Photon surface extensions in general spherical dust collapse
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\^ıtre--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 [gr-qc]), we derive the photon surface equations for the LTB metric with k(x)≠ 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 inclosed form when k(x)≠ 0, the implicit solutions available in the literature, together with comparison theorems for ordinary differential equations, are sufficient to reduce the problem to a single scalar equation, governed by one quantity which vanishes identically for homogeneous data. Using this reduction we show that the extension of the exterior photon sphere r=3M into the collapsing cloud is unique, and null, if and only if the central singularity is covered. When the singularity is naked, uniqueness fails: besides the null hypersurface there exists a one-parameter family of genuinely timelike photon surfaces matching r=3M, all of them reaching the central singularity. The threshold separating the two behaviours coincides with the one separating black hole and naked singularity end states. We also show that the null photon surface reaches the central singularity if and only if the singularity is naked. The structural dichotomy between the naked and covered end states is discussed in connection with possible observational signatures in the early-time evolution of black-hole shadows.
Related
- Critical and extremal gravitational collapse of a spherical charged scalar field in 4+1 spacetime dimensions
- Studying spherical collapse and its implications in the Eddington-inspired Born–Infeld gravity theory
- Soft-photon theorem near static spherical horizons in generic dimensions
- Bounds on the minimum orbital period of general static spherical black holes: a sharp upper bound and a conditional lower bound
- Improved constraints on higher-derivative effective field theory extensions of general relativity with GW230627_015337
BibTeX
@misc{giambo2026photon,
title = {{Photon surface extensions in general spherical dust collapse}},
author = {Roberto Giambò and Camilla Lucamarini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.19809},
eprint = {2606.19809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19809},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Photon surface extensions in general spherical dust collapse
AU - Roberto Giambò
AU - Camilla Lucamarini
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.19809
UR - https://arxiv.org/abs/2606.19809
AB - 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\^ıtre--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 [gr-qc]), we derive the photon surface equations for the LTB metric with k(x)≠ 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 inclosed form when k(x)≠ 0, the implicit solutions available in the literature, together with comparison theorems for ordinary differential equations, are sufficient to reduce the problem to a single scalar equation, governed by one quantity which vanishes identically for homogeneous data. Using this reduction we show that the extension of the exterior photon sphere r=3M into the collapsing cloud is unique, and null, if and only if the central singularity is covered. When the singularity is naked, uniqueness fails: besides the null hypersurface there exists a one-parameter family of genuinely timelike photon surfaces matching r=3M, all of them reaching the central singularity. The threshold separating the two behaviours coincides with the one separating black hole and naked singularity end states. We also show that the null photon surface reaches the central singularity if and only if the singularity is naked. The structural dichotomy between the naked and covered end states is discussed in connection with possible observational signatures in the early-time evolution of black-hole shadows.
ER -