11 research outputs found
Quantization of Black Holes in the Wheeler-DeWitt Approach
We discuss black hole quantization in the Wheeler-DeWitt approach. Our
consideration is based on a detailed investigation of the canonical formulation
of gravity with special considerations of surface terms. Since the phase space
of gravity for non-compact spacetimes or spacetimes with boundaries is
ill-defined unless one takes boundary degrees of freedom into account, we give
a Hamiltonian formulation of the Einstein-Hilbert-action as well as a
Hamiltonian formulation of the surface terms. It then is shown how application
to black hole spacetimes connects the boundary degrees of freedom with
thermodynamical properties of black hole physics. Our treatment of the surface
terms thereby naturally leads to the Nernst theorem. Moreover, it will produce
insights into correlations between the Lorentzian and the Euclidean theory.
Next we discuss quantization, which we perform in a standard manner. It is
shown how the thermodynamical properties can be rediscovered from the quantum
equations by a WKB like approximation scheme. Back reaction is treated by going
beyond the first order approximation. We end our discussion by a rigorous
investigation of the so-called BTZ solution in 2+1 dimensional gravity.Comment: 28 pages, 2 figure
Semiclassical Black Hole States and Entropy
We discuss semiclassical states in quantum gravity corresponding to
Schwarzschild as well as Reissner Nordstr\"om black holes. We show that reduced
quantisation of these models is equivalent to Wheeler-DeWitt quantisation with
a particular factor ordering. We then demonstrate how the entropy of black
holes can be consistently calculated from these states. While this leads to the
Bekenstein-Hawking entropy in the Schwarzschild and non-extreme
Reissner-Nordstr\"om cases, the entropy for the extreme Reissner-Nordstr\"om
case turns out to be zero.Comment: Revtex, 15 pages, some clarifying comments and additional references
included, to appear in Phys. Rev.
On Modular Invariance and 3D Gravitational Instantons
We study the modular transformation properties of Euclidean solutions of 3D
gravity whose asymptotic geometry has the topology of a torus. These solutions
represent saddle points of the grand canonical partition function with an
important example being the BTZ black hole, and their properties under modular
transformations are inherited from the boundary conformal field theory encoding
the asymptotic dynamics. Within the Chern Simons formulation, classical
solutions are characterised by specific holonomies describing the wrapping of
the gauge field around cycles of the torus. We find that provided these
holonomies transform in an appropriate manner, there exists an associated
modular invariant grand canonical partition function and that the spectrum of
saddle points naturally includes a thermal bath in as discussed by
Maldacena and Strominger. Indeed, certain modular transformations can naturally
be described within classical bulk dynamics as mapping between different
foliations with a "time" coordinate along different cycles of the asymptotic
torus.Comment: 14 pages, RevTeX, typos corrected, to appear in Phys. Rev.