In this talk we address some aspects of Einstein gravity + matter in asymptotically AdS spacetimes: 1) The solution and reduced phase space dynamics of 2+1 dim Einstein gravity with a cylindrical (wormhole) topology within the Hamiltonian formulation of General Relativity in the maximal slicing gauge;  2) The dynamics of a probe scalar field in the background of the solution in (1) that corresponds after a coordinate transformation to the extended static BTZ black hole solution. The bulk Hamiltonian that evolves the scalar field is a finite Hermitian operator expressed entirely in terms of the modes of the generalized free fields on the two asymptotic boundaries of the AdS spacetime. We present correlation functions in the Hartle-Hawking vacuum in the maximal slicing gauge which do not indicate any thermal behavior.  The thermal description emerges only after a singular coordinate transformation to BTZ coordinates that are only valid outside the horizon of the BH.