Black holes in a box: Toward the numerical evolution of black holes in AdS space-times

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 nonglobally hyperbolic space-times and the Cauchy problem may only be well defined if the initial data are supplemented by boundary conditions at the timelike 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 mirrorlike 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 observed only in the first interaction. We discuss the possible role of superradiant scattering for this result. Unlike the studies with outgoing boundary conditions, both of the Newman-Penrose scalars ${\Psi }_4$ and ${\Psi }_0$ are nontrivial in our setup, and we show that the numerical data verifies the expected relations between them.

Figure 1

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) disc.

Figure 2

Illustration of a (Lego-)spherical outer boundary.

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 ( center), and IN2.3 (right panel) simulations, obtained by time-shifting such that the maxima overlap.

Figure 6

The energy spectrum for the $l=m=2$ multipole of the outgoing scalar ${\Psi }_4$, for model IN2.3. This spectrum corresponds solely to the initial wave packet, i.e., the waveform has been truncated immediately before the first reflection off the boundary. The vertical line marks the threshold frequency for superradiance.

Figure 7

Time evolution of the area of the apparent horizon for the head-on and inspiral simulation HD1 and IN1. The area of the rotating BH increases at regular intervals corresponding to the propagation time of the pulse between the hole and the reflective boundary at $r_{\mathrm{ex}}=35M$. Because of the small amount of radiation generated during the plunge in the head-on case (HD1), the variation in the AH area is buried in numerical noise.

Figure 8

Time evolution of the (relative) mass of the BH (solid line) computed by $M=C_e/4\pi$, the irreducible mass (dashed line) and the total spin $J=j{M}^2$ (dashed-dotted line).

Figure 9

Snapshots of $\Re ({\Psi }_4)$ superposed by $\Re ({\Psi }_0)$. The snapshots show the evolution from $t=150M$ until $t=540M$ and have a time interval of $\Delta t=10M$. We show a slice of the orbital plane with both coordinates going from $-48M,\dots ,48M$.

Figure 10

Comparison of the $l=2$, $m=2$ modes of ${\Psi }_4$ obtained for models C1.1 (solid) and O1 (dashed curve). The expected ranges in time for subsequent wave pulses resulting from first and second reflections are indicated in the figure.

Figure 11

Convergence analysis of the $l=m=2$ mode of ${\Psi }_4$ obtained for model C1 of Table .

Figure 12

Comparison of the $l=m=2$ (left) and $l=m=4$ (right) modes of ${\Psi }_4$ obtained for models C1.3 (cubic boundary), IN1 (spherical boundary), and O1 (outgoing condition).