Quasinormal modes for the Vaidya metric

We consider here scalar and electromagnetic perturbations for the Vaidya metric in double-null coordinates. Such an approach allows one to go a step further in the analysis of quasinormal modes for time-dependent spacetimes. Some recent results are refined, and a new nonstationary behavior corresponding to some sort of inertia for quasinormal modes is identified. Our conclusions can enlighten some aspects of the wave scattering by black holes undergoing some mass accretion processes.

Figure 1

Lines of constant $u$ for the solution of (6) with the hyperbolic mass function (10), with $m_2>m_1$ and $\rho >0$. All solutions in the region below the line $r=2m(v)$ (the apparent horizon) have $r^{\prime }<0$, where the prime denotes the derivative with respect to $v$. Any solution that enters into this region will reach the singularity at $r=0$ with finite $v$. On the other hand, solutions confined to the $r^{\prime }>0$ region always escape from the singularity and reach ${\mathcal{I}}^{+}$. In the present case, there exists an event horizon (the dashed line on the inserted conformal diagram) close to the solutions $r_{\mathrm{H}}(v)$.

Figure 2

Values of the electromagnetic perturbations $\psi (u_{\max },v)$ with $\ell =2$ for a Schwarzschild black hole with mass $m=0.5$ and for the Vaidya metric with the hyperbolic mass function (10), where $v_1=75$, $\rho =0.08$, $m_1=0.5$, and $m_2=0.65$ (Hyperbolic 3). One can clearly appreciate, for the time-dependent case, the slowing down of the oscillation frequency and damping taking place after $v_1$.

Figure 3

The real (${\omega }_{\mathrm{R}}$) part of the frequency for scalar perturbations with $\ell =2$ as a function of $v$ for decreasing linear and hyperbolic mass functions. For all the cases, $m_1=0.5$ and $m_2=0.35$. Linear 1: $v_1=60$, $v_2=90$. Linear 2: $v_1=74.5$, $v_2=75.5$. Hyperbolic 1: $v_1=75$, $\rho =0.08$. Hyperbolic 2: $v_1=75$, $\rho =0.8$. The imaginary part ${\omega }_{\mathrm{I}}$ has similar behavior for the increasing mass case (Fig. 4); see the text.

Figure 4

The real (${\omega }_{\mathrm{R}}$) and imaginary (${\omega }_{\mathrm{I}}$) parts of the frequency for electromagnetic perturbations with $\ell =2$ as a function of $v$ for increasing linear and hyperbolic mass functions. For all the cases, $m_1=0.5$ and $m_2=0.65$. Linear 3: $v_1=60$, $v_2=90$. Linear 4: $v_1=74.5$, $v_2=75.5$. Hyperbolic 3: $v_1=75$, $\rho =0.08$. Hyperbolic 4: $v_1=75$, $\rho =0.8$.