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Isolated Horizons: Hamiltonian Evolution and the First Law

Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan

TL;DR

The paper generalizes black hole mechanics by formulating a quasi-local framework of isolated horizons that can be distorted and rotating, removing the requirement of global Killing fields. Using a real tetrad–Lorentz action and a covariant phase space, it shows that a Hamiltonian evolution exists if and only if the first law holds at the horizon, thereby linking horizon dynamics directly to energy, area, and charge variations. The authors define a local horizon mass in Einstein–Maxwell theory, extend the first law to Yang–Mills and dilatonic couplings, and demonstrate how horizon quantities (area, charges, potentials) determine the evolution and energy without reference to infinity. They also reveal an infinite family of first laws corresponding to different permissible horizon evolutions, with canonical choices recovering familiar static results and providing a framework for analyzing strongly gravitating, dynamical black holes, including numerical simulations. The work thus unifies and extends black hole thermodynamics to non-stationary, distorted horizons in multiple matter couplings, offering practical tools for strong-field physics and further theoretical insights into horizon energetics.

Abstract

A framework was recently introduced to generalize black hole mechanics by replacing stationary event horizons with isolated horizons. That framework is significantly extended. The extension is non-trivial in that not only do the boundary conditions now allow the horizon to be distorted and rotating, but also the subsequent analysis is based on several new ingredients. Specifically, although the overall strategy is closely related to that in the previous work, the dynamical variables, the action principle and the Hamiltonian framework are all quite different. More importantly, in the non-rotating case, the first law is shown to arise as a necessary and sufficient condition for the existence of a consistent Hamiltonian evolution. Somewhat surprisingly, this consistency condition in turn leads to new predictions even for static black holes. To complement the previous work, the entire discussion is presented in terms of tetrads and associated (real) Lorentz connections.

Isolated Horizons: Hamiltonian Evolution and the First Law

TL;DR

The paper generalizes black hole mechanics by formulating a quasi-local framework of isolated horizons that can be distorted and rotating, removing the requirement of global Killing fields. Using a real tetrad–Lorentz action and a covariant phase space, it shows that a Hamiltonian evolution exists if and only if the first law holds at the horizon, thereby linking horizon dynamics directly to energy, area, and charge variations. The authors define a local horizon mass in Einstein–Maxwell theory, extend the first law to Yang–Mills and dilatonic couplings, and demonstrate how horizon quantities (area, charges, potentials) determine the evolution and energy without reference to infinity. They also reveal an infinite family of first laws corresponding to different permissible horizon evolutions, with canonical choices recovering familiar static results and providing a framework for analyzing strongly gravitating, dynamical black holes, including numerical simulations. The work thus unifies and extends black hole thermodynamics to non-stationary, distorted horizons in multiple matter couplings, offering practical tools for strong-field physics and further theoretical insights into horizon energetics.

Abstract

A framework was recently introduced to generalize black hole mechanics by replacing stationary event horizons with isolated horizons. That framework is significantly extended. The extension is non-trivial in that not only do the boundary conditions now allow the horizon to be distorted and rotating, but also the subsequent analysis is based on several new ingredients. Specifically, although the overall strategy is closely related to that in the previous work, the dynamical variables, the action principle and the Hamiltonian framework are all quite different. More importantly, in the non-rotating case, the first law is shown to arise as a necessary and sufficient condition for the existence of a consistent Hamiltonian evolution. Somewhat surprisingly, this consistency condition in turn leads to new predictions even for static black holes. To complement the previous work, the entire discussion is presented in terms of tetrads and associated (real) Lorentz connections.

Paper Structure

This paper contains 26 sections, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of isolated horizons and the zeroth law
  3. Non-expanding Horizons
  4. Weakly Isolated horizons
  5. Electromagnetic Field
  6. Other definitions and remarks
  7. Action
  8. Preliminaries
  9. Internal boundary $\Delta$
  10. Covariant Phase space
  11. Hamiltonian evolution and the first law
  12. Yang-Mills Field
  13. Preliminaries
  14. Action and Phase Space
  15. Hamiltonian and First Law
  16. ...and 11 more sections

Figures (1)

  • Figure 1: The region of space-time ${\mathcal{M}}$ under consideration has an internal boundary $\Delta$ and is bounded by two partial Cauchy surfaces $M^{\pm}$ which intersect $\Delta$ in the $2$-spheres $S^{\pm}$ and extend to spatial infinity $i^{o}$.