Photon surfaces extensions for dynamical gravitational collapse

Abstract

The equations for the photon surface in spherical symmetry are worked out, starting from arXiv:gr-qc/0005050, in the most general dynamical setting. We show that the condition for a timelike hypersurface to be a photon surface can be reformulated as a non-autonomous dynamical system, whose analysis reveals that the same condition also holds when the surface is generated by a null radial geodesic. As an application, we consider a well-known model of a spherical dust cloud undergoing gravitational collapse. In the marginally bound case the photon surface condition reduces to an exact planar dynamical system, whose analysis shows that the extension of the exterior photon surface $r=3M$ into the cloud is unique, and null, if and only if the central singularity is covered, an open set of timelike extensions being available otherwise. Comparing our findings with those in arXiv:1910.13758, we then investigate analytically whether the null extension covers the singularity developing in the Lemaitre-Tolman-Bondi (LTB) model.