcation 1-sphere as the left boundary, and anti-de Sitter infinity as the right one. The metric functions on the hypersurfaces and the radial component of the vector potential one-form are the canonical coordinates. The Hamiltonian action is written, the Hamiltonian being a sum of constraints. One finds a new action which yields an unconstrained theory with two pairs of canonical coordinates ${M,P_M; Q,P_Q}$, where $M$ is the mass parameter, which needs renormalization, $P_M$ its conjugate momenta, $Q$ is the charge parameter, and $P_Q$ its conjugate momentum. The resulting Hamiltonian is a sum of boundary terms only. A quantization of the theory is performed. The Schr\"odinger evolution operator is constructed, the trace is taken, and the partition function of the grand canonical ensemble is obtained, the chemical potential being the electric potential. The charged black hole entropies differ, in general, from the usual quarter of the horizon area due to the dilaton.

The action for a class of three-dimensional dilaton-gravity theories with a cosmological constant can be recast in a Brans-Dicke type action, with its free $\omega$ parameter. These theories have static spherically symmetric black holes. Those with well formulated asymptotics are studied through a Hamiltonian formalism, and their thermodynamical properties are found out. The theories studied are general relativity ($\omega\to\infty$), a dimensionally reduced cylindrical four-dimensional general relativity theory ($\omega=0$), and a theory representing a class of theories ($\omega=-3$). The Hamiltonian formalism is setup in three dimensions through foliations on the right region of the Carter-Penrose diagram, with the bifurcation 1-sphere as the left boundary, and anti-de Sitter infinity as the right boundary. The metric functions on the foliated hypersurfaces are the canonical coordinates. The Hamiltonian action is written, the Hamiltonian being a sum of constraints. One finds a new action which yields an unconstrained theory with one pair of canonical coordinates $\{M,P_M\}$, $M$ being the mass parameter and $P_M$ its conjugate momenta The resulting Hamiltonian is a sum of boundary terms only. A quantization of the theory is performed. The Schr\"odinger evolution operator is constructed, the trace is taken, and the partition function of the canonical ensemble is obtained. The black hole entropies differ, in general, from the usual quarter of the horizon area due to the dilaton.

We construct thin-shell electrically charged wormholes in d-dimensional general relativity with a cosmological constant. The wormholes constructed can have different throat geometries, namely, spherical, planar and hyperbolic. Unlike the spherical geometry, the planar and hyperbolic geometries allow for different topologies and in addition can be interpreted as higher-dimensional domain walls or branes connecting two universes. In the construction we use the cut-and-paste procedure by joining together two identical vacuum spacetime solutions. Properties such as the null energy condition and geodesics are studied. A linear stability analysis around the static solutions is carried out. A general result for stability is obtained from which previous results are recovered.

A calculation of the entropy of static, electrically charged, black holes with spherical, toroidal, and hyperbolic compact and oriented horizons, in D spacetime dimensions, is performed. These black holes live in an anti-de Sitter spacetime, i.e., a spacetime with negative cosmological constant. To find the entropy, the approach developed by Solodukhin is followed. The method consists in a redefinition of the variables in the metric, by considering the radial coordinate as a scalar field. Then one performs a 2+(D-2) dimensional reduction, where the (D-2) dimensions are in the angular coordinates, obtaining a 2-dimensional effective scalar field theory. This theory is a conformal theory in an infinitesimally small vicinity of the horizon. The corresponding conformal symmetry will then have conserved charges, associated with its infinitesimal conformal generators, which will generate a classical Poisson algebra of the Virasoro type. Shifting the charges and replacing Poisson brackets by commutators, one recovers the usual form of the Virasoro algebra, obtaining thus the level zero conserved charge eigenvalue L_0, and a nonzero central charge c. The entropy is then obtained via the Cardy formula.

The Hamiltonian thermodynamics formalism is applied to the general $d$-dimensional Reissner-Nordstr\"om-anti-de Sitter black hole with spherical, planar, and hyperbolic horizon topology. After writing its action and performing a Legendre transformation, surface terms are added in order to guarantee a well defined variational principle with which to obtain sensible equations of motion, and also to allow later on the thermodynamical analysis. Then a Kucha\v{r} canonical transformation is done, which changes from the metric canonical coordinates to the physical parameters coordinates. Again a well defined variational principle is guaranteed through boundary terms. These terms influence the fall-off conditions of the variables and at the same time the form of the new Lagrange multipliers. Reduction to the true degrees of freedom is performed, which are the conserved mass and charge of the black hole. Upon quantization a Lorentzian partition function $Z$ is written for the grand canonical ensemble, where the temperature $\bf T$ and the electric potential $\phi$ are fixed at infinity. After imposing Euclidean boundary conditions on the partition function, the respective effective action $I_*$, and thus the thermodynamical partition function, is determined for any dimension $d$ and topology $k$. This is a quite general action. Several previous results can be then condensed in our single general formula for the effective action $I_*$. Phase transitions are studied for the spherical case, and it is shown that all the other topologies have no phase transitions. A parallel with the Bose-Einstein condensation can be established. Finally, the expected values of energy, charge, and entropy are determined for the black hole solution.

Using a Hamiltonian treatment, charged thin shells in spherically symmetric spacetimes in d dimensional Lovelock-Maxwell theory are studied. The coefficients of the theory are chosen to obtain a sensible theory, with a negative cosmological constant appearing naturally. After writing the action and the Lagrangian for a spacetime comprised of an interior and an exterior regions, with a thin shell as a boundary in between, one finds the Hamiltonian using an ADM description. For spherically symmetric spacetimes, one reduces the relevant constraints. The dynamic and constraint equations are obtained. The vacuum solutions yield a division of the theory into two branches, d-2k-1>0 (which includes general relativity, Born-Infeld type theories) and d-2k-1=0 (which includes Chern-Simons type theories), where k gives the highest power of the curvature in the Lagrangian. An additional parameter, chi, gives the character of the vacuum solutions. For chi=1 the solutions have a black hole character. For chi=-1 the solutions have a totally naked singularity character. The integration through the thin shell takes care of the smooth junction. The subsequent analysis is divided into two cases: static charged thin shell configurations, and gravitationally collapsing charged dust shells. Physical implications are drawn: if such a large extra dimension scenario is correct, one can extract enough information from the outcome of those collapses as to know, not only the actual dimension of spacetime, but also which particular Lovelock gravity, is the correct one.