J. Gegenberg

A Finite action for three-dimensional gravity with a minimally coupled scalar field

Three-dimensional gravity with a minimally coupled self-interacting scalar is considered. The fall-off of the fields at infinity is assumed to be slower than that of a localized distribution of matter, so that the asymptotic symmetry group is the conformal group. The counterterm Lagrangian needed to render the action finite is found by demanding that the action attain an extremum for the boundary conditions implied by the above fall-off of the fields at infinity. These counterterms explicitly depend on the scalar field. As a consequence, the Brown-York stress-energy tensor acquires a non trivial contribution from the matter sector. Static circularly symmetric solutions with a regular scalar field are explored for a one-parameter family of potentials. Their masses are computed via the Brown-York quasilocal stress-energy tensor, and they coincide with the values obtained from the Hamiltonian approach. The thermal behavior, including the transition between different configurations, is analyzed, and it is found that the scalar black hole can decay into the BTZ solution irrespective of the horizon radius. It is also shown that the AdS/CFT correspondence yields the same central charge as for pure gravity.

Black holes in three-dimensional topological gravity.

We investigate the black hole solution to (2+1)-dimensional gravity coupled to topological matter, with a vanishing cosmological constant. We calculate the total energy, angular momentum, and entropy of the black hole in this model and compare with results obtained in Einstein gravity. We find that the theory with topological matter reverses the identification of energy and angular momentum with the parameters in the metric, compared with general relativity, and that the entropy is determined by the circumference of the inner rather than the outer horizon. We speculate that this results from the contribution of the topological matter fields to the conserved currents. We also briefly discuss two new possible (2+1)-dimensional black holes.

Thermodynamics and statistical mechanics of induced Liouville gravity

In this paper we describe a Liouville gravity which is induced by a set of quantum fields (constituents) and represents a two-dimensional analog of Sakharovs induced gravity. The important feature of the considered theory is the presence of massless constituents which are responsible for the appearance of the induced Liouville field. The role of the massive constituents is only to induce the cosmological constant. We consider the instanton solutions of the Euclidean Liouville gravity with negative and zero cosmological constants, some instantons being interpreted as two-dimensional anti-de Sitter

Using 3D string-inspired gravity to understand the Thurston conjecture

AdS_2

Quantum theory of black holes

black holes. We study thermodynamics of all the solutions and conclude that their entropy is completely determined by the statistical-mechanical entropy of the massless constituents. This shows, in particular, that the constituents of the induced gravity are the true degrees of freedom of

Gravitational non-commutativity and Gödel-like spacetimes

AdS_2

Quantum structure of space near a black hole horizon

black holes. Special attention is also paid to the induced Liouville gravity with zero cosmological constant on a torus. We demonstrate the equivalence of its thermodynamics to the thermodynamics of BTZ black holes and comment on computations of the BTZ black hole entropy.

Tomimatsu–Sato geometries, holography and quantum gravity

We present a string-inspired three-dimensional (3D) Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell–Chern–Simons field. For constant dilaton, the theory appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed 3-sphere. It therefore appears that one can construct modified Ricci flow equations in which the topology of the geometry is encoded in the parameters of an underlying field theory.

Yang–Mills flow and uniformization theorems

A solvable two-dimensional conformally invariant midisuperspace model for black holes is obtained by imposing spherical symmetry in four-dimensional conformally invariant Einstein gravity. The Wheeler-DeWitt equation for the theory is solved exactly to obtain the unique quantum wave functional for an isolated black hole with a fixed mass. By suitably relaxing the boundary conditions, a nonperturbative ansatz is obtained for the wave functional of a black hole interacting with its surroundings

An instability of hyperbolic space under the Yang-Mills flow

We derive general conditions under which geodesics of stationary spacetimes resemble trajectories of charged particles in an electromagnetic field. For large curvatures (analogous to strong magnetic fields), the quantum mechanicical states of these particles are confined to gravitational analogs of lowest Landau levels. Furthermore, there is an effective non-commutativity between their spatial coordinates. We point out that the Som–Raychaudhuri and Gödel spacetime and its generalisations are precisely of the above type and compute the effective non-commutativities that they induce. We show that the non-commutativity for Gödel spacetime is identical to that on the fuzzy sphere. Finally, we show how the star product naturally emerges in Som–Raychaudhuri spacetimes.