A 3+1 perspective on null hypersurfaces and isolated horizons

TL;DR

This paper integrates the null hypersurface formalism with the 3+1 spacetime slicing approach to better understand black hole horizons, linking isolated horizons, membrane paradigm, and numerical relativity.

Contribution

It reformulates the isolated horizon and related structures within a 3+1 framework, bridging null and spatial geometries for improved black hole analysis.

Findings

Reformulation of isolated horizons in 3+1 formalism

Explicit geometrical objects in terms of 3+1 quantities

Application to Schwarzschild and Kerr spacetime slicings

Abstract

The isolated horizon formalism recently introduced by Ashtekar et al. aims at providing a quasi-local concept of a black hole in equilibrium in an otherwise possibly dynamical spacetime. In this formalism, a hierarchy of geometrical structures is constructed on a null hypersurface. On the other side, the 3+1 formulation of general relativity provides a powerful setting for studying the spacetime dynamics, in particular gravitational radiation from black hole systems. Here we revisit the kinematics and dynamics of null hypersurfaces by making use of some 3+1 slicing of spacetime. In particular, the additional structures induced on null hypersurfaces by the 3+1 slicing permit a natural extension to the full spacetime of geometrical quantities defined on the null hypersurface. This 4-dimensional point of view facilitates the link between the null and spatial geometries. We proceed by reformulating the isolated horizon structure in this framework. We also reformulate previous works, such as Damour's black hole mechanics, and make the link with a previous 3+1 approach of black hole horizon, namely the membrane paradigm. We explicit all geometrical objects in terms of 3+1 quantities, putting a special emphasis on the conformal 3+1 formulation. This is in particular relevant for the initial data problem of black hole spacetimes for numerical relativity. Illustrative examples are provided by considering various slicings of Schwarzschild and Kerr spacetimes.