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Isolated Horizons: the Classical Phase Space

A. Ashtekar, A. Corichi, K. Krasnov

TL;DR

Isolated horizons provide a quasi-local setting for black hole thermodynamics in non-stationary space-times. The authors develop a Hamiltonian formulation based on SL(2,$\mathbb{C}$) self-dual variables, showing that horizon boundary conditions induce a universal Chern–Simons boundary term with coefficient $\frac{a_\Delta}{4\pi}$, yielding a horizon symplectic structure encoded by a U(1) connection $V$ on $\Delta$. This boundary structure, together with the bulk constraints, furnishes a natural arena for quantization and the derivation of generalized black hole laws and entropy, with the entropy ultimately depending on the horizon area $a_\Delta$ via $S_{\rm bh}=a_\Delta/(4\ell_P^2)$ (up to units). The framework elegantly accommodates Maxwell and dilatonic hair and extends to cosmological horizons, with potential applications to numerical relativity and higher-derivative theories.

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.

Isolated Horizons: the Classical Phase Space

TL;DR

Isolated horizons provide a quasi-local setting for black hole thermodynamics in non-stationary space-times. The authors develop a Hamiltonian formulation based on SL(2,) self-dual variables, showing that horizon boundary conditions induce a universal Chern–Simons boundary term with coefficient , yielding a horizon symplectic structure encoded by a U(1) connection on . This boundary structure, together with the bulk constraints, furnishes a natural arena for quantization and the derivation of generalized black hole laws and entropy, with the entropy ultimately depending on the horizon area via (up to units). The framework elegantly accommodates Maxwell and dilatonic hair and extends to cosmological horizons, with potential applications to numerical relativity and higher-derivative theories.

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.

Paper Structure

This paper contains 19 sections, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Review of connection dynamics
  3. Self-dual connections
  4. Real connections
  5. Boundary conditions for isolated horizons
  6. Definition
  7. Symmetries and Gauge on $\Delta$
  8. Consequences of boundary conditions
  9. Action and phase space
  10. Action Principle
  11. Legendre transform, constraints and gauge
  12. Passage to real variables
  13. Inclusion of hair: electric, magnetic and dilatonic charges
  14. Maxwell Fields
  15. Dilatonic Couplings
  16. ...and 4 more sections

Figures (2)

  • Figure 1: A typical gravitational collapse. The portion $\Delta$ of the horizon at late times is isolated. The space-time $\mathcal{M}$ of interest is the triangular region bounded by $\Delta$, ${\cal I}^+$ and a partial Cauchy slice $M$.
  • Figure 2: A spherical star of mass $M$ undergoes collapse. Later, a spherical shell of mass $\delta{M}$ falls into the resulting black hole. While $\Delta_1$ and $\Delta_2$ are both isolated horizons, only $\Delta_2$ is part of the event horizon.