Isolated horizons: The classical phase space

Abstract

A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are "isolated" near future time-like infinity or for a finite time interval. The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration. A precise notion of non-rotating isolated horizons is formulated to capture these ideas. With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon. The required term turns out to be precisely the Chern-Simons action for the self-dual connection. The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure. We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics. In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons. It may also have applications in classical general relativity, particularly in the investigation of of analytic issues that arise in the numerical studies of black hole collisions.

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

“…Note that this result applies to the four-dimensional SchwarzschildAdS as well as the Schwarzschild black holes since the formulation depends crucially on the local properties of the horizon [57]. However, when we compute the usual micro-canonical entropy S = lnΩ(E), there is a sharp difference due to the additional term of (7.25).…”

“…where S QG = lnN (E(p)), (7.26) which is the entropy defined in quantum geometry approach [10,57,41]. As we shall see, the additional term in (7.25) is the key ingredient which resolves the above mentioned confusions.…”

“…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.…”

Black hole entropy from loop quantum gravity in higher dimensions

“…We will neglect the detailed treatment of spatial infinity in this paper, as it is not relevant for the discussion, see e.g. [15]. The main result is that one can perform a phase space extension from the Lorentzian ADM phase space [16] in D + 1 spacetime dimensions to a canonical pair consisting of an SO(D +1) connection A aIJ and its conjugate momentum π aIJ .…”

“…In order to be able to quantize them, we will select a maximally commuting subset adapted to the puncturing spin network in the next section, that is, we perform gauge unfixing [28]. This subset does however not form a closing algebra with the Gauß constraints, since the smearing functions Λ IJ are not constant on H. We thus further restrict to constant Λ IJ on H and note that also in the U(1) framework in 3 + 1 dimensions [15,24], the restriction to constant Λ IJ on H becomes necessary, however for different reasons [7].…”