2+1 dimensional gravity in AAdS spacetimes with spatial wormhole slices: Reduced phase space dynamics and the BTZ black hole

Abstract: We solve Einstein's equations with negative cosmological constant in $2+1$ dimensions in the Hamiltonian formulation. The spacetime has the topology of $\Sigma \times \mathbf{R}$ where $\mathbf{R}$ corresponds to the time direction and $\Sigma$ is a cylinder $\mathbf{R} \times \mathbf{S}^1$ and the spacetime metric satisfies asymptotically AdS (AAdS) boundary conditions. We address the question of gauge invariance by fixing the maximal slicing and spatial harmonic gauge conditions and demonstrate that there are no residual small diffeomorphisms in this gauge. We explicitly solve the Hamiltonian and momentum constraints, and the gauge conditions to obtain a two dimensional reduced phase space. For simplicity, and with the BTZ black hole in mind, we restrict the solution of the momentum constraints to be independent of $\mathbf{S}^1$. In AAdS spacetimes besides the standard Wheeler-deWitt equations there is a Schroedinger equation corresponding to the boundary ADM Hamiltonian. We express this Hamiltonian in terms of the reduced phase space variables and discuss its classical solutions and quantization. We exhibit the wave functions and a continuous positive energy spectrum. Each energy eigenvalue $E$ corresponds to a BTZ black hole of mass $M=E/2$. This identification is based on the fact that the classical solution of the reduced phase space dynamics gives rise to a spacetime that is related to the two-sided BTZ black hole by a diffeomorphism.