Black holes in a box: towards the numerical evolution of black holes in AdS

TL;DR

This work explores the nonlinear evolution of black hole binaries confined inside a reflective boundary to mimic Anti-de Sitter space, testing the well-posedness and dynamics of such a setup. Using a 3+1 BSSN-based numerical relativity framework with a spherical mirror boundary, the authors track both outgoing and ingoing gravitational radiation via Ψ4 and Ψ0 and monitor horizon properties through the merger and up to two boundary reflections. They find that roughly 15% of the radiated energy is absorbed by the remnant per interaction, with the first interaction increasing the spin by about 5%; the evolution remains formally convergent for a few reflections but shows deteriorating convergence thereafter, highlighting boundary-induced challenges. The results demonstrate the feasibility and richness of BH dynamics in AdS-like boxes and point to future work on longer evolutions, boundary conditions, and direct AdS backgrounds to fully understand phenomena such as potential superradiant instabilities and BH bombs.

Abstract

The evolution of black holes in "confining boxes" is interesting for a number of reasons, particularly because it mimics the global structure of Anti-de Sitter geometries. These are non-globally hyperbolic space-times and the Cauchy problem may only be well defined if the initial data is supplemented by boundary conditions at the time-like conformal boundary. Here, we explore the active role that boundary conditions play in the evolution of a bulk black hole system, by imprisoning a black hole binary in a box with mirror-like boundary conditions. We are able to follow the post-merger dynamics for up to two reflections off the boundary of the gravitational radiation produced in the merger. We estimate that about 15% of the radiation energy is absorbed by the black hole per interaction, whereas transfer of angular momentum from the radiation to the black hole is only observed in the first interaction. We discuss the possible role of superradiant scattering for this result. Unlike the studies with outgoing boundary conditions, both the Newman-Penrose scalars Ψ_4 and Ψ_0 are non-trivial in our setup, and we show that the numerical data verifies the expected relations between them.

Figures (12)

• Figure 1: Illustration of a (Lego-)spherical outer boundary.
• Figure 2: Sketch of the foliation for the numerical evolution of BH binaries in a (spherical) box. The location of the considered numerical domain on each spatial hypersurface is shown as a dark (red) sphere.
• Figure 3: Convergence analysis of the outgoing Weyl scalar $\Psi_4$ (left panel) and the ingoing Weyl scalar $\Psi_0$ (right panel) for the IN2 runs. We show the differences of the $l=m=2$ mode between the coarse and medium and the medium and fine resolution run. The latter has been amplified by the factors $Q=1.47$ (fourth-order convergence) and $Q =1.26$ (second-order convergence). We observe fourth-order convergence in the signal due to the merger whereas the first and second after-merger cycles show only second-order convergence. The first two reflected and ingoing wave pulses show second-order convergence.
• Figure 4: Real part of the $l=m=2$ mode of $rM\Psi_0$ and $rM\Psi_4$ of run IN1. The ingoing signal $rM\Psi_0$ has been shifted in time by $\Delta t = 10M$ and in phase by $\pi$ (thus equivalent to an extra minus sign) to account for the additional propagation time and the reflection.
• Figure 5: Overlap of the amplitudes of successive pulses of the same waveform; $l=2$,$m=0$ for the HD1 run (left), $l=m=2$ for the IN1 (centre) and IN2.3 (right panel) simulations, obtained by time-shifting such that the maxima overlap.