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 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.
Highlights
The radius r = 3rg/2 and it is densely filled by light rings located at different values of the polar angle θ
Situation becomes more complicated in stationary axisymmetric spacetimes with rotation, when circular orbits typically exist in the equatorial plane in presence of Z2 symmetry θ → π − θ
Due to existence of the Carter integral, the geodesic equations give rise to independent equations for r and θ motion, from which one finds that the orbits with constant r exist in the interval r p < r < rr for which θ oscillates between some bounds, so that the orbits lie on the some spherical surface
Summary
In non-spherical static spacetimes, properties of the photon spheres can be shared by the photon surfaces of nonspherical form. The existence of the photon sphere is related to spherical symmetry of spacetime It is worth noting, that the photon sphere is not destroyed by the Newman–Unti– Tamburino (NUT) parameter, in which case the so(3) algebra still holds locally, though metric is already non-static. Note that in rotating spacetimes the photon orbits with constant Boyer–Lindquist radius may exist, but they do not fill densely any spheres, since their existence requires certain relation between the constants of motion Such orbits fill the three-dimensional PR which in this case can be interpreted as a set of non-closed timelike hypersurfaces, parameterized by the value of the azimuthal impact parameter ρ = L/E, where L , E are the motion integrals corresponding to timelike and azimuthal Killing vectors [26,27]. For some of them the mapping {ρ} → P R not always univalent, contrary to the Kerr case [11, 12]
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