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arXiv · 2606.19809

Photon surfaces extension in general spherical dust collapse

Roberto Giambò·Camilla Lucamarini
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Abstract

We extend the analysis of photon surfaces in spherical dust collapse to the general, non-marginally bound case, i.e.\ allowing the \textit{energy function} $k(x)$ of the Lema\^{\i}tre--Tolman--Bondi (LTB) model to be non-zero. Starting from the dynamical-systems formulation developed in our previous work for the marginally bound case (arXiv:2509.01368), we derive the photon surface equations for the LTB metric with $k(x)\neq 0$, and we recast the geometric condition for a timelike hypersurface to be a photon surface as a non-autonomous first-order dynamical system. Even though the LTB evolution equation does not integrate in closed form when $k(x)\neq 0$, the implicit solutions available in the literature, together with comparison theorems for ordinary differential equations, are sufficient to show that the only physically acceptable extension of the exterior photon sphere $r=3M$ into the collapsing dust cloud is a null hypersurface, generated by outgoing radial null geodesics. Combining this fact with the known results on the causal structure of the general LTB model, we then establish that the photon surface reaches the central singularity if and only if the singularity is naked, thereby extending to the general case the picture already known in the marginally bound regime. The structural dichotomy between the naked and covered end states is also discussed in connection with possible observational signatures in the early-time evolution of black-hole shadows.

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Roberto Giambò, Camilla Lucamarini. 2026-06-18. Photon surfaces extension in general spherical dust collapse. https://arxiv.org/abs/2606.19809

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