Photon regions and umbilic conditions in stationary axisymmetric spacetimes

Abstract

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 $r_p<r<r_r$ the spherical orbits with constant $r$ exist which are no more planar, but filling some spheres. These spheres, however, do not correspond to umbilic hypersurfaces. In more general stationary axisymmetric spacetimes not allowing for complete integration of geodesic equations, the numerical integration show the existence of PR as well, but the underlying geometric structure was not fully identified so far. Here we suggest geometric description of PR in generic stationary axisymmetric spacetimes, showing that PR can be foliated by {\em partially umbilic hypersurfaces}, such that the umbilic condition holds for classes of orbits defined by the foliation parameter. New formalism opens a way of analytic description of PR in stationary axisymmetric spacetimes with non-separable geodesic equations.

References in corpus (16)

• Relativistic images of Schwarzschild black hole lensing
• Chaotic lensing around boson stars and Kerr black holes with scalar hair
• Fundamental photon orbits: black hole shadows and spacetime instabilities
• Shape and position of the shadow in the $\delta =2$ Tomimatsu-Sato space-time
• General relativistic polytropes with a repulsive cosmological constant
• On conformal Killing-Yano tensors for Plebanski-Demianski family of solutions
• The non-integrability of the Zipoy-Voorhees metric
• Free motion around black holes with discs or rings: between integrability and chaos - I
• Aspherical Photon and Anti-Photon Surfaces
• Efficiency of the Keplerian accretion in the Braneworld Kerr--Newman Spacetimes and mining instability of some naked singularity spacetimes
• Black Hole Shadows and Invariant Phase Space Structures
• Completing characterization of photon orbits in Kerr and Kerr-Newman metrics
• Uniqueness of static photon surfaces: Perturbative approach
• Photon surfaces as pure tension shells: Uniqueness of thin shell wormholes
• Rigidly rotating ZAMO surfaces in the Kerr spacetime
• Escape probability of a photon emitted near the black hole horizon

Cited by in corpus (13)

• Dynamical photon sphere and time evolving shadow around black holes with temporal accretion
• Photon surfaces in less symmetric spacetimes
• Killing tensors and photon surfaces in foliated spacetimes
• Galactic wormholes: Geometry, stability, and echoes
• Slice-reducible conformal Killing tensors, photon surfaces and shadows
• Classification of static asymptotically flat spacetimes with a photon sphere in Einstein-multiple-scalar field theory
• Quasi-local studies of the particle surfaces and their stability in general spacetimes
• Uniqueness of photon sphere for Reissner-Nordström electric-magnetic system
• Shadow formation in gravitational collapse: Redshift and blueshift by spacetime dynamics
• Photon surfaces, shadows and accretion disks in gravity with minimally coupled scalar field
• Massive particle surfaces and black hole shadows from intrinsic curvature
• Higher-dimensional electric-magnetic photon sphere uniqueness
• Uniqueness of electric-magnetic spacetimes with massive particle