Quasinormal ringing of general spherically symmetric parametrized black holes

The general parametrization of spherically symmetric and asymptotically flat black hole spacetimes in arbitrary metric theories of gravity was suggested in Rezzolla and Zhidenko [Phys. Rev. D 90, 084009 (2014)]. The parametrization is based on the continued fraction expansion in terms of the compact radial coordinate and has superior convergence and a strict hierarchy of parameters. It is known that some observable quantities, related to particle motion around the black hole, such as the eikonal-quasinormal modes, radius of the shadow, frequency at the innermost stable circular orbit, and others, depend mostly on only a few of the lowest coefficients of the parametrization. Here we continue this approach by studying the dominant (low-lying) quasinormal modes for such generally parametrized black holes. We show that, due to the hierarchy of parameters, the dominant quasinormal frequencies are also well determined by only the first few coefficients of the expansion for the so-called moderate black hole geometries. The latter are characterized by a relatively slow change of the metric functions in the radiation zone near the black hole. The nonmoderate metrics, which change strongly between the event horizon and the innermost stable circular orbit, are usually characterized by echoes or by the distinctive (from the Einstein case) quasinormal ringing which does not match the current observational data. Therefore, the compact description of a black hole spacetime in terms of the truncated general parametrization is an effective formalism for testing strong gravity and imposing constraints on allowed black hole geometries.

Figure 1

Real and imaginary parts of the fundamental quasinormal mode ($\ell =1$) of the electromagnetic ($s=1$, top panels) and scalar field ($s=0$, bottom panels) as a function of $\epsilon$ for various $a_1$ and $b_1$: $a_1=0.9$, $b_1=0.5$ (cyan); $a_1=0.1$, $b_1=0.2$ (red); $a_1=0.2$, $b_1=-0.1$ (blue); $a_1=b_1=0$ (black); $a_1=-0.5$, $b_1=-0.1$ (magneta).

Figure 2

Real and imaginary parts of the fundamental quasinormal mode ($s=1$, $\ell =1$, $n=0$) for $b_1=0.5$ and various $\epsilon$ as a function of $a_1$, and the relative deviation from ${\omega }_0$, corresponding to $a_1=0$: $\epsilon =-0.5$ (blue); $\epsilon =0$ (black); $\epsilon =0.25$ (red); $\epsilon =0.5$ (magenta).

Figure 3

Real and imaginary parts of the fundamental quasinormal mode ($s=1$, $\ell =1$, $n=0$) for $a_1=0.5$ and various $\epsilon$ as a function of $b_1$, and the relative deviation from ${\omega }_0$, corresponding to $b_1=0$: $\epsilon =0.5$ (magenta); $\epsilon =0.25$ (red); $\epsilon =0$ (black); $\epsilon =-0.25$ (cyan); $\epsilon =-0.5$ (blue).

Figure 4

Real and imaginary parts of the fundamental quasinormal mode ($n=0$, top panels) and the first overtone ($n=1$, bottom panels) of the electromagnetic field ($s=1$, $\ell =1$) for $a_1=b_1=0.5$, $b_2=0$ and various $\epsilon$ as a function of $a_2$, and the relative deviation from ${\omega }_0$, corresponding to $a_2=0$: $\epsilon =0.5$ (red), $\epsilon =0$ (green), $\epsilon =-0.5$ (blue).

Figure 5

Real and imaginary parts of the fundamental quasinormal mode ($s=1$, $\ell =1$, $n=0$) for $a_1=b_1=a_2=0.5$ and various $\epsilon$ as a function of $b_2$, and the relative deviation from ${\omega }_0$, corresponding to $b_2=0$: $\epsilon =0.5$ (red), $\epsilon =0$ (green), $\epsilon =-0.5$ (blue).

Figure 6

Effective potentials and time-domain profiles ($s=1$, $\ell =1$, $r_0=1$) for $b_1=a_2=b_2=\dots =0$ and $\epsilon =-0.5$: $a_1=0$ ($\omega \approx 0.562-0.301i$, red) and $a_1=20$ ($\omega \approx 0.3–0.6i$, blue).

Figure 7

Effective potentials and time-domain profiles ($s=0$, $\ell =0$, $r_0=1$) for $b_1=a_2=b_2=\cdots =0$ and $\epsilon =0.5$: $a_1=0$ ($\omega \approx 0.153-0.120i$, red) and $a_1=4$ ($\omega \approx 0.13–0.21i$, blue).

Figure 8

Effective potentials and time-domain profiles ($s=0$, $\ell =2$, $r_0=1$) for $\epsilon =a_1=a_2=b_2=\cdots =0$, $b_1=-0.3$ ($\omega \approx 0.9667-0.2237i$, red) and $\epsilon =a_1=a_2=b_3=\cdots =0$, $b_1=-0.99$ and $b_2=10$ ($\omega \approx 0.9762-0.2230i$, blue).

Figure 9

Effective potentials and time-domain profiles ($s=0$, $\ell =1$, $M=0.5$) for $r_0=0.435$, $\epsilon =1.3$, $a_0=1.5$, $a_1=0.13$, $a_2=0$ $b_1=b_2=\cdots =0$ ($\omega \approx 0.791-0.172i$, red) and $r_0=0.2$, $\epsilon =4$, $a_0=7$, $a_1=0.5$, $a_2=-8$, $a_3=8$, $a_4=-0.1$, $a_5=0$, $b_0=b_1=b_2=\cdots =0$ (blue, echo).