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

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.

💡 Research Summary

The paper investigates how the ringdown phase of gravitational waves is altered when the background spacetime is not static but instead described by the Vaidya metric, which models a spherically symmetric black hole whose mass changes with advanced time V. In stationary, spherically symmetric black holes it is well‑known that, in the eikonal (high‑ℓ) limit, the complex quasinormal mode (QNM) frequency can be expressed in terms of two geometric quantities associated with the unstable circular photon orbit (UCOP): the angular velocity Ω of the orbit and the Lyapunov exponent λ that measures the instability of radial perturbations. This QNM‑geodesic correspondence can be derived elegantly using the Penrose limit (PL): by zooming in on the infinitesimal neighbourhood of the null geodesic, the metric reduces to a plane‑wave (pp‑wave) form whose wave equation separates into an inverted harmonic oscillator (radial direction) and a regular harmonic oscillator (angular direction). The eigenvalues of these oscillators reproduce the familiar eikonal QNM formula ω≈Ωℓ−iλ(n+½).

The authors aim to extend this correspondence to a dynamical setting. They consider the Vaidya line element
ds² = −(1−2M(V)/r)dV² + 2 dV dr + r² dΩ²,
with a linearly growing mass function M(V)=M₀+M′V (M′>0). In this spacetime a “dynamical photon sphere’’ exists: its radius r_c(V) follows the instantaneous mass and therefore dr_c/dV≠0. The key idea is to apply the Penrose limit to a null geodesic that lies on this moving photon sphere. By introducing adapted coordinates (u,v,x,y) and taking the ε→0 limit, the metric becomes the Brinkmann pp‑wave
ds̄² = du dv + (α²x²−β²y²)du² + dx² + dy²,
where the coefficients α and β are directly related to the instantaneous Lyapunov exponent and angular velocity, α=E f_c λ, β=E f_c Ω (E is the conserved energy in the static limit, f_c≡f(r_c)). Because the Vaidya background is conformally related to a static metric, the null geodesic equations are unchanged under the conformal factor, allowing the same PL construction.

To make the formalism predictive, the authors adopt an adiabatic approximation: if the mass changes slowly compared with the QNM period, one can treat M(V) as quasi‑static at each instant, compute r_c(V), Ω(V) and λ(V) from the static formulas, and then define a time‑dependent QNM frequency
ω(t) ≈ Ω(t) ℓ − i λ(t) (n+½).
They explicitly derive Ω(t) and λ(t) for the Vaidya case, obtaining analytic expressions (Eqs. 3.12‑3.15) that reduce to the Schwarzschild values when M′→0.

The theoretical predictions are tested against full numerical simulations. Two families of simulations are performed:

1. Constant accretion rate (M′=const). Tensor perturbations are evolved on the full Vaidya background using a characteristic or Cauchy code. The resulting waveform is extracted at a large radius, and a time‑frequency analysis (Hilbert transform and sliding‑window Fourier transform) yields instantaneous ω(t). For modest accretion rates (M′≲10⁻³ M₀), the adiabatic PL frequencies match the numerical ω(t) to within a few percent for ℓ≥4, confirming the validity of the correspondence.

2. Time‑dependent accretion rate. The authors consider scenarios where M′(V) varies sharply (e.g., a Gaussian burst of accretion or a rapid mass loss). In these cases the adiabatic condition breaks down. The numerical waveforms display noticeable deviations: the real part of the frequency drifts faster than the PL prediction, while the damping rate λ exhibits overshoots and undershoots. For very rapid changes (timescale comparable to 1/Ω), the PL estimate can be off by 20‑30 % or more, and the waveform develops a pronounced tail that is not captured by the plane‑wave approximation.

A further observation concerns the angular momentum number ℓ. The analysis focuses on ℓ≥4 because the eikonal approximation is more reliable there. For lower ℓ (ℓ=2,3) the power‑law tail dominates the late‑time signal, and the PL‑based QNM estimate becomes inaccurate, a limitation the authors acknowledge and plan to address in future work.

The paper concludes that the Penrose‑limit approach provides a powerful geometric tool to understand ringdown in slowly evolving spacetimes. It confirms that, under adiabatic evolution, the instantaneous photon‑sphere properties continue to dictate the QNM spectrum, just as in the static case. However, the method’s reliance on an infinitesimal neighbourhood of the geodesic means it cannot capture global, non‑adiabatic effects such as rapid mass jumps or strong tail contributions. The authors suggest several extensions: developing a non‑adiabatic PL framework (perhaps via a time‑dependent pp‑wave background), incorporating back‑reaction of the perturbations on the background mass function, and applying the technique to more realistic astrophysical scenarios (e.g., accretion onto supermassive black holes, post‑merger remnants surrounded by dense matter). Overall, the work bridges a gap between geometric optics intuition and full numerical relativity for dynamical black holes, offering a promising avenue for interpreting future gravitational‑wave observations where the final black hole may still be accreting or evaporating.