This paper studies how a small scalar field that grows around some charged black holes can produce a large and universal change deep inside the hole. In earlier work, a quantity that describes the final spacelike singularity — the Kasner parameter β — was found to blow up near a critical charge-to-mass ratio qc, with a power-law β ∝ (q/qc − 1)−γ and an exponent γ = 1/2. The authors show that γ need not always be 1/2. They find a family of different universality classes with other γ values, and they explain why these different behaviors all come from the same emergent symmetry at the would-be inner horizon of the background Reissner–Nordström (RN) solution.

The researchers work in a class of Einstein–Maxwell–scalar (EMS) models. These add a neutral real scalar field to Einstein gravity and couple it to the electromagnetic field through a coupling function Z(φ). They require Z to be analytic and to satisfy simple monotone conditions so that the ordinary RN black hole becomes unstable and a “hairy” solution develops. They analyze both the external bifurcation where hairy black holes first appear and the interior evolution after crossing the event horizon. They complement analytical arguments with numerical solutions to test their predictions.

At a high level the mechanism has two parts. First, the exterior transition supplies an arbitrarily small scalar hair at the event horizon: the horizon value of the scalar follows a power law in q − qc. Second, inside the black hole the would-be inner (Cauchy) horizon becomes a special point where an emergent scaling symmetry appears. This produces a logarithmic branch for the collapsing scalar and creates a narrow “boundary layer” near the would-be horizon. Inside that layer the equations become autonomous and independent of the detailed form of Z(φ). The layer acts like an amplifier: a smaller hair at the horizon can produce a larger scalar field just behind the would-be horizon. Through this “seesaw amplification,” the same power law that controls the exterior hair also controls the divergence of the Kasner parameter at the singularity.