Quantum Geometry and Black Hole Entropy

Abstract

A `black hole sector' of non-perturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large non-rotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula S = A/4*l_p^2. With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal.Comment: Revtex, 8 pages, 1 figur

Inflationary universe in loop quantum cosmology

“…Likewise, LQC is a canonical quantization of homogeneous spacetimes based upon techniques used in LQG. In LQC, due to the symmetries of the homogeneous and isotropic spacetime, the phase space structure is simplified, i.e., the connection is determined by a single quantity labeled c and likewise the triad is determined by a parameter p. The variables c and p are canonically conjugate with Poisson bracket {c, p} = γκ/3, where κ = 8πG (G is the Newton's gravitational constant) and γ is the dimensionless Barbero-Immirzi parameter which is set to be γ ≈ 0.2375 by the black hole thermodynamics in LQG [19]. For the spatially flat model of cosmology, the new variables have the relations with the metric components of the Friedmann-Robertson-Walker (FRW) universe as…”

Testing Holographic Principle from Logarithmic and Higher Order Corrections to Black Hole Entropy

“…But there has been no general proof for more general contexts yet. 10 There are some mismatches in c eff and ∆ eff with other alternative approaches [16,17]. But, presumably similar logarithmic corrections would be obtained also if the vacuum is chosen properly [45].…”

“…where A n is the area eigenvalue of the horizon with n punctures each of which carries a spin 1/2, and N (E(A n )) is the degeneracy of the energy level E(A n ) [10,41]. In the very large p limit, this may be approximated as an integral…”

“…But one can still assume safely δE ≪ any available energy fluctuation, by considering large black holes. Moreover, the relation (2.8) between the micro-canonical entropy and the canonical entropy S c shows a perturbative correction [25] to the BH entropy 3 [23,24] 10) where A is the horizon area, and G is the Newton's constant; when there is rotational degrees of freedom, one can generalize straightforwardly to the grand-canonical partition function and its associated density of states, but this will not be considered in this paper.…”

“…b Quantum Geometry Approach of the Horizon For four-dimensional non-rotating black holes with or without a cosmological constant or electric, magnetic, and dilatonic charges, there is an alternative derivation of black hole entropy from the quantum geometry approach of Ashtekar et al that counts the boundary states of a three-dimensional Chern-Simons theory at the horizon [10,57,41]. There were some confusions about the logarithmic-correction term in this approach, by identifying the entropy in this approach with the usual (micro-canonical) entropy S = lnΩ(E) and generalizing this result in d = 4 to other dimensions [40]; as a result of this, a wrong behavior of central charge c (or P ) in the Cardy's formula (4.16) has been speculated in Ref.…”