Notes on black holes and three dimensional gravity

These notes are the written version of two lectures delivered at the VIII Mexican School on Particles and Fields on November 1998. The level of the notes is basic assuming only some knowledge on Statistical Mechanics, General Relativity and Yang-Mills theory. After a brief introduction to the classical and semiclassical aspects of black holes, we review some relevant results on 2+1 quantum gravity. These include the Chern-Simons formulation and its affine Kac-Moody algebra, the asymptotic algebra of Brown and Henneaux, and the statistical mechanics description of 2+1 black holes. Hopefully, this contribution will be complementary with the review paper hep-th/9901148 by the same author, and perhaps, a shortcut to some recent developments in three dimensional gravity.Comment: 20 pages. Uses aipproc. To appear in the proceedings of the VIII Mexican School on Particles and Field

Solitons and Black Holes

We explore the relationship between black holes in Jackiw-Teitelboim(JT) dilaton gravity and solitons in sine-Gordon field theory. Our analysis expands on the well known connection between solutions of the sine-Gordon equation and constant curvature metrics. In particular, we show that solutions to the dilaton field equations for a given metric in JT theory also solve the sine-Gordon equation linearized about the corresponding soliton. Since the dilaton generates Killing vectors of the constant curvature metric, it is interesting that it has an analoguous interpretation in terms of symmetries of the soliton solution. We also show that from the B${\ddot a}$cklund transformations relating different soliton solutions, it is possible to construct a flat SL(2,R) connection which forms the basis for the gauge theory formulation of JT dilaton gravity.Comment: Substantial revisions in sections 4-5, Latex file, 13 pages. Typographical errors corrected; accepted for publication in Phys. Lett.

Black Holes Without Coordinates

These lectures describe how to study the geometry of some black holes without the use of coordinates.Comment: 10 pages, 2 figures. Lectures presented at the Cargese Summer School, "Theory and Particle Physics: the LHC perspective and beyond" (June 16 to June 28, 2008). This paper is a sequel to gr-qc/070111

Three-dimensional quantum geometry and black holes

We review some aspects of three-dimensional quantum gravity with emphasis in the `CFT -> Geometry' map that follows from the Brown-Henneaux conformal algebra. The general solution to the classical equations of motion with anti-de Sitter boundary conditions is displayed. This solution is parametrized by two functions which become Virasoro operators after quantisation. A map from the space of states to the space of classical solutions is exhibited. Some recent proposals to understand the Bekenstein-Hawking entropy are reviewed in this context. The origin of the boundary degrees of freedom arising in 2+1 gravity is analysed in detail using a Hamiltonian Chern-Simons formalism.Comment: 39 pages, Latex, no figures. Invited talk at the Second Meeting "Trends in Theoretical Physics", held in Buenos Aires, December, 1998. v2: References added and minor corrections. v3: An incorrect statement about the sign of the Chern-Simons level erased. Extended (and in some cases modified) discussions in most sections. References adde

A Note on the Positive Constant Curvature Space

We construct a positive constant curvature space by identifying some points along a Killing vector in a de Sitter Space. This space is the counterpart of the three-dimensional Schwarzschild-de Sitter solution in higher dimensions. This space has a cosmological event horizon, and is of the topology ${\cal M}_{D-1}\times S^1$, where ${\cal M}_{D-1}$ denotes a $(D-1)$-dimensional conformal Minkowski spacetime.Comment: Revtex, 12 pages with two eps figures, to appear in PL

Black hole solutions in 2+1 dimensions

We give circularly symmetric solutions for null fluid collapse in 2+1-dimensional Einstein gravity with a cosmological constant. The fluid pressure $P$ and energy density $\rho$ are related by $P=k\rho$ $(k\le 1)$. The long time limit of the solutions are black holes whose horizon structures depend on the value of $k$. The $k=1$ solution is the Banados-Teitelboim-Zanelli black hole metric in the long time static limit, while the $k<1$ solutions give other, `hairy' black hole metrics in this limit.Comment: 8 pages, RevTeX (to appear in Phys. Rev. D) References to Mann and Ross, and Mann, Chan and Chan adde

Chern-Simons Gravity and Holographic Anomalies

We present a holographic treatment of Chern-Simons (CS) gravity theories in odd dimensions. We construct the associated holographic stress tensor and calculate the Weyl anomalies of the dual CFT.Comment: Added references, and minor corrections. 21 pages, havmac, no figure