Exact Formulas for Spherical Photon Orbits Around Kerr Black Holes

Abstract

Exact formulas relating the radii of the spherical photon orbits to the black hole's rotation parameter and the effective inclination angle of the orbit have been known only for equatorial and polar orbits up to now. Here we provide exact formulas for nonequatorial orbits that lie between these extreme limits. For a given rotation parameter of the black hole, there is a critical inclination angle below which there are four null photon orbits two of which are in the exterior region. At the critical angle, the radii of these orbits are explicitly found. We also provide approximate but very accurate formulas for any orbit. To arrive at these analytical solutions, we carefully study the sextic polynomial in the radius that arises for null spherical geodesics.

arXiv comment6 pages, 5 figures and 2 tables; v2 matches the published version

References in corpus (8)

• Observation of Gravitational Waves from a Binary Black Hole Merger
• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• GW190521: A Binary Black Hole Merger with a Total Mass of $150M_{\odot }$
• Stationary black holes and light rings
• Fundamental photon orbits: black hole shadows and spacetime instabilities
• Spherical null geodesics of rotating Kerr black holes
• Effect of gravitational radiation reaction on circular orbits around a spinning black hole
• Where is the extremal Kerr ISCO?

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• Gravitational Lensing in the Kerr Spacetime: An Analytic Approach for Light and High-Frequency Gravitational Waves
• Light propagation around a Kerr-like black hole immersed in an inhomogeneous anisotropic plasma in Rastall gravity: Analytical solutions to the equations of motion
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