An Introduction to Local Black Hole Horizons in the 3+1 Approach to General Relativity

Abstract

We present an introduction to dynamical trapping horizons as quasi-local models for black hole horizons, from the perspective of an Initial Value Problem approach to the construction of generic black hole spacetimes. We focus on the geometric and structural properties of these horizons aiming, as a main application, at the numerical evolution and analysis of black hole spacetimes in astrophysical scenarios. In this setting, we discuss their dual role as an a priori ingredient in certain formulations of Einstein equations and as an a posteriori tool for the diagnosis of dynamical black hole spacetimes. Complementary to the first-principles discussion of quasi-local horizon physics, we place an emphasis on the rigidity properties of these hypersurfaces and their role as privileged geometric probes into near-horizon strong-field spacetime dynamics.

Initial data for perturbed Kerr black holes on hyperboloidal slices

“…In this paper, we compute initial data corresponding to a perturbed Kerr black hole in vacuum. In order to describe the black hole, we excise its singular interior and require the excision boundary H to be a marginally outer-trapped surface [23,32]. We thus obtain an inner boundary of our computational domain.…”

“…However, this picture changes as we allow strong perturbations, since they lead to the formation of new closed 2-surfaces S + and S − inside the numerical grid, satisfying Θ + = 0 and Θ − = 0 respectively. The marginally trapped surface condition, given by Θ + Θ − = 0 [23,32], is satisfied when either Θ + or Θ − vanishes. To locate both kinds of 2-surfaces, S + and S − , we implemented a horizon finder (see e.g.…”

“…For example, for j = 0.5 we obtain ∆σ II H ≈ 0.074 and ∆σ III H ≈ −0.030. Region II is classified as future trapped region and is associated with the interior of the black hole, since both outgoing and ingoing light rays converge [23]. As we are interested in initial data outside the black hole region, one can restrict the numerical domain up to S + and consider this surface as the inner numerical boundary for perturbations in the regime ∆σ H > ∆σ II H .…”

“…We find distinct differences with respect to the corresponding Kerr expressions, which confirms that our data describe perturbed rotating black holes (and not other different slices of the Kerr space-time). Finally, we compute the Bondi Mass of our slice and, together with the black hole angular momentum, we look at some aspects in the standard picture of gravitational collapse (see [23] for an overview). Specifically we probe some black hole inequalities that were conjectured with the assumption of the Weak Cosmic Censorship.…”

Curvature Invariants and the Geometric Horizon Conjecture in a Binary Black Hole Merger

“…This approach requires an alternative description of BH horizons which is not dependent of the BH's future. [56,55,3,4,31,28].…”