Reconstruction of Black Hole Metric with Gravitational Shadows

Abstract

Based on a two-parameter perturbative framework, we derive perturbative formulae for the radii of massive particle spheres and massive shadow radii, where an extremal black hole is employed as the perturbative background. Adopting extremal black holes as the perturbative background enables energy-dependent scans to probe stronger gravitational regimes. Using these formulae, we perform perturbative analyses of the massive shadows for several non-standard black holes, and obtain the expansion expressions near the photon sphere of the extremal black hole. It is found that, although the dependence on the energy parameter $ε$ is the same across models, the coefficients associated with the deviation parameter $δ$ differ significantly, which can serve to distinguish among different black hole models. Furthermore, we construct a two-point Padé approximant in the form of a continued fraction which achieves a globally accurate approximation for the massive shadow radius of the black hole over the entire parameter range. Numerical tests show that the relative errors based on this approximant are small enough to provide a reliable foundation for model-independent reconstruction of metric parameters.