Nonlinear scalar perturbations of extremal Reissner-Nordstr\"{o}m spacetimes

TL;DR

This paper proves the persistence of horizon instability in extremal Reissner-Nordström black holes for nonlinear wave equations, demonstrating global existence with asymptotic blow-up without symmetry assumptions.

Contribution

It provides the first rigorous analysis of nonlinear wave equations on extremal black holes without symmetry, introducing a new vector field method for decay estimates.

Findings

Horizon instability persists in nonlinear models.

Global existence with asymptotic blow-up is established.

A new vector field method enables sharp decay estimates.

Abstract

We present the first rigorous study of nonlinear wave equations on extremal black hole spacetimes without any symmetry assumptions on the solution. Specifically, we prove global existence with asymptotic blow-up for solutions to nonlinear wave equations satisfying the null condition on extremal Reissner-Nordstr\"{o}m backgrounds. This result shows that the extremal horizon instability persists in model nonlinear theories. Our proof crucially relies on a new vector field method that allows us to obtain almost sharp decay estimates.