Figure 1

Left: A Reissner-Nordström black hole formed by the collapse of a thin spherical charged shell and subject to spherical perturbations by a massless scalar field. Two types of singularities form: A null, weak singularity at the CH (the diagonal dashed line) and a spacelike $r=0$ singularity (the solid horizontal line, though see footnote ). Depending on the strength of the initial perturbation, the spacelike piece of the singularity might also form much earlier. It may intersect the worldline describing the shell or even entirely remove the would-be outgoing inner horizon. Right: Our proposed effective geometry of the perturbed (but spherically symmetric) collapsing shell spacetime as seen by late-time observers. It consists precisely of the region of the unperturbed eternal Reissner-Nordström black hole with $r>r_{-}$, together with two types of singularities on its future boundary (at $r=r_{-}$): The diagonal dashed line at the upper right represents the mass inflation singularity, a null, weak singularity located at the CH. The diagonal solid line at the upper left represents a null shock-wave singularity, where the metric tensor is effectively discontinuous.Reuse & Permissions

Figure 2

Left: The eternal Reissner-Nordström black hole. The vertical lines are singularities at $r=0$, while the diagonal lines describe copies of asymptotically flat future or past null infinities ($I^{+}$ and $I^{-}$), outer horizons ($r_{+}$), and inner horizons ($r_{-}$). The full analytically extended solution consists of a periodic vertical array of copies of the regions shown. Right: A conformal diagram for the spacetime of a collapsing (but otherwise unperturbed) spherically symmetric, massive charged thin shell. The $r=0$ curve is a regular origin in the flat region inside the shell. The Cauchy horizon is also indicated.Reuse & Permissions

Figure 3

The typical behavior of $\varphi$ as a function of the observer’s proper time is depicted for several values of $v_{\mathrm{eh}}$.Reuse & Permissions

Figure 4

Left: Some lines of constant $u$, $v$ drawn on the spherical spacetime describing a collapsing shell perturbed by a massless scalar field. The long-dashed line (top right boundary) is the mass-inflation singularity. In contrast, the short dashes mark coordinate lines lying outside the physical spacetime (since $r<0$). The vector field ${\partial }_u+{\partial }_v$ is tangent to any surface $u=\pm \infty$ or $v=\pm \infty$. The arrows indicate the direction of this vector field along such surfaces. The point marked $i^{+}$ is fixed under the action of this vector field. Right: The effective geometry for late-time observers obtained by flowing the interior of our spacetime backward along the vector field ${\partial }_u+{\partial }_v$, while similarly deforming the boundaries. As with Fig. 1 (right), it consists of the region of the unperturbed eternal Reissner-Nordström black hole with $r>r_{-}$, together with certain shock-wave singularities. The shock along the lower-left boundary descirbes a jump from $r=r_{+}$ to $r=0$ (corresponding to the regular center of spherical symmetry shown in the left diagram). Though it arises only in the formal late-infall limit, this shock is not actually accessible to any late-infall observer. In this sense the shock is fictitious. In contrast, the shock along the upper-left boundary is accessible to late-time observers. It contains the final piece of the shell worldline and describes a jump from $r=r_{-}$ to $r=0$ (corresponding to the spacelike singularity shown in the left diagram).Reuse & Permissions