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=3rg/2 r=3rg/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, rₚ rp , and retrograde, rᵣ rr , motion (where r is Boyer–Lindquist radial variable), while for $$r_p rp<r