General relativistic limit of f(R) gravity in the horizon scale of black hole
The environment of the Galactic Centre (GC) black hole, Sgr A* gives us new opportunities to test black hole physics and deviation from General Relativity. In this analytical study we calculate null geodesics around the GC black hole for the recently developed stationary, axisymmetric and vacuum metric of f(R) gravity theory. We study the impact of scalaron degree of freedom of f(R) gravity theory on size and shape of the black hole shadow. A minimum bound of $10^{-17}$ eV for scalaron mass has been obtained by using $1 \sigma$ upper bounds on deviation parameter measured by the Keck telescope and the VLT. Scalarons lighter than this bound are found to increase shadow size beyond that measured by the Event Horizon Telescope. We calculate shadow displacement and asymmetry and infer that $10^{-16}$ eV scalarons which produce exact Kerr sized shadow yield 6\% departure from Kerr quadrupole. From asymmetry we infer preservation of black hole no-hair theorem for such scalaron mass and further examine possibility of violation of the theorem. The same mass scale is found to reproduce the PPN parameter ($\gamma$) constrained in the weak field limit of the solar system. Gravitational identifiers, the Kretschmann scalar ($\kappa$) and gravitational potential ($\phi$) have been used to infer scalaron masses in the regime of S-stars near Sgr A* which are found to be consistent with the limits obtained by using shadow scales. We ensure existence of an appropriate general relativistic limit of f(R) gravity scalaron mass in the horizon scale of the black hole.