Lorentz Invariance in Shape Dynamics

Shape dynamics is a reframing of canonical general relativity in which time reparametrization invariance is "traded" for a local conformal invariance. We explore the emergence of Lorentz invariance in this model in three contexts: as a maximal symmetry, an asymptotic symmetry, and a local invariance.Comment: v2: discussion of light cone structure added; minor typos fixed; 14 page

Quantum Bubble Dynamics in 2+1 Dimensional Gravity I: Geometrodynamic Approach

The Dirac quantization of a 2+1 dimensional bubble is performed. The bubble consists of a string forming a boundary between two regions of space-time with distinct geometries. The ADM constraints are solved and the coupling to the string is introduced through the boundary conditions. The wave functional is obtained and the quantum uncertainty in the radius of the ring is calculated; this uncertainty becomes large at the Planck scale.Comment: 14 pages, Latex (\cite typos corrected

Three-Dimensional Gravity and String Ghosts

It is known that much of the structure of string theory can be derived from three-dimensional topological field theory and gravity. We show here that, at least for simple topologies, the string diffeomorphism ghosts can also be explained in terms of three-dimensional physics.Comment: 6 page

The Modular Group, Operator Ordering, and Time in (2+1)-Dimensional Gravity

A choice of time-slicing in classical general relativity permits the construction of time-dependent wave functions in the ``frozen time'' Chern-Simons formulation of $(2+1)$-dimensional quantum gravity. Because of operator ordering ambiguities, however, these wave functions are not unique. It is shown that when space has the topology of a torus, suitable operator orderings give rise to wave functions that transform under the modular group as automorphic functions of arbitrary weights, with dynamics determined by the corresponding Maass Laplacians on moduli space.Comment: 8 pages, LaTe

The Spin Holonomy Group In General Relativity

It has recently been shown by Goldberg et al that the holonomy group of the chiral spin-connection is preserved under time evolution in vacuum general relativity. Here, the underlying reason for the time-independence of the holonomy group is traced to the self-duality of the curvature 2-form for an Einstein space. This observation reveals that the holonomy group is time-independent not only in vacuum, but also in the presence of a cosmological constant. It also shows that once matter is coupled to gravity, the "conservation of holonomy" is lost. When the fundamental group of space is non-trivial, the holonomy group need not be connected. For each homotopy class of loops, the holonomies comprise a coset of the full holonomy group modulo its connected component. These cosets are also time-independent. All possible holonomy groups that can arise are classified, and examples are given of connections with these holonomy groups. The classification of local and global solutions with given holonomy groups is discussed.Comment: 21 page

Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations

We extend the discrete Regge action of causal dynamical triangulations to include discrete versions of the curvature squared terms appearing in the continuum action of (2+1)-dimensional projectable Horava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are 2-spheres, we employ Markov chain Monte Carlo simulations to study the path integral defined by this extended discrete action. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry, and we present preliminary evidence for the consistency of these phases with solutions to the equations of motion of classical Horava-Lifshitz gravity. Apparently, the phase diagram contains a phase transition between a time-dependent de Sitter-like phase and a time-independent phase. We speculate that this phase transition may be understood in terms of deconfinement of the global gravitational Hamiltonian integrated over a spatial 2-sphere.Comment: 24 pages; 10 figure

Global constants in (2+1)--dimensional gravity

The extended conformal algebra (so)(2,3) of global, quantum, constants of motion in 2+1 dimensional gravity with topology R x T^2 and negative cosmological constant is reviewed. It is shown that the 10 global constants form a complete set by expressing them in terms of two commuting spinors and the Dirac gamma matrices. The spinor components are the globally constant holonomy parameters, and their respective spinor norms are their quantum commutators.Comment: 14 pages, to appear in Classical and Quantum Gravity, Spacetime Safari: Essays in Honor of Vincent Moncrief on the Classical Physics of Strong Gravitational Field

Comparative Quantizations of (2+1)-Dimensional Gravity

We compare three approaches to the quantization of (2+1)-dimensional gravity with a negative cosmological constant: reduced phase space quantization with the York time slicing, quantization of the algebra of holonomies, and quantization of the space of classical solutions. The relationships among these quantum theories allow us to define and interpret time-dependent operators in the ``frozen time'' holonomy formulation.Comment: 24 pages, LaTeX, no figure

Geometrical Finiteness, Holography, and the BTZ Black Hole

We show how a theorem of Sullivan provides a precise mathematical statement of a 3d holographic principle, that is, the hyperbolic structure of a certain class of 3d manifolds is completely determined in terms of the corresponding Teichmuller space of the boundary. We explore the consequences of this theorem in the context of the Euclidean BTZ black hole in three dimensions.Comment: 6 pages, Latex, Version to appear in Physical Review Letter

Black Hole Entropy from Conformal Field Theory in Any Dimension

Restricted to a black hole horizon, the ``gauge'' algebra of surface deformations in general relativity contains a Virasoro subalgebra with a calculable central charge. The fields in any quantum theory of gravity must transform accordingly, i.e., they must admit a conformal field theory description. Applying Cardy's formula for the asymptotic density of states, I use this result to derive the Bekenstein-Hawking entropy. This method is universal---it holds for any black hole, and requires no details of quantum gravity---but it is also explicitly statistical mechanical, based on counting microscopic states.Comment: 9 pages, LaTeX, no figures. Slightly shortened and polished for journal; no significant changes in substanc