Metric-based Hamiltonians, null boundaries, and isolated horizons

TL;DR

This paper extends the Hamiltonian formulation of general relativity to null boundaries, enabling the study of isolated horizons and deriving their first law from off-shell variations, with results consistent with known energy definitions.

Contribution

It introduces a generalized Brown-York formalism for null boundaries, deriving the first law of isolated horizon mechanics from off-shell Hamiltonian variations.

Findings

Derived the first law of isolated horizon mechanics from Hamiltonian variation.

Defined energy and angular momentum for isolated horizons consistent with Komar and ADM/Bondi quantities.

Compared isolated horizon thermodynamics with Brown-York thermodynamics.

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.