Photon trapping in static axially symmetric spacetime

Abstract

Recently, several new characteristics have been introduced to describe null geodesic structure of strong gravitational field, such as photon regions, transversely trapping surfaces and some generalizations. They give an alternative and concise way to describe lensing and shadow features of compact objects with strong gravitational field without recurring to complete integration of the geodesic equations. Here we test this construction in the case of the Weyl metrics when geodesic equations are non-separable, and thus can not be integrated analytically, while the above characteristic surfaces and regions can be described in a closed form. We develop further our formalism for a class of static axially symmetric spacetimes introducing more detailed specification of transversely trapping surfaces in terms of their principal curvatures. Surprisingly, we find in the static case without spherical symmetry certain features, such as photon regions, previously known in the Kerr space. These photon regions can be regarded as photon spheres, "thickened" due to oblateness of the metric.including the dilaton field [34]).Meanwhile, in the stationary spacetimes photon surfaces generically do not exist, being, in some sense, disintegrated by rotation. The Taub-NUT metric, though belongs to this class, is an exception, whose existence is explained by local SO(3) symmetry which is preserved. In the Kerr metric the circular photon orbits exist in the equatorial plane [35]. Non-equatorial orbits with constant Boyer-Lindquist radii no longer belong to any plane, but lie on the surface of a sphere instead ("spherical photon orbits" [36]). But such surfaces are not photon spheres, which by definition should be densely filled. In the Kerr case every spherical orbit corresponds to certain value of the impact parameter defined as ratio of the angular momentum to the energy. Altogether spherical orbits now will a three-dimensional photon region (PR) [37][38][39], which can be regarded as "thickened" photon sphere. Note that the closed photon orbits may exist in more general spacetimes in which case the name of fundamental or/and spheroidal photon orbits was suggested [40][41][42].More general trapping surfaces, proposed in [43] and further studied in [44], were called transversely trapping surfaces (TTS). In Schwarzschild case TTS are the spheres with the radii r ≤ 3M . These are defined as surfaces such that initially tangent photons either remain in them or move inward; these do exist in Kerr and more general stationary axially symmetric metrics.The totality of TTSs form a three-dimensional region (TTR), which apparently will be invisible if one looks at the Schwarzschild black hole illuminated from behind. Further generalizations suggested in [44] include partial TTS (PTTS) which are non-closed surfaces (contrary to the definition in [43]) of the shape of a spherical cap. Finally, it is reasonable to introduce anti-TTS and PTTS (abbreviated as ATTS and APPTS, the corresponding regions -TTR and PTTR respoctively) replacing inward to outwa...

Photon regions and umbilic conditions in stationary axisymmetric spacetimes

“…A similar design has been proposed in [37] to investigate geometry and topology of PR in Kerr gravitational field. Our purpose here is to investigate the tangent space structure of PRs in more general stationary axisymmetric spacetimes based on previous work [10][11][12]26,27].…”

“…An important feature of this equation in the static case (ω = 0) is the parity in the parameter ρ (Compare with [26]). In particular, every S ρ will coincide with S −ρ .…”

“…In particular, every S ρ will coincide with S −ρ . Note also the possibility of the presence of boundaries at the spatial sections of fundamental photon hypersurfaces where the relation ϕ-TTS holds [26]:…”

“…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].…”

“…3.5. It is shown that they have PRs of three different types, and as a consequence, different optical shadows [11,26,42,43]). For some of them the mapping {ρ} → P R not always univalent, contrary to the Kerr case [11,12].…”