A 3+1 perspective on null hypersurfaces and isolated horizons

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.Comment: numerous corrections, references added, 224 pages, 21 figures, Physics Reports, in pres

Angular momentum conservation for dynamical black holes

“…In fact, these conditions do hold on a null trapping horizon under the dominant energy condition, as shown below. This treatment also turns out to be compatible with the definition of weakly isolated horizon [29,31,32,33,34,35]. Consider a null trapping horizon, assumed henceforth in this section to be given by θ + ∼ = 0.…”

“…It is difficult to find a practical formalism describing all cases without some degeneracy arising in the null case, but the dual-null formalism appears to be adequate; one fixes the additional freedom in the null case by (74). In particular, no additional conditions need be imposed on the horizon itself, as compared with the variety of definitions of isolated horizons [27,28,29,30,31,32,33,34,35]. Numerical evidence that such horizons exist in practice has been given by Dreyer et al [34], who looked for and found approximately null trapping horizons, under the name non-expanding horizons.…”

“…Such equations actually hold not just on a trapping horizon, but anywhere in the space-time, energy conservation reducing at null infinity to the Bondi energy equation. Contemporaneously, Ashtekar et al and others [27,28,29,30,31,32,33,34,35] developed a theory of null trapping horizons with various additional conditions, under the names non-rotating isolated horizons, non-expanding horizons, weakly isolated horizons, rigidly rotating horizons and (strongly) isolated horizons. Each is intended to capture the idea that the black hole is quasi-stationary in some sense.…”

“…The dual-null foliation becomes non-unique for a null hypersurface, with ω becoming non-unique. This is reflected in the fact that different 1-forms were used for isolated horizons [27,28,29,30,31,32,33,34,35] and dynamical horizons [36,37,38]. Neither 1-form is necessarily preserved along a null trapping horizon, but a certain linear combination is so preserved.…”

Spacetimes containing slowly evolving horizons

“…In the last decade an alternative approach to this problem has been proposed. The isolated horizon programme (see [2,3,4] for reviews) explicitly studies the quasilocal conditions under which a black hole may be said to be in equilibrium with its (possibly dynamic) surroundings and then studies consequences of those conditions -as opposed to studying equilibrium spacetimes which also contain black holes.…”