Does infinitesimal rotation prevent naked singularity formation in nonspherical relativistic collapse? The open reach of the Apostolatos–Thorne cylindrical bounce

Apostolatos and Thorne (1992) proved that an arbitrarily small amount of rotation halts the relativistic gravitational collapse of an infinitely long, thin cylindrical shell of counter-rotating dust (zero net angular momentum): by C-energy arguments the shell can never collapse below a nonzero minimum radius, $R\ge4\alpha\lambda$ (equivalently, a minimum circumference $8\pi\alpha\Lambda$), and after damped oscillations it should settle to a static equilibrium. The result stands in tension with the Shapiro–Teukolsky simulations of highly prolate (spindle-shaped) collisionless clouds, which produced divergent curvature outside the matter with no apparent horizon — a candidate naked singularity. The open problem, acknowledged in and around the Apostolatos–Thorne paper, is the reach of the rotation-halting mechanism beyond its idealization: whether the bounce and its quantitative lower bound survive shells with net angular momentum, realistic pressure, and gravitational radiation; whether arbitrarily small rotation or pressure prevents the vacuum curvature blowup in finite-length prolate collapse, or whether that vacuum singularity forms independently of the matter's own bounce; and what the causal character of any such vacuum blowup is and whether a horizon eventually cloaks it.

Background

General relativity predicts that sufficiently concentrated matter collapses under its own gravity, and the classical singularity theorems assert that complete collapse generically produces spacetime singularities. Whether such singularities are hidden from distant observers inside horizons, or can be visible as "naked" singularities, is the subject of the weak cosmic censorship conjecture, one of the central open problems of the theory. A related quantitative proposal is the hoop conjecture, formulated by Kip S. Thorne in 1972 (for a modern statement and discussion see Bronnikov, Santos and Wang 2020): horizons form when, and only when, a mass $M$ is compacted into a region whose circumference in every direction is $\lesssim 4\pi M$.

Two early results set the stage. First, Shapiro and Teukolsky (1991) simulated, in full general relativity, the collapse of collisionless (mutually noninteracting) particle clouds initially shaped as prolate — elongated, spindle-like — spheroids, and found that sufficiently large ones develop divergent curvature outside the matter with no apparent horizon (an apparent horizon is the outermost marginally trapped surface, a slice-dependent local proxy for a black-hole boundary; an event horizon is the global boundary of the region that can ever send signals to infinity). Because their code could not be continued past the blowup, a later-forming event horizon could not be excluded; the configurations nonetheless became leading candidate counterexamples to weak cosmic censorship. Second, Apostolatos and Thorne (1992) analyzed the opposite idealization: collapse in cylindrical symmetry, i.e. an infinitely long distribution possessing both a translational and a rotational Killing symmetry. Cylindrical spacetimes admit genuine gravitational waves (Einstein–Rosen waves, the simplest radiative solutions of Einstein's equations) and a quasi-local energy per unit length, the C-energy, introduced by Kip S. Thorne in 1965; in one standard convention it is $C=\frac{1}{8}(1-e^{-2\gamma})$ (the overall normalization is conventional), where $\gamma$ is a metric function in Einstein–Rosen coordinates. Apostolatos and Thorne treated the matter as an infinitesimally thin shell — matched across a hypersurface by Israel's junction conditions, which relate the jump in the shell's extrinsic curvature to its surface stress-energy — composed of counter-rotating dust: pressure-free particles, half orbiting the axis with conserved angular momentum per unit rest mass $+\alpha$ and half with $-\alpha$, so the net angular momentum, and with it frame dragging (the dragging of inertial frames by rotating mass), vanishes and the exterior vacuum is of the static Levi-Civita type (the static cylindrically symmetric vacuum solution) rather than the rotating Lewis type (its stationary, frame-dragging generalization). For momentarily static, radiation-free (MSRF) initial data — data with vanishing initial velocities and no incident gravitational radiation — and with the space assumed not closed in the radial direction, they proved by C-energy bookkeeping that the shell bounces: the shell can never collapse to a circumference smaller than $8\pi\alpha\Lambda$, where $\Lambda$ is its (nonconserved) mass per unit proper length; equivalently, the quantity $R\equiv\Vert\partial/\partial\varphi\Vert\,\Vert\partial/\partial z\Vert$ — the product of the lengths of the rotational and translational Killing vectors at the shell's location — can never fall below $4\alpha\lambda$, where $\lambda$ is the shell's conserved rest mass per unit Killing length $z$. After its centrifugal bounce the shell oscillates, emits gravitational waves that carry away C-energy, and was argued to settle down to a static, equilibrium radius. Thus infinitesimal rotation halts cylindrical collapse.

The two results stand in tension, and each analysis lacks exactly what the other has: the cylindrical model has rotation but is infinitely long, dust-only, and free of net angular momentum, while the prolate simulations are finite and realistic in shape but had no rotation or pressure and could not be evolved to late times. The open questions acknowledged in and around the Apostolatos–Thorne paper concern the reach of the rotation-halting mechanism across this gap: the net-angular-momentum extension of the bounce and its bound; the role of small rotation and realistic pressure in finite prolate spindles, where the numerically observed vacuum curvature blowup (monitored via the Kretschmann invariant $K=R_{abcd}R^{abcd}$, a curvature-squared scalar) might either be prevented or form independently of the matter's bounce; the causal character of any such blowup; whether horizons eventually form around it; and the effect of radiation and realistic matter, including whether the shell's post-bounce settling to equilibrium is dynamically realized at all.

Problem Statement

Determine the reach of the rotation-induced bounce in relativistic gravitational collapse — that is, whether, and under what precise conditions, rotation (by itself, or together with realistic pressure and gravitational radiation) prevents curvature-singularity formation in strongly nonspherical collapse, in place of the naked singularities suggested by the Shapiro–Teukolsky prolate-spindle simulations. The problem comprises four linked components that share this single joint objective: (1) Net angular momentum: do the centrifugal bounce and its quantitative lower bound — the bound $R\ge4\alpha\lambda$ on the product of the rotational and translational Killing-vector lengths of a collapsing cylindrical shell of counter-rotating dust, equivalently the circumference bound $8\pi\alpha\Lambda$ (with $\alpha$ the conserved angular momentum per unit rest mass of each particle stream and $\lambda$ the conserved rest mass per unit Killing length) — extend to cylindrical shells with nonzero net angular momentum, whose exteriors are frame-dragging (Lewis-type) rather than the static exteriors of the zero-net-rotation model? (2) Finite prolate configurations: for a finite-length, highly prolate imploding body (the spindle scenario), does an arbitrarily small rotation, or realistic pressure, prevent the vacuum curvature blowup outside the matter that numerical simulations appear to show — or does that vacuum singularity form independently of the matter's own bounce? (3) Causal structure and horizons: what is the causal character of any such vacuum blowup (spacelike or timelike, causally prior to or causally decoupled from the matter singularity), and does an apparent or event horizon eventually form around it at late times? (4) Realistic matter and radiation, comprising four parts: (a) how does pressure change the bounce; (b) how does particle emission change the bounce; (c) how does gravitational-wave content change the bounce; and (d) is the post-bounce settling of the shell to a static equilibrium dynamically realized — given that the stability analyses of the original model conflict? The open problem is which mechanisms — angular momentum, pressure, radiative damping, or horizon formation — decide the outcome across the cylindrical, prolate, and radiative regimes, and thereby the role of rotation vis-à-vis weak cosmic censorship in nonspherical collapse.

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