Quasinormal modes in a time-dependent black hole background

We have studied the evolution of the massless scalar field propagating in a time-dependent charged Vaidya black hole background. A generalized tortoise coordinate transformation was used to study the evolution of the massless scalar field. It is shown that, for the slowest damped quasinormal modes, the approximate formulas in the stationary Reissner-Nordström black hole turn out to be a reasonable prescription, showing that results from quasinormal mode analysis are rather robust.

Figure 1

The behavior of $r_{+}$ and $r_{-}$ for the black hole with $q=0$, $q=0.7$, and $q=0.999$, respectively. The linear model is used with $m_0=0.5$, $\lambda =0.002$, $v_0=0$, $v_1=150$ (correspondingly, $m_1=0.35$ and $\dot{M}=-0.001$).

Figure 2

Temporal evolution of the field in the background of Vaidya metric ($q=0$) for $l=2$, evaluated at $r=5$. The mass of the black hole is $M(v)=0.5\pm 0.001v$. The field evolution for $M(v)=0.5+0.001v$ and $M(v)=0.5-0.001v$ is shown as the top curve and the bottom curve, respectively. For comparison, the oscillations for $M=0.5$ are given in the middle line.

Figure 3

The frequency ${\omega }_R$ for $l=2$, evaluated at $r=5$ in the linearly absorbing black hole $M(v)=0.5+0.001v$.

Figure 4

The frequency ${\omega }_I$ for $l=2$, evaluated at $r=5$ in the linearly absorbing black hole $M(v)=0.5+0.001v$.

Figure 5

The frequency ${\omega }_R$ for $l=2$, evaluated at $r=5$ in the linearly evaporating black hole $M(v)=0.5-0.001v$.

Figure 6

The frequency ${\omega }_I$ for $l=2$, evaluated at $r=5$ in the linearly evaporating black hole $M(v)=0.5-0.001v$.

Figure 7

$c_R(l,q)/{\omega }_R$ as the function of time $v$ for $l=2$, $r=5$, and $M(v)=0.5+0.001v$.

Figure 8

$c_I(l,q)/{\omega }_I$ as the function of time $v$ for $l=2$, $r=5$, and $M(v)=0.5+0.001v$.

Figure 9

$c_R(l,q)/{\omega }_R$ as the function of time $v$ for $l=2$, $r=5$, and $M(v)=0.5-0.001v$.

Figure 10

$c_I(l,q)/{\omega }_I$ as the function of time $v$ for $l=2$, $r=5$, and $M(v)=0.5-0.001v$.

Figure 11

${\omega }_R$ as the function of time $v$ for $l=2$, $q=0$, $M(v)=0.5-0.001v$, evaluated at different position $r=10$, $r=20$, and $r=30$. The temporal evolution of the field $\Psi (r,v)$ at radius $r=20$ (or $r=30$) has a retard to the position $r=10$.

Figure 12

Considering the black hole mass: $q=0.999$, $M(v)=0.5$ for $v<60$, $M(v)=0.5+0.05(v-60)$ for $60\le v\le 63$, and $M(v)=0.65$ for $v>60$.

Figure 13

At $r=5$, the observed behavior of the quasinormal ringing for $l=2$.

Figure 14

Change of the real part of the quasinormal frequency. The black hole mass starts to increase when $v=60$, and the real part of the QNM frequency starts to decrease at the same moment. It is clear that the time scale of the change of the QNM frequency is much longer than the change of the black hole mass ($dv=3$).

Figure 15

Change of the imaginary part of the QNM frequency. The black hole mass starts to increase when $v=60$, and the imaginary part of the QNM frequency starts to increase at the same moment. It is clear that the time scale of the change of the QNM frequency is much longer than the change of the black hole mass ($dv=3$).

Figure 16

$c_R(l,q)/{\omega }_R$ and $c_I(l,q)/{\omega }_I$ as the functions of time $v$ for the linear and exponential model. In the linear model, $\lambda =-1/150$, and in the exponential model $\alpha =-(\ln 2)/150$. Both models describe the absorption process of the black hole, where the mass increases from $M(v_0)=0.5$ to $M(v_1)=1$. The relative parameters in both models are selected as $l=2$, $r=5$, $q=0.999$, $v_0=0$, and $v_1=150$. The two dotted lines are the mass functions $M(v-v^{\prime })$ for models (26, 30) with $v^{\prime }=8$.

Figure 17

$c_R(l,q)/M{\omega }_R$ and $c_I(l,q)/M{\omega }_I$ as the functions of time $v$. The corresponding parameters are the same as chosen in Fig. 16.

Figure 18

$c_R(l,q)/{\omega }_R$ and $c_I(l,q)/{\omega }_I$ as the functions of time $v$ in the case of $q=0$. The corresponding parameters are the same as chosen in Fig. 16.

Figure 19

$c_R(l,q)/M{\omega }_R$ and $c_I(l,q)/M{\omega }_I$ as the functions of time $v$. The corresponding parameters are the same as chosen in Fig. 16.