Structure of trapped regions at the extremal threshold

Determine whether the two trapped regions observed in genuine charged-scalar collapse remain disconnected all the way to the Cauchy horizon as the extremal threshold is approached, or whether they are connected outside the finite numerical domain.

Background

Near the extremal threshold, the numerical evolutions display an early cigar-shaped trapped region and a later trapped region. On the finite computational domain these regions can appear either disconnected or joined, depending on the initial-data parameter and the domain size.

The authors cannot determine from their finite-domain simulations whether the apparent bridge between the regions persists in the continuum spacetime or whether the regions remain separate until the Cauchy horizon. This affects the global interpretation of the extremal horizon and the trapped-region geometry.

References

On the other hand, on our finite numerical domain we cannot see what happens as $p\to p_{1e}$, so we do not know if there is always a single trapped region (connected outside our numerical domain) or if there really are two trapped regions that remain separate all the way to the Cauchy horizon.

— Critical and extremal gravitational collapse of a spherical charged scalar field in 4+1 spacetime dimensions  (2609.17207 - Mittal et al., 15 Sep 2026) in Section 4.3, Theoretical understanding

We have no explanation for this, but are not concerned, as we expect the power laws to hold increasingly well as extremality is approached with increasing fine-tuning.

— Critical and extremal gravitational collapse of a spherical charged scalar field in 4+1 spacetime dimensions  (2609.17207 - Mittal et al., 15 Sep 2026) in Section 5.2, Scaling of $\lambda_{\text{trap}}(p)$