Rotating thin shells in (2+1)-dimensional asymptotically AdS spacetimes: the overspinning regime and admissible matter models

In (2+1)-dimensional general relativity with a negative cosmological constant, Lemos, Lopes and Minamitsuji (2015) characterized stationary, circularly symmetric rotating thin shells (rings) with a rotating Bañados-Teitelboim-Zanelli (BTZ) exterior and a non-rotating anti-de Sitter (AdS) interior, using the Darmois-Israel junction conditions. They computed the shell's rest-energy density, pressure, and angular-momentum flux, exhibited Machian dragging of inertial frames, and showed that the dominant energy condition is satisfied only in limiting branches (large-radius shells with mass growing in proportion to the radius, and the extremal case). Their analysis was restricted to $|J|\le m\ell$, where $m$ and $J$ are the exterior ADM mass and angular momentum and $\ell$ is the AdS curvature radius. The open completion of that program comprises two parts: (i) the junction-condition, energy-condition, and causal-structure analysis of overspun shells with $|J|>m\ell$, whose exterior has no horizons and a naked singularity; and (ii) the identification of physically motivated equations of state realizing the energy-condition-allowed parameter-space branches.

Background

In (2+1)-dimensional Einstein gravity with a negative cosmological constant $\Lambda=-1/\ell^{2}$, the unique circularly symmetric, asymptotically anti-de Sitter (AdS) black hole is the Bañados-Teitelboim-Zanelli (BTZ) solution, characterized by an asymptotic (ADM) mass $m$ and angular momentum $J$. For $|J|\le m\ell$ it has an outer gravitational radius $r_{+}$ and an inner (Cauchy) radius $r_{-}$; the case $|J|=m\ell$ is extremal, while for $|J|>m\ell$ both radii become imaginary, there are no horizons, and the geometry contains a naked singularity at $r=0$ (the overspun, or overspinning, regime). A thin shell is an idealization in which matter is confined to a hypersurface: matching two vacuum metrics across the shell by the Darmois-Israel junction conditions equates the jump in extrinsic curvature to the shell's surface stress-energy tensor, whose circularly symmetric rotating components in (2+1) dimensions are the rest-energy density $\lambda$, the pressure $p$, and the angular-momentum flux density $j$. The weak energy condition (WEC) requires the energy density to be non-negative for all physical observers, and the dominant energy condition (DEC) additionally requires the energy-momentum flux to be non-spacelike. Lemos, Lopes and Minamitsuji (2015) studied such a shell: a rotating ring with a rotating BTZ exterior and the empty non-rotating AdS spacetime inside. They obtained $\lambda$, $p$ and $j$ as functions of $m$, $J$ and the shell radius $R$, found the frame in which the matter is a perfect fluid rotating at a definite angular velocity, demonstrated Machian effects (inertial observers in the interior are dragged into rotation by the shell, corotating with it as $R\to r_{+}$), and showed that the DEC holds only in two limiting settings: the $R\to\infty$ branches, where finite shell density requires the shell's proper mass to grow in proportion to $R$, and the extremal case $|J|=m\ell$. Their mechanical analysis was confined to stationary configurations with $|J|\le m\ell$ and $R\ge r_{+}$; the overspun case $|J|>m\ell$ was set aside by the authors as too simple to warrant detailed treatment, no perturbative or collapse dynamics was considered, and no classification of the equations of state able to realize the permitted branches was attempted. Related results exist around this setup: the same authors and collaborators extended the analysis to the thermodynamics of the rotating shells (including power-law equations of state and thermodynamic stability) and to the extremal limit; the full time-dependent collapse of rotating shells in (2+1) dimensions with dust and polytropic matter was studied by Mann, Oh and Park in a related but different matching setup; and the overspun BTZ geometry itself (without a shell) has been analyzed in the cosmic-censorship and semiclassical-backreaction literature.

Problem Statement

Complete the junction-condition theory of circularly symmetric rotating thin shells in (2+1)-dimensional asymptotically AdS spacetimes (rotating BTZ exterior with ADM mass $m$ and angular momentum $J$, non-rotating AdS interior, AdS radius $\ell$), which Lemos, Lopes and Minamitsuji established only for stationary configurations with $|J|\le m\ell$. The common objective is to determine which rotating shell spacetimes in this family are supported by physically admissible matter. Concretely, two linked deliverables: (i) Overspun shells: solve the Darmois-Israel matching for a stationary rotating shell with $|J|>m\ell$, where the exterior has no horizons and a naked singularity at $r=0$ and the constraint $R\ge r_{+}$ is replaced by $R\ge0$; determine the shell stress-energy components ($\lambda$, $p$, $j$) as functions of $m$, $J$ and $R$, map the regions of parameter space obeying the weak and dominant energy conditions, characterize the causal structure of the matched spacetime (including the possible presence of closed timelike curves, i.e., worldlines returning to their own past, and the $R\to0$ point-particle limit), and state whether overspun shells can be supported by energy-condition-respecting matter at all. (ii) Matter models: identify physically motivated equations of state or matter models that realize the energy-condition-allowed branches of the stationary solution space, in particular the large-radius branch requiring the shell's proper mass to grow in proportion to $R$ for finite densities; an admissible matter model must be given as a constitutive relation between the shell's fluid variables (for instance between the perfect-fluid pressure $\bar p$ and density $\bar\lambda$ in the corotating frame) that holds across the branch it realizes and is consistent with the first-law integrability conditions of the companion thermodynamic analysis, not a relation reverse-defined from a single equilibrium configuration. A complete resolution requires all of the sub-goals listed under (i) and (ii) to be addressed jointly; no proper subset suffices.

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