Photon Sphere and Shadow of a Perturbative Black Hole in $f(R,\mathcal{G})$ Gravity

TL;DR

This paper analyzes how higher-curvature corrections in a perturbative $f(R, G)$ gravity framework affect black-hole observables, including photon spheres and shadows, with implications for testing gravity theories via observations.

Contribution

It provides analytic expressions for metric deviations and photon-sphere shifts in $f(R, G)$ gravity, highlighting the impact of higher-curvature terms on black-hole shadows and lensing.

Findings

Higher-curvature terms modify the photon-sphere radius and black-hole shadow size.

Gauss--Bonnet sector has a more significant effect than mixed curvature terms.

Observables like shadow radius and lensing are sensitive to higher-curvature corrections.

Abstract

We investigate the impact of higher-curvature corrections on black-hole observables within a perturbative $f(R, G)$ gravity framework. Working in a static, spherically symmetric spacetime, we construct leading-order deviations from the Schwarzschild solution by expanding the field equations in small coupling parameters associated with quadratic curvature invariants. The resulting metric corrections are obtained as asymptotic expansions and used to analyze null geodesics. We derive analytic expressions for the shift in the photon-sphere radius and show that higher-curvature terms modify the location of unstable photon orbits, with the Gauss--Bonnet sector producing a more significant contribution than mixed curvature terms. These modifications propagate to observable quantities, leading to corrections in the black-hole shadow radius. We identify the distinct roles of photon-sphere displacement and direct metric perturbations in determining the shadow size. We further discuss the implications of these corrections for strong gravitational lensing and quasinormal modes, highlighting the enhanced sensitivity of strong-field observables to higher-curvature effects. While the present analysis is based on an asymptotic perturbative treatment, our results provide a consistent framework for estimating leading-order deviations from general relativity and suggest that high-resolution observations, including very-long-baseline interferometry and gravitational-wave measurements, may offer constraints on modified gravity models.