Perturbation theory for gravitational shadows in Kerr-like spacetimes

TL;DR

This paper develops an analytical perturbation framework to compute gravitational shadows in Kerr-like spacetimes, incorporating plasma dispersion via a separable Benenti-type metric formalism. Shadow observables $D_X$, $D_Y$, $X_C$, $\bar{R}$, $\delta C$, and $\delta K$ are obtained as polynomial expansions in the Kerr spin parameter $a$ up to $\mathcal{O}(a^5)$, avoiding repeated numerical integration of the shadow boundary. The method is demonstrated on Kerr-Newman, Kerr-Newman with plasma, and Modified Kerr/Kerr-Sen spacetimes, with explicit coefficient structures ($S$, $S^{\Delta}$, $L_0$, $L_1$, $L_2$, $N$) that encode charge and deformation parameters and exhibit a clear plasma frequency dependence through $\omega$. The results enable efficient parameter reconstruction from shadow data and motivate shadow spectroscopy across multiple frequencies, though multi-parameter degeneracies may require additional frequency information to disentangle parameters robustly.

Abstract

We present a fully analytical method for calculating the key parameters of a Kerr-like gravitational shadow, including its horizontal and vertical diameters, $D_X$ and $D_Y$, the coordinates of its center $X_{C}$, the average radius $\bar{R}$, the deviation from sphericity $δC$, and the mean deviation from the Kerr shadow $δK$. Developed within the framework of perturbation theory, this approach yields all characteristic parameters as simple polynomial expressions with an accuracy of $\sim a^5$, where $a$ is the Kerr spin parameter. This eliminates the need for repeated numerical integration of cumbersome parametric equations. Furthermore, our derived formulas account for the effects of a plasma medium - a feature of particular relevance given the prospect of multi-frequency astrophysical observations.

Figures (2)

• Figure 1: Gravitational shadow for a Kerr-Newman black hole, observed at an inclination angle $\bar{\theta} = \pi/2$, for various combinations of the spin $a$ and charge $Q$ parameters. In Figs. \ref{['shb0']} and \ref{['shb1']}, the shadow shape is shown in coordinates centered on the shadow's geometric center. The exact shape is depicted by solid lines, while the fifth-order approximation is indicated by dotted lines. The dependence of the principal size parameters ($D_X$, $D_Y$, $2\bar{R}_A$ ) on the spin $a$ and the dimensionless charge $q = Q/M$ is illustrated in Figs. \ref{['shb2']} and \ref{['shb4']}, respectively.
• Figure 2: The deviation from perfect sphericity $\delta C_A$ for a Kerr-Newman black hole as a function of the spin parameter $a$, shown for different values of the dimensionless charge $q = Q/M$. Solid lines represent the exact values, while dotted lines correspond to the fifth-order approximation.