Quantum gravitational stellar evolution beyond shell-crossing singularities

TL;DR

The paper addresses extending spacetime after shell-crossing singularities in loop-quantum-gravity-inspired stellar collapse by enforcing Darmois-Israel junction conditions on a non-isolated thin shell. The interior, post-bounce region is matched to an exterior effective Schwarzschild geometry in Painlevé-Gullstrand coordinates, ensuring a timelike shell with a continuous induced metric while allowing mass exchange with the interior. The authors derive the energy-balance equation $M-\frac{4}{3}\pi\rho R^3=2\pi\sigma R^2(\sqrt{\dot{R}^2+F_+}+\sqrt{\dot{R}^2+F_-})$ and the shell-mass evolution $\frac{d}{d\tau}(4\pi R^2\sigma)=-4\pi R^2\rho\dot{T}_-(\dot{R}+\dot{T}_-N_-^r)$, and provide a toy numerical model showing the shell grows and drives the system toward an inter-universal, white-hole-driven exit from the first asymptotic region. The resulting conformal diagrams illustrate the causal structure and demonstrate a physically consistent extension beyond SCS applicable to a broad class of effective collapse models. This framework lays the groundwork for more complete IVP analyses and potential inclusion of semi-classical effects such as Hawking radiation.

Abstract

Models of effective stellar collapse inspired by loop quantum gravity predict a bounce when the stellar energy density reaches the Planck scale, typically followed by the formation of shell-crossing singularities. This work aims to extend the spacetime beyond these singularities by employing a Hamiltonian formulation of the Darmois-Israel junction conditions, treating the singularity as a non-isolated thin dust shell. By construction, the shell's motion remains timelike throughout the entire evolution, regardless of the amount of initial stellar mass, and the induced metric on the shell remains continuous. The resulting stellar evolution produces an inter-universal wormhole, analogous to the simpler Oppenheimer-Snyder scenario. The proposed approach provides a general framework for any effective (or classical) theory of stellar collapse characterized by shell-crossing singularities.

Figures (3)

• Figure 1: Schematic plots illustrating two dynamical extensions of LTB solutions beyond shell-crossing singularities (left and right diagrams) of the same solutions to the LTB equations of motion are shown, where the trajectories of the LTB shells are evolved from a initial time slice and displayed on the areal radius $r$–proper time $t$ plane. Left diagram. The dynamical extension in which shell crossings among the LTB shells are allowed and therefore manifest. In this picture, different shells may intersect in the $(r,t)$ plane, leading to the formation of shell-crossing singularities. In particular, we can distinguish between a family of shells that undergo an early bounce and another family that bounces at later times. An inconsistency arises because the latter family encodes a spacetime geometry (metric) that differs from that of the former family in the region where their domains of validity overlap. In other words, at spacetime points $(r,t)$ within the overlapping region, two distinct spacetime metrics are assigned by the LTB shells belonging to the respective families. Importantly, the proper time $t$ on the LTB shells remains continuous across the shock surface (shown in black). The picture of this type was explicitly obtained in Fazzini:2023ovaBobula:2024chrCafaro:2024lreLiu:2025fil within the model originally formulated in Kelly:2020lecHusain:2021ojzHusain2022Giesel:2023hys. Right diagram. We propose the following dynamical extension of the LTB solutions. Starting from the trajectories of the LTB shells, we first identify the causally earliest shell-crossing singularity. To the future of this event, we investigate the existence of a junction surface that separates the families of LTB shells such that the induced metric remains continuous across the surface and the surface acts as a boundary between distinct shell evolutions. The shells either emanate from or terminate at this junction surface. Critically, the proper time $t_\pm$ on the LTB shells is continuous prior to shock formation, i.e. $t_+ = t_-$. After the shock forms, however, this proper time becomes discontinuous across the junction surface, as indicated in the plot. Since no further shell crossings occur beyond this point, the spacetime contains no additional SCSs, strictly speaking, apart from the first one. In general, it is possible to find a junction surface with a continuous induced metric, while the extrinsic curvature is discontinuous across it. This discontinuity signals the presence of a thin shell localized on the junction surface. Importantly, the depicted trajectories of each LTB shell are fully determined by the initial data, similarly as in the case illustrated by the left diagram. However, in the right diagram, the junction conditions determine which segments of these trajectories are physically realized at a given time. This picture serves as a precursor to a more detailed and fully consistent analysis of the initial-value problem in gravitational theories that are prone to the development of shell-crossing singularities.
• Figure 2: Numerical results for the non-isolated thin shell as a future of a shell crossing singularity. Top left. Time evolution of areal radius at the non-isolated thin shell. Top right. The time evolution of the FLRW mass $M_\mathrm{FLRW}$ along with the inertial thin shell mass $4 \pi \sigma R^2$. Bottom left. The coordinate relative velocity of the thin shell with respect to the FLRW interior. Bottom right. The discontinuity between the time coordinates $T_-$ and $T_+$ across the non-isolated thin shell characterized by the function $\mathrm{d}T_-/\mathrm{d}T_+$. Since this function is not constant, the two time coordinates do not evolve uniformly. For each plot we have taken $M=10$, $\gamma=1$, $T_{\text{ini}}=1$, $R_\text{ini}=r_\text{min}+0.01$, where $r_\text{min}=r_\text{min}(x=1, t=0)$ and $\dot{R}_\mathrm{ini}=0.01$ in Planck units.
• Figure 3: Left diagram. The conformal diagram for an inhomogeneous dust collapse with an initial Gaussian density profile, obtained within the analysis of the LQG-inspired model Kelly:2020lecHusain:2021ojzHusain2022Giesel:2023hys, presented in Bobula:2024chr (see also Figs. 3 and 9 therein). The LTB shells evolve from the initial time surface and subsequently undergo shell crossings in the deep Planck regime. Shell-crossing singularities are therefore present, and the future light cone emanating from the causally first shell-crossing singularity defines the shadow—the greyed-out region—in which the employed dynamical extension beyond the first SCS is expected to be unreliable. The spacetime structure in this region constitutes a concrete realization of the schematically depicted dynamical extension shown in the left diagram of Fig. \ref{['schematic_plot']}. Right diagram. The numerically computed conformal diagram for the toy model analysed in this work illustrates a consistent dynamical extension beyond shell-crossing singularities, compatible with the right schematic plot in Fig. \ref{['schematic_plot']}, and covering the grayed-out region in the left diagram. A non-isolated thin shell emerges at the spacetime point where the causally first shell-crossing singularity occurs. In the toy model considered here, we employ an approximation in which the interior LTB shells are treated as homogeneous and isotropic FLRW shells. Furthermore, the shells constituting the tail of the initial Gaussian density profile are assumed to have already been collected into the thin shell at the initial time of the evolution (shown in Fig. \ref{['3plots']}). The non-isolated thin shell emerges from the deep Planck-regime region, crosses the first inner null white-hole horizon (which also constitutes a Cauchy horizon), then crosses the outer white-hole horizon, and finally approaches the vicinity of timelike infinity in the new asymptotic region. Inset. As shown in the right schematic plot in Fig. \ref{['schematic_plot']}, the non-isolated thin shell can either collect or release LTB shells. Here we illustrate a single instance of this process, in which the thin shell collects one of the interior FLRW shells. This diagram is generated using the same model parameters and initial conditions as in Fig. \ref{['3plots']}, in addition, we set $\kappa = 0.05$. The interior FLRW shells shown correspond to comoving radii $x \in \{0,0.25,0.5,0.75,0.91\}$. All quantities are expressed in Planck units.