Shell-Crossing Singularities

• Shell-crossing singularities are weak spacetime divergences occurring when neighboring dust shells cross, leading to caustics and density spikes.
• They are mathematically identified by the vanishing Jacobian determinant between Lagrangian and Eulerian coordinates, central to gravitational collapse and cosmological models.
• These singularities trigger multi-stream regions and vorticity generation, and their analysis is extended via quantum gravity and weak solution frameworks.

Shell-crossing singularities are weak spacetime singularities that generically arise in self-gravitating, pressureless (dust) systems due to the overtaking or crossing of neighboring matter shells. Their key signature is the vanishing of the Jacobian determinant relating comoving (Lagrangian) coordinates to physical (Eulerian) positions, resulting in a breakdown of the single-stream fluid regime, divergence of density, and the onset of multi-stream (caustic) regions. These singularities play a central role not only in cosmological structure formation, but also in gravitational collapse, the late-time evolution of cosmological models with cold dark matter, and the dynamics of semiclassical or quantum-corrected spacetimes.

1. Formal Definition and Physical Manifestation

Shell crossing is mathematically defined by the vanishing of the radial derivative of the areal radius in Lagrangian (comoving) coordinates: $\partial_R r = 0$ where $R$ labels the dust shell, and $r(t, R)$ is the areal radius (Delliou et al., 2011). In multidimensional Lagrangian frameworks—such as for cosmological fluids—the singularity occurs when the Jacobian $J = \det(\partial x / \partial q)$ vanishes, with $x$ denoting Eulerian position and $q$ the initial Lagrangian label (Rampf et al., 2020). This condition signals a loss of invertibility: multiple comoving fluid elements are mapped to the same physical location, giving rise to caustics where density formally diverges.

These "weak" singularities do not crush physical volumes to zero (unlike shell-focusing singularities); rather, they are characterized by diverging density or curvature invariants, but are extendable in the sense that geodesics can, under some circumstances, traverse them unimpeded (Joshi et al., 2012). In Newtonian cosmology, shell-crossings correspond to the points at which the mapping from initial to final positions—treated in the Zel'dovich approximation—ceases to be monotonic, resulting in an S-shaped phase-space configuration.

2. Mathematical Frameworks and Core Equations

Spherical, Inhomogeneous Λ-CDM Models

Shell-crossing singularities naturally arise in spherically symmetric, inhomogeneous Λ–CDM models described by the Generalised Lemaître–Tolman–Bondi (GLTB) class of metrics: $$ds^2 = -dt^2 + \frac{(\partial_R r)^2}{1 + E(R)} dR^2 + r^2 d\Omega^2$$ with dynamics governed by the generalized energy equation: $$\dot{r}^2 = \frac{2M}{r} + E + \frac{\Lambda}{3} r^2$$ Here $M(R)$ is the Misner–Sharp mass, $E(R)$ the energy (curvature) profile, and $\Lambda$ the cosmological constant. Shell crossing occurs precisely when $\partial_R r = 0$ (Delliou et al., 2011).

General Relativistic Collapse

In general spherically symmetric collapse, coordinates are chosen so that $R(t, r)$ is the physical radius satisfying: $$R' = v + r v' \quad \text{with} \quad R(t, r) = r v(t, r)$$ Shell crossing occurs for the first time when $R' = 0$ (Joshi et al., 2012). Notably, it is possible to show that for any spherically symmetric matter configuration, there always exists a finite comoving neighborhood around the center ($r=0$) free of shell-crossings at all times.

Multi-Dimensional and Quasi-1D Flows

In cosmological perturbation theory, especially when pushing beyond shell crossing, the Euler–Poisson equations are recast into Lagrangian coordinates: $x(q; \tau) = q + \xi(q; \tau)$ with $\tau$ the linear growth time (Rampf et al., 2017). Shell crossing is then detected by the vanishing of $J = \det(\partial x / \partial q)$. In the quasi-one-dimensional (Q1D) limit, all-order recursion relations for the time-Taylor coefficients of the displacement field can be developed, facilitating highly accurate computations of the shell-crossing time and location.

3. Role in Cosmological Structure Formation

Shell-crossing is fundamental to the formation of non-linear cosmic structures in cold dark matter cosmology. Before shell crossing, the density and velocity fields remain single-valued and can be accurately described by Lagrangian perturbation theory (LPT), notably the Zel'dovich approximation. The onset of shell crossing marks the appearance of caustics, where multi-stream regions (with overlapping dark matter flows) develop and density formally diverges (Rampf et al., 2020).

After shell crossing, the Eulerian fields develop non-analytic or divergent features, signaling the breakdown of perturbative expansions. Extensions such as post-collapse perturbation theory (Taruya et al., 2017) deploy local Taylor expansions about shell-crossing points to include multi-stream effects, often supplemented by adaptive smoothing schemes that coarse-grain phase-space structures over a scale tied to the post-collapse “next crossing time.”

In $1+1$ dimensions, the system evolves toward attractors in the multi-streaming regime: the mapping $x(q,\tau)$ “flattens” near shell-crossing points, particles increasingly cluster towards shocks, and the phase-space configuration exhibits the “adhesion” model’s asymptotic geometry (Pietroni, 2018). In Q1D flows, even an infinitesimal departure from strict one-dimensionality leads not only to the earlier onset of shell crossing but also to pronounced spatial localization of the first caustic.

