Stability of extremal critical collapse for charged scalar fields

Determine whether stable extremal critical collapse occurs for the spherical Einstein–Maxwell–charged scalar field system, meaning whether a neighbourhood of the collapse threshold contains initial data whose evolutions approach black holes with charge-to-mass ratio tending to unity.

Background

The paper discusses the stable extremal critical-collapse conjecture formulated for related Einstein–Maxwell matter models. In the conjectured scenario, extremality is reached at the threshold of genuine collapse and persists in a neighbourhood of the threshold, with the final charge-to-mass ratio approaching one.

The simulations reported in the paper do not find evidence for this behaviour in 4+1-dimensional genuine collapse of a charged scalar field. The authors distinguish it from the codimension-1 extremal collapse they do observe inside the collapse region, where the transition is from an extremal to a subextremal black hole rather than from an extremal black hole to dispersion.

References

It is presently unknown whether this bound is sharp.

— The third law of black hole thermodynamics and the mass-to-charge ratio of scalar fields  (2609.30170 - Hod et al., 24 Sep 2026) in Section 3, immediately after Eq. (2)

More specifically, we conjecture that, in the Einstein-Maxwell-massive-scalar field theory, an extremal (zero-temperature) Reissner-Nordström black hole of charge $Q$ cannot form dynamically from a charged massive scalar field whose parameters satisfy

— The third law of black hole thermodynamics and the mass-to-charge ratio of scalar fields  (2609.30170 - Hod et al., 24 Sep 2026) in Section 3, paragraph containing Eq. (3)

We further conjecture that the bound (\ref{Eq4}) is sharp: that is, extremal Reissner-Nordström black holes can only be formed classically from charged scalar fields with $m/e$ arbitrarily close to, but below, the threshold (\ref{Eq4}).

— The third law of black hole thermodynamics and the mass-to-charge ratio of scalar fields  (2609.30170 - Hod et al., 24 Sep 2026) in Section 3, final paragraph before the acknowledgments

In particular, we have found no evidence, (in 4+1 dimensions), for the stable extremal critical collapse conjecture of , which says that there is a neighbourhood within the threshold of collapse where $|Q|/\mathcal{M}\to 1$.

— 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 6, Conclusions

It is not clear to us, however, how relevant those results are for the scalar field, as Vlasov matter is very different.

— 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 6, Conclusions