Asymptotic Schwarzschild solutions in $f(R)$ gravity and their observable effects on the photon sphere of black holes

We investigate asymptotic Schwarzschild exterior solutions in the context of modified gravity theories, specifically within the framework of $f(R)$ gravity, where the asymptotic behavior recovers the standard Schwarzschild solution of General Relativity. Unlike previous studies that rely mainly on analytical approximations, our approach combines asymptotic analysis with numerical integration of the underlying differential equations. Using these solutions, we analyze strong lensing effects to obtain the photon sphere radius and the corresponding capture parameter. Considering rings produced by total reflection, we define the photon sphere width as the difference between the first total reflection and the capture parameter; and study how it is modified in the $f(R)$ scenario. Our results show that the photon sphere width increases in the presence of $f(R)$-type modifications, indicating deviations from GR that could be observable in the strong-field regime.

💡 Research Summary

This paper investigates static, spherically symmetric black‑hole solutions in the simplest viable class of modified gravity, namely quadratic f(R) gravity with Lagrangian f(R)=R + a R². The authors require that the exterior metric asymptotically approach the Schwarzschild solution of General Relativity (GR) while allowing for small curvature corrections parameterised by the constant a. By writing the line element as
 ds² = B(r) dt² − A(r) dr² − r²(dθ² + sin²θ dφ²)
and introducing dimensionless perturbation functions m(r) and U(r) through
 B(r)=1−r_S