Boundary conditions for Black Holes using the Ashtekar Isolated and Dynamical Horizons formalism

TL;DR

This paper develops boundary conditions for black holes using the Ashtekar isolated and dynamical horizons formalism, deriving Hamiltonian equations and applying them to various spacetime examples.

Contribution

It introduces a method to generate boundary conditions for black holes within the Ashtekar formalism and derives Hamiltonian equations using both direct and standard approaches.

Findings

Derived Hamiltonian equations for general relativity with horizon boundary conditions

Applied formalism to Kerr, Schwarzschild-De Sitter, McVittie, and Vaiyda spacetimes

Compared results with existing models and examples

Abstract

Isolated and Dynamical horizons are used to generate boundary conditions upon the lapse and shift vectors. Numerous results involving the Hamiltonian of General relativity are derived, including a self-contained derivation of the Hamiltonian equations of general relativity using both a direct 'brute force' method of directly computing Lie derivatives, as well as the standard Hamil- tonian approach. Conclusions are compared to numerous examples, including the Kerr, Schwarzschild-De Sitter, McVittie, and Vaiyda spacetimes.