  We study the thermodynamics of an uncharged, non-rotating BTZ black hole. In
addition to the thermal properties, we are interested in the phase transition
between two locally thermodynamic stable phases which may arise from different
areas of validity for the ads/CFT correspondence. We report on the calculation
of entanglement entropies for quantum field theory defined on two dimensional
Minkowski space by using its geometric realization as the hyperbolic plane. For
free quantum fields in one or more spatial dimensions, we show that a geometric
approach reproduces some of the recent results found by studying explicit field
correlators. Using this technique, we further explore the geometrical behavior
when gravity is turned on and study how it affects thermalization properties
such as decoherence times and out-of-equilibrium phase transitions. We show
that the microstates for both phases can be described by three macroscopic
quantities, thereby allowing us to derive a first order equation for these
variables, whereupon they solve this equation at extremality. Superluminal
signalling is ruled out if and only if one assumes that no superradiance occurs
during evaporation; but it is allowed within the thin membrane paradigm already
introduced earlier in this context. Our result holds independent on how long it
takes for radiation to escape from near extremal configurations; while there
are reasonable expectations regarding this time scale our derivation is
completely model-independent.
