On the horizon instability of an extreme Reissner-Nordstr\"om black hole

TL;DR

This paper demonstrates that horizon instabilities in extreme Reissner-Nordström black holes extend beyond massless scalar fields to massive scalar and gravitational-electromagnetic perturbations, supported by numerical and analytical evidence.

Contribution

It shows the presence of horizon instabilities for various perturbations in extreme RN black holes, extending previous results and connecting conserved quantities at the horizon and null infinity.

Findings

Instabilities occur for initial perturbations outside the horizon.

Numerical results match analytical calculations in the near-horizon geometry.

Aretakis' conserved quantities relate to Newman-Penrose quantities at null infinity.

Abstract

Aretakis has proved that a massless scalar field has an instability at the horizon of an extreme Reissner-Nordstr\"om black hole. We show that a similar instability occurs also for a massive scalar field and for coupled linearized gravitational and electromagnetic perturbations. We present numerical results for the late time behaviour of massless and massive scalar fields in the extreme RN background and show that instabilities are present for initial perturbations supported outside the horizon, e.g.\ an ingoing wavepacket. For a massless scalar we show that the numerical results for the late time behaviour are reproduced by an analytic calculation in the near-horizon geometry. We relate Aretakis' conserved quantities at the future horizon to the Newman-Penrose conserved quantities at future null infinity.