4. Shell-Crossing and the Dynamics of Gravitational Collapse

Within the context of gravitational collapse in general relativity, shell-crossing presents both a practical and theoretical challenge. While classical collapse models like Oppenheimer–Snyder, which feature discontinuous density profiles, can evade crossings, any continuous density profile of compact support generically leads to shell-crossing singularities (Fazzini et al., 2023).

In effective models incorporating quantum-gravitational corrections (such as loop quantum gravity, LQG), shell crossing remains generic even as the dynamics at high density are modified to include a quantum-gravitational bounce. Following the matter bounce, characteristics of the effective PDEs governing the LTB spacetime cross, necessitating the transition from classical (differentiable) solutions to weak (distributional) ones. These weak solutions necessarily admit discontinuities—shocks—analogous to those in nonlinear fluid dynamics (Fazzini et al., 2023). The LQG effects effectively convert the classical divergence into the formation and propagation of a thin shell, whose subsequent evolution (governed by polymerized Hamiltonian Israel junction conditions) remains regular and timelike throughout (Fazzini, 5 Sep 2025).

The effective Hamiltonian dynamics for a collapsing thin shell are expressed as: $$M = \frac{m}{2}\left[\sqrt{\dot{R}^2+1} + \sqrt{\dot{R}^2+1-\frac{R_S}{R} + \frac{\gamma^2\Delta R_S^2}{R^4}}\right]$$ with evolution giving rise to a bounce at radius $R_{\text{bounce}} \sim (\gamma^2 \Delta R_S)^{1/3}$, beyond which the shell re-expands and emerges from a white-hole vacuum region (Fazzini, 5 Sep 2025).

5. Shear, Trapped Matter Shells, and Relativistic Effects

Shear plays an essential role in the dynamics near shell crossing. In spherical, inhomogeneous Λ–CDM models, nonzero shear—arising from gradients in the energy and mass profiles—favors the cracking or splitting of trapped matter shells (Delliou et al., 2011). The latter are defined as the loci where the combination $(\Theta/3 + a) = 0$, ($\Theta$: expansion, $a$: shear) and their time derivative vanishes, marking surfaces separating collapsing and expanding regions. Shell-crossing generically converts a unique separating shell into two limit shells (an inner and an outer trapped matter shell), thus “shielding” the collapse region from the expanding universe and vice versa.

Comparisons to Newtonian theory show that while the qualitative conditions for caustic formation (vanishing of the Jacobian determinant) are analogous, the relativistic (GLTB) framework includes global or non-local effects via the integral character of $M$ and the inclusion of shear, resulting in a richer phenomenology (Delliou et al., 2011). Classification of singularities in related settings (e.g., Wigner caustic singularities in semiclassical mechanics) reveals analogous mathematical structures such as fold and cusp caustics (Domitrz et al., 2012).

6. Impact on Vorticity and Magnetic Weyl Structure

Shell-crossing does not merely mark the end of the single-stream regime; it also has profound implications for the generation of vorticity and the evolution of higher-order gravitational degrees of freedom. High-resolution general relativistic N-body simulations reveal that, on scales corresponding to shell-crossing, the vorticity power spectrum of the dark matter fluid dominates over that of the velocity divergence (Umeh, 2023). Analytical models show that vorticity is generated precisely at the boundaries between expanding and collapsing regions, with the amplitude set by discontinuities in the gradients of the gravitational potential, pressure, and expansion rate.

This generation of vorticity further implies the emergence of a non-vanishing magnetic part of the Weyl tensor, with potential consequences for cluster-scale magnetic fields and the detailed relativistic dynamics of halo formation (Umeh, 2023). The presence of such vorticity, as a result of shell crossing, provides a relativistic mechanism for the decoupling of halos from the Hubble flow and may influence the overall acceleration dynamics in a spatially averaged universe.

7. Resolution and Interpretation in Quantum-Corrected and Geometric Approaches

Shell-crossing singularities, being weak singularities, are not necessarily pathological and often admit physically meaningful continuations. In effective quantum gravitational models (notably effective LQG), the regime beyond the bounce—where shell crossing becomes inevitable for inhomogeneous dust—is best described by weak solutions in which energy density and geometry experience shocks. In the Hamiltonian framework, these correspond to formation and propagation of a thin shell, whose timelike evolution remains regular, thus providing a mechanism for extending the spacetime through the would-be singular region (Fazzini, 5 Sep 2025).

Analyses in semiclassical quantum systems demonstrate that analogous “crossing singularities” in path-integral saddle-point approximations require analytic continuation of complex classical paths through singularities or branch cuts in the potential. These continuations modify the tunneling amplitude by extra action terms and reflect a universal structure in singularity formation that bridges cosmological shell crossings and quantum tunneling phenomena (Feldbrugge et al., 2023).

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In summary, shell-crossing singularities are generically encountered in non-linear gravitational evolution, demarcating the transition from single-stream to multi-stream flow or collapse. They are central to the formation of caustics, act as sources for vorticity and potentially the magnetic Weyl tensor, and serve as critical markers for the limits of classical perturbative theories. With advances in both analytic and numerical methods, including weak solution frameworks and quantum gravity-inspired models, it is possible to rigorously analyze, characterize, and even extend the spacetime evolution through these singularities, providing deeper insight into both cosmic structure formation and the ultimate fate of gravitationally collapsing systems.