The Two-Dimensional Analogue of General Relativity

General Relativity in three or more dimensions can be obtained by taking the limit $\omega\rightarrow\infty$ in the Brans-Dicke theory. In two dimensions General Relativity is an unacceptable theory. We show that the two-dimensional closest analogue of General Relativity is a theory that also arises in the limit $\omega\rightarrow\infty$ of the two-dimensional Brans-Dicke theory.Comment: 8 pages, LaTeX, preprint DF/IST-17.9

The Tolman-Bondi--Vaidya Spacetime: matching timelike dust to null dust

The Tolman-Bondi and Vaidya solutions are two solutions to Einstein equations which describe dust particles and null fluid, respectively. We show that it is possible to match the two solutions in one single spacetime, the Tolman-Bondi--Vaidya spacetime. The new spacetime is divided by a null surface with Tolman-Bondi dust on one side and Vaidya fluid on the other side. The differentiability of the spacetime is discussed. By constructing a specific solution, we show that the metric across the null surface can be at least $C^1$ and the stress-energy tensor is continuous.Comment: 5 pages, no figur

Two-Dimensional Black Holes and Planar General Relativity

The Einstein-Hilbert action with a cosmological term is used to derive a new action in 1+1 spacetime dimensions. It is shown that the two-dimensional theory is equivalent to planar symmetry in General Relativity. The two-dimensional theory admits black holes and free dilatons, and has a structure similar to two-dimensional string theories. Since by construction these solutions also solve Einstein's equations, such a theory can bring two-dimensional results into the four-dimensional real world. In particular the two-dimensional black hole is also a black hole in General Relativity.Comment: 11 pages, plainte

Gravitational collapse to toroidal, cylindrical and planar black holes

Gravitational collapse of non-spherical symmetric matter leads inevitably to non-static external spacetimes. It is shown here that gravitational collapse of matter with toroidal topology in a toroidal anti-de Sitter background proceeds to form a toroidal black hole. According to the analytical model presented, the collapsing matter absorbs energy in the form of radiation (be it scalar, neutrinos, electromagnetic, or gravitational) from the exterior spacetime. Upon decompactification of one or two coordinates of the torus one gets collapsing solutions of cylindrical or planar matter onto black strings or black membranes, respectively. The results have implications on the hoop conjecture.Comment: 6 pages, Revtex, modifications in the title and in the interpretation of some results, to appear in Physical Review

Hamiltonian thermodynamics of three-dimensional dilatonic black holes

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.Comment: 34 pages, 3 figures, references added, minor changes in the revised versio

Conformal entropy from horizon states: Solodukhin's method for spherical, toroidal, and hyperbolic black holes in D-dimensional anti-de Sitter spacetimes

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.Comment: 21 page

Pair creation of higher dimensional black holes on a de Sitter background

We study in detail the quantum process in which a pair of black holes is created in a higher D-dimensional de Sitter (dS) background. The energy to materialize and accelerate the pair comes from the positive cosmological constant. The instantons that describe the process are obtained from the Tangherlini black hole solutions. Our pair creation rates reduce to the pair creation rate for Reissner-Nordstrom-dS solutions when D=4. Pair creation of black holes in the dS background becomes less suppressed when the dimension of the spacetime increases. The dS space is the only background in which we can discuss analytically the pair creation process of higher dimensional black holes, since the C-metric and the Ernst solutions, that describe respectively a pair accelerated by a string and by an electromagnetic field, are not know yet in a higher dimensional spacetime.Comment: 10 pages; 1 figure included; RexTeX4. v2: References added. Published version. v3: Typo in equation (46) fixe

The Three-Dimensional BTZ Black Hole as a Cylindrical System in Four-Dimensional General Relativity

It is shown how to transform the three dimensional BTZ black hole into a four dimensional cylindrical black hole (i.e., black string) in general relativity. This process is identical to the transformation of a point particle in three dimensions into a straight cosmic string in four dimensions.Comment: Latex, 9 page

Charged shells in Lovelock gravity: Hamiltonian treatment and physical implications

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.Comment: 25 pages, 9 figure

Collapsing shells of radiation in anti-de Sitter spacetimes and the hoop and cosmic censorship conjectures

Gravitational collapse of radiation in an anti-de Sitter background is studied. For the spherical case, the collapse proceeds in much the same way as in the Minkowski background, i.e., massless naked singularities may form for a highly inhomogeneous collapse, violating the cosmic censorship, but not the hoop conjecture. The toroidal, cylindrical and planar collapses can be treated together. In these cases no naked singularity ever forms, in accordance with the cosmic censorship. However, since the collapse proceeds to form toroidal, cylindrical or planar black holes, the hoop conjecture in an anti-de Sitter spacetime is violated.Comment: 4 pages, Revtex Journal: to appear in Physical Review