Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity

Abstract

Abstract. We review the black hole entropy calculation in the framework of Loop Quantum Gravity based on the quasi-local definition of a black hole encoded in the isolated horizon formalism. We show, by means of the covariant phase space framework, the appearance in the conserved symplectic structure of a boundary term corresponding to a Chern-Simons theory on the horizon and present its quantization both in the U (1) gauge fixed version and in the fully SU (2) invariant one. We then describe the boundary degrees of freedom counting techniques developed for an infinite value of the Chern-Simons level case and, less rigorously, for the case of a finite value. This allows us to perform a comparison between the U (1) and SU (2) approaches and provide a state of the art analysis of their common features and different implications for the entropy calculations. In particular, we comment on different points of view regarding the nature of the horizon degrees of freedom and the role played by the Barbero-Immirzi parameter. We conclude by presenting some of the most recent results concerning possible observational tests for theory.

CFT/gravity correspondence on the isolated horizon

“…The dimension of the resulting Chern-Simons Hilbert space of the punctured two-sphere has been computed in different statistical mechanical ensembles and shown to generate a leading term linear in the IH area [5][6][7][8][9] (see [10] for a review) which matches the Bekenstein-Hawking formula for a specific numerical real value γ 0 of the Barbero-Immirzi parameter γ . However, this feature of the standard LQG black hole entropy calculation has raised concern in the past years.…”

Gibbs paradox, black hole entropy, and the thermodynamics of isolated horizons

“…in order to couple them again [7,8,17,22,23]. The level of the CS-theory is denoted by j p1 ), ..., (p N , j pN )} runs over all finite puncture sets andF i ab is the quantization of the curvature 2-form of the Ashtekar-connection A i a being pulled-back to the horizon surface S and is therefore the curvature of the SU (2) CS-connection.…”

“…Usually, but not necessarily, one considers the level to scale with the macroscopical classical horizon area k ∝ a H /ℓ 2 p setting k → ∞. The case where k is fixed can be found in [9,17,21,23].…”

Black hole entropy from loop quantum gravity in higher dimensions

“…(5) is the analogue of the isolated horizon boundary condition F IJ (A) ∝ e I ∧ e J familiar from the 3 + 1-dimensional treatment [8]. In fact, (5) can be rewritten in the form F IJ (A) ∝ e I ∧ e J in 3 + 1 dimensions [9].…”

“…Here, the calculation of black hole entropy is translated to the calculation of the entropy of isolated horizons, see [8] for a review. The advantage of this viewpoint is that isolated horizons provide local definitions of the relevant horizons (e.g.…”