Horizon dynamics of distorted rotating black holes

We present numerical simulations of a rotating black hole distorted by a pulse of ingoing gravitational radiation. For strong pulses, we find up to five concentric marginally outer trapped surfaces. These trapped surfaces appear and disappear in pairs, so that the total number of such surfaces at any given time is odd. The world tubes traced out by the marginally outer trapped surfaces are found to be spacelike during the highly dynamical regime, approaching a null hypersurface at early and late times. We analyze the structure of these marginally trapped tubes in the context of the dynamical horizon formalism, computing the expansion of outgoing and incoming null geodesics, as well as evaluating the dynamical horizon flux law and the angular momentum flux law. Finally, we compute the event horizon. The event horizon is well-behaved and approaches the apparent horizon before and after the highly dynamical regime. No new generators enter the event horizon during the simulation.

Figure 1

Convergence of the elliptic solver for different amplitudes $A$. Plotted is the square-sum of the Hamiltonian and momentum constraints, Eqs. (5, 6), as a function of numerical resolution, measured here by the number of radial basis functions in the spherical shell containing the gravitational waves.

Figure 2

ADM energy $E_{\mathrm{ADM}}$ and Christodoulou mass $M_i$ of the initial data sets, versus the gravitational wave amplitude $A$. The inset shows the Ricci scalar $R$ along the $x$-axis. All quantities are given in units of the mass of the background Kerr-Schild metric.

Figure 3

Constraint violations for the evolution with $A=0.5$. Plotted is the $L^2$ norm of all constraints, normalized by the $L^2$ norm of the spatial gradients of all dynamical fields.

Figure 4

The solid curves are the Christodoulou masses $M(t)$ divided by their initial values $M_i$ for the five evolutions with different amplitudes $A=0.1$, 0.2, 0.3, 0.4, and 0.5 for the ingoing pulse of gravitational waves. The horizontal dotted lines denote the ADM energy of each data set, $E_{\mathrm{ADM}}/M_i$.

Figure 5

Irreducible mass $M_{\mathrm{H}}$ divided by its initial value $M_{\mathrm{H},i}$ for the evolution with $A=0.5$. The solid circles are the values of $M_{\mathrm{H}}$ for MOTSs found during the evolution. The completed curve is traced out by open circles. The vertical shaded region indicates when five MOTSs exist at the same time.

Figure 6

The solid red line denotes the apparent horizon for the evolution with $A=0.5$. The solid blue circles denote an erroneous “apparent horizon,” which is found when the apparent horizon finder is run during the evolution in larger time intervals. The black dashed lines denotes all five MTTs as shown in Fig. 5.

Figure 7

Extrema of the intrinsic scalar curvature $\overline{R}$ of MOTSs during the evolution with $A=0.5$. The horizontal dotted lines are the values for the apparent horizon in the initial data. Around $t=14.25M_i$, the MOTSs have regions of negative $\overline{R}$.

Figure 8

Extrema of ${\theta }_{(k)}$ on each MOTS along the MTTs during the evolution with $A=0.5$. For the time shown, ${\theta }_{(k)}<0$.

Figure 9

Extrema of ${\mathcal{L}}_k{\theta }_{(l)}$ on each MOTS along the MTTs during the evolution with $A=0.5$. For the time shown, ${\mathcal{L}}_k{\theta }_{(l)}<0$.

Figure 10

Extrema of $-{\mathcal{L}}_l{\theta }_{(l)}$ on each MOTS along the MTTs during the evolution with $A=0.5$. Near $t=14M_i$, ${\mathcal{L}}_l{\theta }_{(l)}\ne 0$ somewhere on ${\mathcal{S}}_v$.

Figure 11

Extremality parameter $e$ along the MTTs during the evolution with $A=0.5$. For the time shown, the MTTs are subextremal with $e<1$, indicating that the MTTs have no timelike sections.

Figure 12

Terms in the dynamical horizon flux law of Eq. (47) plotted against the foliation parameter $v$ along each section of ${\mathcal{S}}_v$. Along MTT1, we choose $v=t$. Along the other MTT sections, we choose $v=0$ on the first ${\mathcal{S}}_v$ we find.

Figure 13

Terms in the angular momentum flux law of Eq. (64) plotted against the foliation parameter $v$ along each section of $\mathcal{H}$. Along MTT1, we choose $v=t$. Along the other MTT sections, we choose $v=0$ on the first ${\mathcal{S}}_v$ we find.

Figure 14

Irreducible masses of the event horizon $M_{\mathrm{EH}}$ and the MOTSs $M_{\mathrm{H}}$ during the evolution with $A=0.5$. At the very beginning of the evolution $M_{\mathrm{EH}}$ is already increasing, while $M_{\mathrm{H}}$ is still fairly constant. As the inset shows, $M_{\mathrm{EH}}$ grows very slightly when $M_{\mathrm{H}}$ changes the most.

Figure 15

Spacetime diagram of the event horizon and dynamical horizons for $A=0.5$. The dotted red lines are the null generators of the event horizon, while the solid grey surface represents the dynamical horizons.

Figure 16

Mass functions $m(v)$ of the Vaidya spacetime for three amplitudes $A=0.25$, 0.5, and 1, along with the straight lines $(v-t)/2$. MOTSs exist at the intersections of these functions. For $A=0.5$ and 1, there are up to three intersections, as illustrated by the dashed black line which intersects the $A=1$ mass curve three times.

Figure 17

Locations of MOTSs (solid lines) and event horizons (dashed lines) in the Vaidya spacetime. For $A=0.25$ there is one MOTS at all times. For $A=0.5$ and 1, up to three MOTSs exist at a time $t$. The event horizons approach the MTTs at very early and late times, and start growing much earlier than the MTTs. The inset shows a larger interval in $t$.