Geometry of generic isolated 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.

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.…”

Insights from Melvin–Kerr–Newman spacetimes

“…Angular momentum The total angular momentum includes matter contributions and has been derived by many different methods including those of Brown-York [40] and isolated and dynamical horizons [24,41]. Calculated on a spacelike two-surface S that respects the rotational symmetry:…”

Non-minimal couplings, quantum geometry and black-hole entropy

“…any isomorphism from the unit 2-sphere in the Lie algebra of SU (2) to S). Then it turns out that the intrinsic geometry of S is completely determined by the pull-back A i r i =: W to S of the (internal-radial component of the) connection A i on M [4]. Furthermore, W is in fact a spin-connection intrinsic to the 2-sphere S. Finally, the fact that S is (the intersection of M with) a type I isolated horizon is captured in a relation between the two canonically conjugate fields:…”