Quasi-local photon surfaces in general spherically symmetric spacetimes

TL;DR

This paper introduces a new quasi-local definition of photon surfaces in spherically symmetric spacetimes, generalizing previous concepts and applying it to gravitational collapse models, revealing insights about photon surface formation and timing.

Contribution

It provides a generalized, gravity-relevant definition of photon surfaces and derives analytic and numerical solutions in collapse models, linking photon surface timing to total mass.

Findings

Analytic photon surface solution in the Oppenheimer-Snyder model

Numerical solutions for marginally bounded collapse in LTB model

Photon surface timing mainly depends on total mass, not size or gravitational strength

Abstract

Based on the geometry of the codimension-2 surface in a general spherically symmetric spacetime, we give a quasi-local definition of a photon sphere as well as a photon surface. This new definition is the generalization of the one by Claudel, Virbhadra, and Ellis but without reference to any umbilical hypersurface in the spacetime. The new definition effectively rules out the photon surface which has noting to do with gravity. The application of the definition to the Lemaitre-Tolman-Bondi (LTB) model of gravitational collapse reduces to a problem of a second order differential equation. We find that the energy balance on the boundary of the dust ball can provide one appropriate boundary condition to this equation. Based on this key investigation, we find an analytic photon surface solution in the Oppenheimer-Snyder (OS) model and reasonable numerical solutions for the marginally bounded collapse in the LTB model. Interestingly, in the OS model, we find that the time difference between the occurrence of the photon surface and the event horizon is mainly determined by the total mass of the system but not the size or the strength of gravitational field of the system.