Entropy of Apparent Horizons and Semiclassical Physics

1. Entropy of Apparent Horizons and Semiclassical Physics AD gr-qc/0505017, AD, HTgr-qc/0602006

2. Origin of Horizon Entropy is in Semiclassical Physics • We donot know whether a Planck area black hole exists. • Causality is a very classical concept, and entropy, temperature and even Hawking radiation have origins in semiclassical physics. • Kinematic coherent states of LQG (non-perturbative gravity) by construction, are peaked at a classical spatial slice. The states should eventually be derived in the physical Hilbert space and evolve in time using the Hamiltonian constraint. Initial value problem

3. The classical variables • Complexification of phase space variables • (x-ip) Variables are defined for a graph with holonomy along edges and momentum along dual 2-surfaces. Thus there is a classical discretisation where vertices sample points of the continuum spatial slice with the length along the edges measured by the classical metric. A typical slicing of the Schwarzschild space-time is in the coordinates where the spatial slices have flat intrinsic curvature

4. Regularisation due to the discretisation

5. The Coherent state Nicely peaked for t=10^(-10), so horizon area is atleast 10^(10) Planck length square. Beyond these semiclassical fluctuations are rather large, and the coherent states no-longer describe classical + quantum corrections. Minumum of area ->Resolution of classical singularity

6. Identifying the apparent horizon Introduces correlations across the apparent horizon and a density matrix calculated by tracing over the states inside the horizon gives the entropy as the logarithm of the number of ways to induce horizon area

7. Entropy Counting • If the edges induce classical area symmetrically, then If one sums over the number of ways to induce horizon area by all possible edges the entropy is