We study the nonlinear stability of the Cauchy horizon in the interior of
extremal
Reissner–Nordström black holes under
spherical symmetry. We consider the
Einstein–Maxwell–Klein–Gordon system such that the charge of the scalar
field is appropriately small in terms of the mass of the background extremal
Reissner–Nordström black hole. Given spherically symmetric characteristic initial
data which approach the event horizon of extremal Reissner–Nordström
sufficiently fast, we prove that the solution extends beyond the Cauchy horizon in
${C}^{0,\frac{1}{2}}\cap {W}_{loc}^{1,2}$,
in contrast to the subextremal case (where generically the solution is
${C}^{0}\setminus \left({C}^{0,\frac{1}{2}}\cap {W}_{loc}^{1,2}\right)$). In
particular, there exist nonunique spherically symmetric extensions which are
moreover solutions to the Einstein–Maxwell–Klein–Gordon system. Finally, in the
case that the scalar field is chargeless and massless, we additionally show that the
extension can be chosen so that the scalar field remains Lipschitz.
