Spherical photon orbits around Kerr-MOG black hole

TL;DR

This paper analyzes photon orbits around Kerr-MOG black holes, deriving equations and classifying orbit types, revealing how the deformation parameter $mbda$ influences orbit structure and stability, with implications for observational signatures.

Contribution

It provides the first detailed derivation of photon orbits in Kerr-MOG black holes, including exact formulas and analysis of the effects of the deformation parameter and rotation on orbit characteristics.

Findings

Polar plane has two effective orbits inside and outside the horizon.

Equatorial plane has four effective orbits, two inside and two outside.

Photon orbits are radially unstable.

Abstract

This study investigates photon orbits around Kerr-MOG black holes. The equation of photon of motion around the Kerr-MOG black hole is derived by solving the Hamilton-Jacobi equation, expressed as a sixth-order polynomial involving the inclination angle $v$, the rotation parameter $u$, and the deformation parameter $\alpha$ that characterizes modified gravity. We find that $\alpha$ constrains the rotation of the black hole, modifying its gravitational field and leading to distinct photon orbital characteristics. Numerical analysis reveals that the polar plane ($v=1$) has two effective orbits: one outside and one inside the event horizon, while the equatorial plane ($v=0$) has four effective orbits: two outside and two inside the event horizon. Moreover, we derive the exact formula for general photon orbits between the polar and equatorial planes ($0<v<1$). In the extremal case, the rotation speed significantly impacts general photon orbits. A slowly rotating extremal black hole has two general photon orbits outside the event horizon, whereas a rapidly rotating extremal black hole has only one such orbit. In the non-extremal case, a critical inclination angle $v_{cr}$ exists in the parameter space $\left(v, u, \alpha \right)$. Below $v_{cr}$, there are four general photon orbits, while above $v_{cr}$, there are two orbits. At the critical inclination angle, three solutions are found: two photon orbits outside and one inside the event horizon. Additionally, the results indicate that all orbits are radially unstable. Furthermore, by analyzing photon impact parameter, we argue that $\alpha$ influences observational properties of the black hole.