Ringdown in Vaidya spacetimes: time-dependent frequencies, Penrose limit and time-domain analyses

TL;DR

The paper investigates ringdown in dynamical Vaidya spacetimes by pairing a Penrose-limit analysis around a moving photon sphere with full numerical simulations. It derives time-dependent QNM-like frequencies in an adiabatic regime and compares Penrose-limit predictions to both frequency-domain and time-domain results for constant and time-dependent accretion. The findings show that the Penrose-limit geometry encodes the local dynamics near the dynamical photon sphere and closely tracks the emitted ringdown once propagation effects such as redshift and scattering are accounted for, though these effects can be significant in the time-dependent case. The ratio of the imaginary to real parts, $\mathcal{R}=\omega^{\mathrm{Im}}/\omega^{\mathrm{Re}}$, emerges as a useful diagnostic that approaches the Penrose-limit prediction in the appropriate adiabatic, high-$\ell$ limit with large observer distance, revealing the evolving geometry of the photon sphere.

Abstract

We examine the possible characterization of ringdown waves in a dynamical Vaidya spacetime using the Penrose limit geometry around the dynamical photon sphere. In the case of a static spherically symmetric black hole spacetime, it is known that the quasinormal frequency in the eikonal limit can be characterized by the angular velocity and the Lyapunov exponent for the null geodesic congruence on the orbit of the unstable circular null geodesic. This correspondence can be further backed up by the analysis of the Penrose limit geometry around the unstable circular null geodesic orbit. We try to extend this analysis to a Vaidya spacetime, focusing on the dynamical photon sphere in it. Then we discuss to what extent the Penrose limit geometry can be relevant to the ringdown waves in the Vaidya spacetime, comparing the results with the numerically calculated waveform in the Vaidya spacetime.

Figures (13)

• Figure 1: The value of $1-(2\ell+1)\mathcal{R}_{\rm PL}$ is shown as a function of $M'$.
• Figure 2: The waveform measured at $x=60$ for the massless scalar field ($s=0$) and $\ell=4$.
• Figure 3: The values of $\omega_n^{\rm Re/Im}$, $\hat{\Omega}^{\rm Re/Im}/(2M(\mathcal{V}))$ and $\Omega^{\rm Re/Im}_{\rm PL}/(2M(\mathcal{V}))$ are plotted for the massless scalar field (top), Maxwell field (middle) and metric perturbation (bottom) with $M'=\pi/2000$ and $\ell=4$.
• Figure 4: The photon sphere trajectory for the Vaidya spacetime with $M_2=3M_1$, $\mathcal{V}_1=0$ and $\mathcal{V}_2=100M_1$. The null dust is accreting in the shaded region.
• Figure 5: The values of $A_{11}$ and $-A_{22}$ with $M_2=3M_1$, $\mathcal{V}_1=0$ and $\mathcal{V}_2=100M_1$. The null dust is accreting in the shaded region.