Padé metrics for black hole perturbations and light rings

TL;DR

This work introduces a two‑point Padé interpolation (M‑fraction) to construct simple yet versatile static, spherically symmetric black hole metrics that smoothly interpolate between near‑horizon and asymptotic regions. It extends prior scalar analyses to axial gravitational perturbations, deriving a Regge–Wheeler form and computing low‑lying quasinormal modes with a matrix method, benchmarking against Schwarzschild, RN, and Bardeen cases. It also analyzes light rings, showing that LR positions are governed by near‑horizon coefficients and robust to far‑field deviations, with negative energy density near the horizon correlating with outward LR shifts. The results demonstrate that Padé metrics provide a broadly applicable, low‑order framework for probing deviations from Schwarzschild and informing gravitational and electromagnetic observational signatures of ultracompact objects. This approach offers a computationally efficient, model‑independent path to connect theoretical deviations with observable QNMs and light rings in current and future data.

Abstract

Most distinguishing features of black holes and their mimickers are concentrated near the horizon. In contrast, astrophysical observations and theoretical considerations primarily constrain the far-field geometry. In this work we develop tools to effectively describe both, using the two-point Padé approximation to construct interpolating metrics connecting the near and far-field. We extend our previous work by computing the quasinormal modes of gravitational perturbations for static, spherically symmetric metrics that deviate from Schwarzschild spacetime. Even at the lowest order, this approach compares well with existing methods in both accuracy and applicability. Additionally, we show that the lowest-order interpolating metric reliably predicts light ring locations. It closely matches exact results, even when unsuitable for quasinormal frequency calculations.

Figures (6)

• Figure 1: The parameter space $(\alpha_1,\alpha_2)$ covered by the two-point Padé approximations $f_{\text{S3}}$ (blue) and the approximations with $n=4$ (orange) and $n=5$ (yellow) of Ref. ST:24. In all expansions we set $\beta_1=1$. For the expansions $n=4$ and $n=5$ we take the Schwarzschild values of the higher order coefficients, $\alpha_3=1, \alpha_4=-1$. Poles in the respective function $f$ develop on the interval $r\in[r_g,\infty)$ for coefficients lying in the shaded region.
• Figure 2: The parameter space $(\alpha_1,\alpha_2)$ covered by the two-point Padé approximation $f_{\text{RN3}}$ for $r_g=1$ and $q=0.1$. In this example, $\beta_1$ and $\beta_2$ are set to their Reissner--Nordström values. Poles on the interval $r\in[r_g,\infty)$ develop for coefficients lying in the shaded region.
• Figure 3: Domains of validity for the lowest order M-fraction (yellow) and RZ (red) approximations of the Bardeen metric. Poles are present for the respective metrics in the shaded region. $\beta_1$ is fixed to its Bardeen value.
• Figure 4: Real, imaginary, and parametric plots of the fundamental $l=2$ mode of the $f_{3}$ metric as a function of $\alpha_1$. The step size is $\Delta\alpha_1=0.00015$. The shaded red region corresponds to a $10\%$ deviation window centered on the known Schwarzschild quasinormal frequency.
• Figure 5: Real, imaginary, and parametric plots of the fundamental $l=2$ mode of the $f_{3}$ metric function as a function of $\alpha_2$. The step size is $\Delta\alpha_2=0.0005$. The shaded red region corresponds to a $10\%$ deviation window centered on the known Schwarzschild quasinormal frequency.