Black holes and their shadows in $F(R)$ gravity

TL;DR

This paper explores how black hole shadows and photon spheres behave in $F(R)$ gravity, deriving key equations and solutions to understand the geometry and curvature effects on black hole imaging.

Contribution

It derives the field equations for spherically symmetric $F(R)$ gravity and provides a method to determine the functional form of $F(R)$ from geometric quantities.

Findings

Derived third-order differential equation for $F_R(r)$

Expressed $F(R)$ as a function of the radial coordinate

Established a procedure to relate $F_R$ to scalar curvature R

Abstract

We investigate the radii of the photon sphere and the black hole shadow in the framework of $F(R)$ gravity. For this purpose, we derive the field equation for the corresponding theory when the general spherically symmetric and static configuration is considered. This equation is the third-order differential equation with respect to $F_R(r)\equiv \left. \frac{dF(R)}{dR}\right|_{R=R(r)}$, where $r$ is the radial coordinate. Solving the equation, we find $F(R)$ as a function of $r$, $F_R=F_R(r)$. By using the assumed and obtained geometry, one can calculate the scalar curvature $R$ as a function of $r$, $R=R(r)$, which could be solved with respect to $r$ as $r=r(R)$. Then one finds the functional form of $F_R$ as a function of the scalar curvature $R$, $F_R=F_R(R)=F_R\left( r=r\left(R\right)\right)$.