Geometry of Generic Isolated Horizons

TL;DR

The paper develops a quasi-local framework for null horizons in general relativity, introducing non-expanding horizons (NEH), weakly isolated horizons (WIH), and isolated horizons (IH) and tying their intrinsic data $(q_{ab}, {\cal D})$ to the spacetime curvature via Einstein's equations. It derives explicit constraint equations and identifies the freely specifiable data, then shows how to select canonical classes of null normals $[\ell]$ in extremal and generic NEHs, including gauge-fixing through divergence-free conditions and trace-free transversal expansions. A key contribution is the notion of good cuts and a geometric characterization of canonical $[\ell]$ via analytic extension to a cross-over sphere, linking near-horizon geometry to structures at null infinity and to Killing horizons. These results have practical impact for numerical relativity, enabling invariant horizon-based measurements of mass, angular momentum, and near-horizon waveforms, and pave the way for perturbation analyses and Kerr-limit studies of black-hole horizons.

Abstract

Geometrical structures intrinsic to non-expanding, weakly isolated and isolated horizons are analyzed and compared with structures which arise in other contexts within general relativity, e.g., at null infinity. In particular, we address in detail the issue of singling out the preferred normals to these horizons required in various applications. This work provides powerful tools to extract invariant, physical information from numerical simulations of the near horizon, strong field geometry. While it complements the previous analysis of laws governing the mechanics of weakly isolated horizons, prior knowledge of those results is not assumed.