New generic ringdown frequencies at the birth of a Kerr black hole

We discuss a new ringdown frequency mode for vacuum perturbations of the Kerr black hole. We evolve initial data for the vacuum radial Teukolsky equation using a near horizon approximation and find a frequency mode analogous to that found in a recent study of radiation generated by a plunging particle close to the Kerr horizon. We discuss our results in the context of that study. We also explore the utility of this mode by fitting a numerical waveform with a combination of the usual quasinormal modes and the new oscillation frequency.

Figure 1

Comparison of the $\Sigma$ and characteristic models. In both figures, the trajectory of the observer discussed and the characteristic boundary surface are plotted, with ($\theta$, $\varphi$) suppressed. The thick line is the interval where the initial data is nonzero, and the two points are ${\xi }_1$ and ${\xi }_2$, respectively. The shaded regions correspond to the nonlinear regime of each model. Top: Comparison in BL coordinates of the two models, with the event horizon illustrated. Bottom: Comparison in ($t$, $r^{*}$) coordinates, where the horizon is pushed to $r^{*}\to \infty$.

Figure 2

A comparison of the real and imaginary parts of ${\omega }_H$ (dashed line) and the first four QNMs, with $\ell =2$, $m=\pm 2$ (solid line).

Figure 3

The section of the numerical waveform given in Scheel et al. over which $t_0$ ranges in the overlap integral. Here we give both the real (solid line) and imaginary (dashed line) portions of $M_{0}r{\Psi }_4$.

Figure 4

Comparisons of $1-{\rho }_{\max }$ for overlaps using just the QNMs (solid line) and those which replace one QNM pair with the HM (dashed line). The top panel compares the first QNM pair (both coand counterrotating) with the HM alone. The second panel compares the first two QNM pairs with the first QNM pair and the HM. Subsequent panels compare the first $n$ ($n=3$, 4, 5) QNM pairs with the first $n-1$ pairs and the HM.

Figure 5

Values for the extracted masses $M_{\mathrm{QNM}}$ (solid line) and $M_{\mathrm{HM}}$ (dashed line) as functions of $t_0$, for three extractions. The top panel compares the first three QNM pairs with the first two and the HM. The middle panel compares the first four QNM pairs with the first three and the HM. The bottom panel compares the first five QNM pairs with the first four and the HM. The solid horizontal line is at $M/M_0=0.95162$, the mass of the final black hole as given in 32. The solid vertical line gives the peak of $|\psi |$.

Figure 6

Values for the extracted spins $a_{\mathrm{QNM}}$ and $a_{\mathrm{HM}}$ as functions of $t_0$, for three extractions. The top panel compares the first three QNM pairs with the first two and the HM. The middle panel compares the first four QNM pairs with the first three and the HM. The bottom panel compares the first five QNM pairs with the first four and the HM. The solid horizontal line is at $a/M_0=0.65325$, the spin of the final black hole as given in 32. The solid vertical line gives the peak of $|\psi |$.

Figure 7

Extraction of the mass $M/M_0$ as a function of $t_0$, using only first three QNM pairs (solid line) or the first two QNM pairs and the HM (dashed line). Here we do not include the corotating $n=1$, $\ell =4$, $m=4$ mode.

Figure 8

Extraction of the spin $a/M_0$ as a function of $t_0$, using only first three pairs QNMs (solid line) or the first two QNM pairs and the HM (dashed line). We do not include the corotating $n=1$, $\ell =4$, $m=4$ mode.