Photon regions and umbilic conditions in stationary axisymmetric spacetimes

Photon region (PR) in the strong gravitational field is defined as a compact region where photons can travel endlessly without going to infinity or disappearing at the event horizon. In Schwarzschild metric PR degenerates to the two-dimensional photon sphere $r=3r_g/2$ where closed circular photon orbits are located. The photon sphere as a three-dimensional hypersurface in spacetime is umbilic (its second quadratic form is pure trace). In Kerr metric the equatorial circular orbits have different radii for prograde, $r_p$, and retrograde, $r_r$, motion (where $r$ is Boyer-Lindquist radial variable), while for $r_p

Keywords:

• Foliation (geology)
• Solving the geodesic equations
• Event horizon
• Classical mechanics
• Kerr metric
• Mathematical physics
• Spacetime
• Hypersurface
• Physics
• Schwarzschild metric
• Photon sphere

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