Alternative mechanism for black hole echoes

Gravitational wave echoes from black holes have been suggested as a crucial observable to probe the spacetime in the vicinity of the horizon. In particular, it was speculated that the echoes are closely connected with specific characteristics of the exotic compact objects, and moreover, possibly provide an access to the quantum nature of gravity. Recently, it was shown that the discontinuity in the black hole metric substantially modifies the asymptotical behavior of quasinormal frequencies. In the present study, we proceed further and argue that a discontinuity planted into the metric furnishes an alternative mechanism for the black hole echoes. Physically, the latter may correspond to an uneven matter distribution inside the surrounding halo. To demonstrate the results, we first numerically investigate the temporal evolution of the scalar perturbations around a black hole that possesses a nonsmooth effective potential. It is shown that the phenomenon persists even though the discontinuity can be located further away from the horizon with rather insignificant strength. Besides, we show that the echoes in the present model can be derived analytically based on the modified pole structure of the associated Green function. The asymptotical properties of the quasinormal mode spectrum and the echoes are found to be closely connected, as both features can be attributed to the same origin. In particular, the period of the echoes in the time domain $T$ is shown to be related to the asymptotic spacing between successive poles along the real axis in the frequency domain $\mathrm{\Delta }(\Re \omega )$ by a simple relation ${\lim }_{\Re \omega \to +\infty }\mathrm{\Delta }(\Re \omega )=2\pi /T$. Moreover, we discuss possible distinguishment between different echo mechanisms. The potential astrophysical implications of the present findings are also addressed.

Figure 1

The effective potential given in Eq. (15), where we take $l=2$, $M_0=1$, and $r_s=100$.

Figure 2

The time-domain profile of scalar field perturbations for the effective potential $V_1$ defined by Eq. (15) (blue solid curve), compared against with that for the Regge-Wheeler potential $V_{\mathrm{RW}}$ (red dashed curve). The calculations are carried out by taking $l=2$, $M_0=1$, and $r_s=100$.

Figure 3

The time-domain profile of scalar field perturbations for the effective potential $V_1$ defined by Eq. (15). The calculations are carried out by taking $l=2$, $M_0=1$, and for $r_s=80$ and $r_s=120$, respectively.

Figure 4

Top: the effective potential given in Eq. (17), where we take $r_s=80$, $\mathrm{\Delta }{r}_s=2$, $l=2$, and $M_0=1$. Bottom: the corresponding time-domain profile of scalar field perturbations, where a second curve in purple corresponds to $\mathrm{\Delta }{r}_s=10$.

Figure 5

The effective potential given in Eq. (21), where we take $r_s=50$, $l=2$, $M_0=1$. In the left panel, we choose $\mathrm{\Delta }{r}_s=2$ with $\mathrm{\Delta }M=1$ and $\mathrm{\Delta }M=5$, while in the right panel, we choose $\mathrm{\Delta }M=1$ with $\mathrm{\Delta }{r}_s=0.5$ and $\mathrm{\Delta }{r}_s=5$.

Figure 6

The time-domain profile of scalar field perturbations for the effective potential $V_3$ defined by Eq. (21). The calculations are carried out by taking $l=2$, $M_0=1$, and $r_s=100$. In the left panel, we show the results by choosing $\mathrm{\Delta }{r}_s=2$ with $\mathrm{\Delta }M=1$ and $\mathrm{\Delta }M=5$, while in the right panel, we choose $\mathrm{\Delta }M=1$ with $\mathrm{\Delta }{r}_s=0.5$ and $\mathrm{\Delta }{r}_s=5$.

Figure 7

The effective potential of the devised wormhole and that given by Eq. (21), shown in the tortoise coordinate. In the left panel, for the effective potential with discontinuity, we choose $M_0=1$, $l=2$, $r_s=100$, $\mathrm{\Delta }{r}_s=2$, and $\mathrm{\Delta }M=1$, and for the wormhole metric, one takes $r_0=2+0.5\times {10}^{-11}$. Such a choice of parameters guarantees that periods of the echoes are identical in the two cases. In the right panel, we take $r_s=300$ and $r_0=2.0001$, while the other parameters remain unchanged. The black double-sided arrows indicate the relevant scales associated with the bouncing of the waveform.

Figure 8

Comparisons between the resultant echoes in the time-domain profiles for the two sets of effective potentials shown in Fig. 7.

Figure 9

A schematic plot of the asymptotical behavior of the quasinormal modes frequencies whose distribution in the complex plane are approximately along the real axis.