Role of angular momentum and cosmic censorship in $(2+1)$-dimensional rotating shell collapse

We study the gravitational collapse problem of rotating shells in three-dimensional Einstein gravity with and without a cosmological constant. Taking the exterior and interior metrics to be those of stationary metrics with asymptotically constant curvature, we solve the equations of motion for the shells from the Darmois-Israel junction conditions in the corotating frame. We study various collapse scenarios with arbitrary angular momentum for a variety of geometric configurations, including anti-de Sitter, de Sitter, and flat spaces. We find that the collapsing shells can form a BTZ black hole, a three-dimensional Kerr-dS spacetime, and an horizonless geometry of point masses under certain initial conditions. For pressureless dust shells, the curvature singularity is not formed due to the angular momentum barrier near the origin. However when the shell pressure is nonvanishing, we find that for all types of shells with polytropic-type equations of state (including the perfect fluid and the generalized Chaplygin gas), collapse to a naked singularity is possible under generic initial conditions. We conclude that in three dimensions angular momentum does not in general guard against violation of cosmic censorship.

Figure 1

Plots of the effective potential for $a_8<0$ as a function of $x$. There are four cases: (a) a single root, (b) a degenerate small root and a large root, (c) three distinct roots, and (d) a degenerate large root and a small root. The arrows denote the possible trajectories of a shell with zero energy. The dotted lines denote the nonrotating cases ($a_2=0$), which exist only for $\Delta <0$ defined in Eq. (3.17), and there are two types, depending on the value of $a_4$. Note that there is a stable equilibrium solution located at the local minimum in (b), and an unstable equilibrium solution located at the local maximum in (d).

Figure 2

Plots of the effective potential as a function of $x$ for $a_8=0$. The arrows denote the possible trajectories of a shell. The dotted lines denote the nonrotating case ($a_2=0$) and there can be two types, depending on the value of $a_4$.

Figure 3

Plots of the effective potentials for the shell with a polytropic equation of state and the exterior AdS black hole (${\alpha }_{o/i}^{+}=+1$, $k_o^{+}=-(x_H^{+}{)}^2=-1$, $k_i^{+}=-1/4$: LHS)/AdS point mass (${\alpha }_{o/i}^{+}=+1$, $k_o^{+}=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters ${\alpha }_o^{-}=+1$, $\omega =+2$, $k_{o/i}^{-}=+1/2$, $m_0=4$, $n=+1$.

Figure 4

Plots of the effective potentials for the shell with a polytropic equation of state and the exterior flat space (${\alpha }_o^{+}=0$, ${\alpha }_i^{+}=+1$, $k_o^{+}=+1$, $k_i^{+}=+1/4$: LHS)/dS space (${\alpha }_o^{+}=+1$, ${\alpha }_i^{-}=-1$, $k_o^{+}=(x_C^{+}{)}^2=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters ${\alpha }_o^{-}=+1$, $\omega =+2$, $k_{o/i}^{-}=+1/2$, $m_0=2$, $n=+1$.

Figure 5

Plots of the effective potentials for the shell with a linear fluid and the exterior AdS black hole ($k_o^{+}=-(x_H^{+}{)}^2=-1$, $k_i^{+}=-1/4$: LHS)/AdS point mass ($k_o^{+}=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters ${\alpha }_o^{-}=+1$, $\omega =+2$, $k_{o/i}^{-}=+1/2$, $m_0=4$.

Figure 6

Plots of the effective potentials for the shell with a linear fluid and the exterior flat space (${\alpha }_o^{+}=0$, ${\alpha }_i^{+}=+1$, $k_o^{+}=+1$, $k_i^{+}=+1/4$: LHS)/dS space (${\alpha }_{o/i}^{+}=+1$, $k_o^{+}=(x_C^{+}{)}^2=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters, ${\alpha }_o^{-}=+1$, $\omega =+2$, $k_{o/i}^{-}=+1/2$, $m_0=4$.

Figure 7

Plots of the effective potentials for the shell with a linear fluid and the exterior AdS black hole ($k_o^{+}=-(x_H^{+}{)}^2=-1$, $k_i^{+}=-1/4$: LHS)/AdS point mass ($k_o^{+}=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters ${\alpha }_o^{-}=+1$, $\omega =+1/2$, $k_{o/i}^{-}=+1/2$, $m_0=4$. The event horizon for the exterior observer is located at $x=1.0$ in the left diagram. However a fluid collapsing through this surface will not bounce from the interior barrier.

Figure 8

Plots of the effective potentials for the shell with a linear fluid and the exterior flat space (${\alpha }_o^{+}=0$, ${\alpha }_i^{+}=+1$, $k_o^{+}=+1$, $k_i^{+}=+1/4$: LHS)/dS space (${\alpha }_{o/i}^{+}=+1$, $k_o^{+}=(x_C^{+}{)}^2=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters, ${\alpha }_o^{-}=+2$, $\omega =+1/2$, $k_{o/i}^{-}=+1/2$, $m_0=4$.

Figure 9

Plots of the effective potentials for the shell with a linear fluid and the exterior AdS black hole ($k_o^{+}=-(x_H^{+}{)}^2=-1$, $k_i^{+}=-1/4$: LHS)/AdS point mass ($k_o^{+}=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters ${\alpha }_o^{-}=+1$, $\omega =+1$, $k_{o/i}^{-}=+1/2$, $m_0=4$.

Figure 10

Plots of the effective potentials for the shell with a linear fluid and the exterior flat space (${\alpha }_o^{+}=0$, ${\alpha }_i^{+}=+1$, $k_o^{+}=+1$, $k_i^{+}=+1/4$: LHS)/dS space (${\alpha }_{o/i}^{+}=+1$, $k_o^{+}=(x_C^{+}{)}^2=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters, ${\alpha }_o^{-}=+1$, $\omega =+1$, $k_{o/i}^{-}=+1/2$, $m_0=1$.

Figure 11

Plots of the effective potentials for the shell with a linear fluid and the exterior flat space (${\alpha }_o^{+}=0$, ${\alpha }_i^{+}=+1$, $k_o^{+}=+1$, $k_i^{+}=+1/4$: LHS)/dS space (${\alpha }_{o/i}^{+}=+1$, $k_o^{+}=(x_C^{+}{)}^2=+1$, $k_i^{+}=+1/4$: RHS). We choose an interior AdS space with a point mass (${\alpha }_i^{-}=+1$), flat (${\alpha }_i^{-}=0$), and dS (${\alpha }_i^{-}=-1$) by setting parameters, ${\alpha }_o^{-}=+1$, $\omega =+1$, $k_{o/i}^{-}=+1/2$, $m_0=4$.

Figure 12

Plots of the effective potentials for the GCG shell with $\lambda =1$ and the exterior/interior AdS spaces with point masses (${\alpha }_{o/i}^{\pm }=+1$, $k_o^{+}=+1$, $k_i^{+}=+1/4$, $k_{o/i}^{-}=+1/2$, $m_0=2$). (1) LHS: the case of nonrotating shell (2) RHS: the case of rotating shell.