Black holes and their shadows in F(R) gravity

Abstract

We investigate the radii of the photon sphere and the black hole shadow in the framework of $F\left(R\right)$ 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 $F_R\left(r\right)\equiv {\left(\frac{dF\left(R\right)}{dR}\right)}_{R=R\left(r\right)}$, where $r$ is the radial coordinate. Solving the equation, we find $F\left(R\right)$ as a function of $r$, $F_R=F_R\left(r\right)$. By using the assumed and obtained geometry, one can calculate the scalar curvature $R$ as a function of $r$, $R=R\left(r\right)$, which could be solved with respect to $r$ as $r=r\left(R\right)$. Then one finds the functional form of $F_R$ as a function of the scalar curvature $R$, $F_R=F_R\left(R\right)=F_R\left(r=r\left(R\right)\right)$.

We then solve the corresponding equation perturbatively by assuming the variation of the geometry from the Schwarzschild spacetime could be small and also the deviation of $F\left(R\right)$ gravity from Einstein’s gravity is small. As a result, we obtain an inhomogeneous linear differential equation and solve the equation in the region around the radius of the photon sphere. This is a quite general approach which may be adopted for any modified gravity. With the help of the obtained solutions, we calculate the radii of the photon sphere and the black hole shadow and find the parameter regions consistent with the observations of M87${}^{\ast }$ and Sgr A${}^{\ast }$.