A 3+1 perspective on null hypersurfaces and isolated horizons

TL;DR

<3-5 sentence high-level summary>This work presents a unified four-dimensional treatment of null hypersurfaces by embedding the 3+1 foliation within the null-horizon geometry, thereby connecting the intrinsic horizon structures used in the isolated horizon framework with the 3+1 formalism central to numerical relativity. By constructing a 3+1-induced foliation of a null hypersurface ${\mathcal{H}}$ and extending key objects like the Weingarten map, the second fundamental form, and the rotation/Hajicek forms to full spacetime, the authors derive explicit 4D expressions that link null geometry to spatial quantities and boundary conditions. They recover and rephrase Damour’s Navier-Stokes equation, the null Raychaudhuri and tidal equations, and present a clear pathway to implement quasi-local horizon data in initial-data problems, illustrated with Schwarzschild and Kerr examples. The NEH (non-expanding horizon) analysis then grounds the quasi-local horizon concept in a local, foliation-independent setting, underpinning practical horizon modeling in numerical relativity and quantum gravity contexts with explicit 3+1–to–null translations.

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.

Figures (20)

• Figure 1.1: Geometrical construction showing that $\bm{\mathcal{L}}_{\bm{\bm{\ell}}}\, \bm{v} \in{\mathcal{T}}({\mathcal{S}}_t)$ for any vector $\bm{v}$ tangent to the 2-surface ${\mathcal{S}}_t$: on ${\mathcal{S}}_t$, a vector can be identified by a infinitesimal displacement between two points, $p$ and $q$ say. These points are transported onto the neighbouring surface ${\mathcal{S}}_{t+\delta t}$ along the field lines of the vector field $\bm{\ell}$ (thin lines on the figure) by the diffeomorphism $\phi_{\delta t}$ associated with $\bm{\ell}$: the displacement between $p$ and $\phi_{\delta t}(p)$ is the vector $\delta t\, \bm{\ell}$. The couple of points $(\phi_{\delta t}(p),\phi_{\delta t}(q))$ defines the vector $\phi_{\delta t} \bm{v}(t)$ tangent to ${\mathcal{S}}_{t+\delta t}$. The Lie derivative of $\bm{v}$ along $\bm{\ell}$ is then defined by the difference between the value of the vector field $\bm{v}$ at the point $\phi_{\delta t}(p)$, i.e. $\bm{v}(t+\delta t)$, and the vector transported from ${\mathcal{S}}_t$ along $\bm{\ell}$'s field lines, i.e. $\phi_{\delta t} \bm{v}(t)$ : $\bm{\mathcal{L}}_{\bm{\bm{\ell}}}\,{\bm{v}}(t+\delta t) = \lim_{\delta t\rightarrow 0} [ \bm{v}(t+\delta t) - \phi_{\delta t} \bm{v}(t)]/\delta t$. Since both vectors $\bm{v}(t+\delta t)$ and $\phi_{\delta t} \bm{v}(t)$ are in ${\mathcal{T}}({\mathcal{S}}_{t+\delta t})$, it is then obvious that $\bm{\mathcal{L}}_{\bm{\bm{\ell}}}\,{\bm{v}}(t+\delta t) \in{\mathcal{T}}({\mathcal{S}}_{t+\delta t})$.
• Figure 2.1: Embedding $\Phi$ of the 3-dimensional manifold ${\mathcal{H}}_0$ into the 4-dimensional manifold ${\mathcal{M}}$, defining the hypersurface ${\mathcal{H}} = \Phi({\mathcal{H}}_0)$. The push-forward $\Phi_*\bm{v}$ of a vector $\bm{v}$ tangent to some curve $C$ in ${\mathcal{H}}_0$ is a vector tangent to $\Phi(C)$ in ${\mathcal{M}}$.
• Figure 2.2: Null hypersurface ${\mathcal{H}}$ with some null normal $\bm{\ell}$ and the null generators (thin lines).
• Figure 2.3: Outgoing light cone in Minkowski spacetime. The null hypersurface ${\mathcal{H}}$ under consideration is the member $u=1$ of the family $({\mathcal{H}}_u)$ of light cones emitted from the origin $(x,y,z)=(0,0,0)$ at successive times $t=1-u$.
• Figure 2.4: Kruskal diagram representing the Schwarzschild spacetime; the hypersurfaces of constant Schwarzschild time $t_{\rm S}$ (dashed lines) do not intersect the future event horizon ${\mathcal{H}}$, except at the bifurcation 2-sphere $B$ (reduced to a point in the figure), whereas the hypersurfaces of constant Eddington-Finkelstein time $t$ (solid lines) intersect it in such a way that $t$ can be used as a regular coordinate on ${\mathcal{H}}$ (figure adapted from Fig. 3.1 of Ref. Thornb93).