Photon surfaces extensions for dynamical gravitational collapse

TL;DR

The paper develops a general dynamical framework for photon surfaces in spherically symmetric spacetimes by reformulating the photon-surface condition as a non-autonomous dynamical system, and shows this condition also governs surfaces generated by null radial geodesics. It applies the framework to a marginally bound LTB dust collapse with exterior Schwarzschild matching, demonstrating that the exterior photon surface at $r=3M$ extends uniquely into the interior as a null hypersurface generated by outgoing radial null geodesics. The analysis reveals a tight link between the central singularity's visibility and the photon surface's interior extension: the surface reaches the center if and only if the central singularity is naked, otherwise it terminates at the regular center, with implications for shadow formation and cosmic censorship. These results refine prior claims and provide a rigorous, geometry-based account of photon-surface behavior in dynamical collapse, suggesting extensions to more general collapse models in future work.

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. Comparing our findings with those in arXiv:1910.13758 we establish that the photon surface uniquely extends in the interior spacetime as a null hypersurface, allowing us to analytically investigate whether it covers the singularity developing in the LTB model.

Figures (5)

• Figure 1: Schematic picture of the collapse of spherical dust. The process begins from a regular configuration at $t=0$. Every shell $x \in [0, x_b]$ within the spherical cloud undergoes collapse, becoming trapped at comoving time $t_h(x)$ and fully collapsing at $t_s(x)$. The symmetry center gets trapped simultaneously as it becomes singular, allowing for the chance that photons might escape from the central singularity and reach the region $r>2m$.
• Figure 2: The behavior of the vector field associated to \ref{['eq:ODE1a-mb']}--\ref{['eq:ODE1b-mb']} is such that the only way to left--extend a solution from $x=x_b$ to $x=0$ is when $y_0=1$. Notice that the field is non autonomous (here is represented the situation at $t=1.01$ only). Other parameters are $n=2,\,a=1.5$.
• Figure 3: The extension of the photon surface in case when the central singularity is naked. The parameters of \ref{['eq:mu']} are set as follows: $n=2$, $a=1.5$, with the star boundary $x_b=0.33$. The curve $t_\xi(x)$, with $\xi=1.35$, is a supersolution of the ODE \ref{['eq:nullradial']} in $[0,x_b]$. The photon surface is extended back to the central singularity.
• Figure 4: Another extension of the photon sphere with a central naked singularity. Here $n=3$, $a=8$ and $x_b=0.3$. The parameter of the supersolution here is $\xi=6$.
• Figure 5: The extension of the photon sphere, in case the central singularity is not naked, is necessarily continued back to the regular centre $t<1$. Here $n=4$, $a=8$, $x_b=0.3$.

Theorems & Definitions (10)

• Definition 1
• Remark 1
• Theorem 1
• Remark 2
• Example 1
• Lemma 1
• proof
• Theorem 2
• Remark 3
• Theorem 3