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Metric-based Hamiltonians, null boundaries, and isolated horizons

Ivan S. Booth

TL;DR

The paper extends the metric-based Hamiltonian formulation of general relativity to regions with null boundaries, enabling a quasilocal treatment of isolated horizons. By performing a Legendre transform of the Einstein–Maxwell action and incorporating a null inner boundary term, it derives a first law of isolated horizon mechanics from Hamiltonian variation, with horizon energy, angular momentum, and charge defined on the horizon and related to Komar/ADM quantities in stationary spacetimes. A Kerr–Newman calibration is used to fix the horizon data–dependent functions κ_ℓ, Ω_φ, and Φ_ℓ so that NQLE on the horizon matches the global energy, while a comparison with Brown–York quasilocal energy highlights key differences in boundary terms and thermodynamic interpretation. The work clarifies how null boundaries influence quasilocal energy definitions and demonstrates how horizon thermodynamics can be consistently derived within a metric-based Hamiltonian framework, connecting to broader results in isolated horizon theory and Komar energies.

Abstract

We extend the quasilocal (metric-based) Hamiltonian formulation of general relativity so that it may be used to study regions of spacetime with null boundaries. In particular we use this generalized Brown-York formalism to study the physics of isolated horizons. We show that the first law of isolated horizon mechanics follows directly from the first variation of the Hamiltonian. This variation is not restricted to the phase space of solutions to the equations of motion but is instead through the space of all (off-shell) spacetimes that contain isolated horizons. We find two-surface integrals evaluated on the horizons that are consistent with the Hamiltonian and which define the energy and angular momentum of these objects. These are closely related to the corresponding Komar integrals and for Kerr-Newman spacetime are equal to the corresponding ADM/Bondi quantities. Thus, the energy of an isolated horizon calculated by this method is in agreement with that recently calculated by Ashtekar and collaborators but not the same as the corresponding quasilocal energy defined by Brown and York. Isolated horizon mechanics and Brown-York thermodynamics are compared.

Metric-based Hamiltonians, null boundaries, and isolated horizons

TL;DR

The paper extends the metric-based Hamiltonian formulation of general relativity to regions with null boundaries, enabling a quasilocal treatment of isolated horizons. By performing a Legendre transform of the Einstein–Maxwell action and incorporating a null inner boundary term, it derives a first law of isolated horizon mechanics from Hamiltonian variation, with horizon energy, angular momentum, and charge defined on the horizon and related to Komar/ADM quantities in stationary spacetimes. A Kerr–Newman calibration is used to fix the horizon data–dependent functions κ_ℓ, Ω_φ, and Φ_ℓ so that NQLE on the horizon matches the global energy, while a comparison with Brown–York quasilocal energy highlights key differences in boundary terms and thermodynamic interpretation. The work clarifies how null boundaries influence quasilocal energy definitions and demonstrates how horizon thermodynamics can be consistently derived within a metric-based Hamiltonian framework, connecting to broader results in isolated horizon theory and Komar energies.

Abstract

We extend the quasilocal (metric-based) Hamiltonian formulation of general relativity so that it may be used to study regions of spacetime with null boundaries. In particular we use this generalized Brown-York formalism to study the physics of isolated horizons. We show that the first law of isolated horizon mechanics follows directly from the first variation of the Hamiltonian. This variation is not restricted to the phase space of solutions to the equations of motion but is instead through the space of all (off-shell) spacetimes that contain isolated horizons. We find two-surface integrals evaluated on the horizons that are consistent with the Hamiltonian and which define the energy and angular momentum of these objects. These are closely related to the corresponding Komar integrals and for Kerr-Newman spacetime are equal to the corresponding ADM/Bondi quantities. Thus, the energy of an isolated horizon calculated by this method is in agreement with that recently calculated by Ashtekar and collaborators but not the same as the corresponding quasilocal energy defined by Brown and York. Isolated horizon mechanics and Brown-York thermodynamics are compared.

Paper Structure

This paper contains 17 sections, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Set-up
  3. The 3D perspective
  4. The 4D perspective
  5. Review of isolated horizons
  6. Definitions
  7. Constant quantities on the horizons
  8. The metric-based Hamiltonian
  9. Physical quantities and thermodynamics
  10. The first law of RRH mechanics
  11. Physical interpretation and calibration
  12. Comparison with Brown-York
  13. Quasilocal energy
  14. Thermodynamics
  15. Discussion
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The region $\Sigma_t$ with its boundaries and normals.
  • Figure 2:

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3