On the horizon instability of an extreme Reissner-Nordström black hole

TL;DR

This work demonstrates that the horizon instability discovered by Aretakis for massless fields on an extreme RN black hole extends to massive scalar fields and to coupled gravitational–electromagnetic perturbations, establishing a tower of horizon-conserved quantities that generically drive blow-up of higher radial derivatives along the horizon. By combining analytic arguments, near-horizon AdS_2 insights, conformal mappings to NP constants, and extensive numerical simulations in double-null coordinates, the authors map out late-time tails and horizon-derivative growth for various multipoles and masses. They show that the instability is intimately tied to the absence of a horizon redshift, persists under ingoing perturbations, and has a universal character at late times governed by near-horizon geometry, with precise decay rates and blow-up patterns depending on the mass and multipole index. The results have implications for the stability of extremal black holes in supergravity and for understanding horizon dynamics in linear perturbations of extremal backgrounds. Overall, the work provides a coherent framework linking horizon conservation laws, AdS_2 near-horizon physics, NP constants, and horizon-instability phenomena across scalar and gravitational-electromagnetic sectors.

Abstract

Aretakis has proved that a massless scalar field has an instability at the horizon of an extreme Reissner-Nordström 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.

Figures (20)

• Figure 1: Penrose diagram for extreme RN black holes. Aretakis took initial data specified on a spacelike surface $\Sigma$ intersecting ${\cal H}^+$ and extending to infinity.
• Figure 2: Schematic plot of the setup used in the numerical calculations.
• Figure 3: Functions $\phi(v,r)$ and $\partial_r\phi(v,r)$ for $l=0$ and $H_0\neq0$ on fixed $v$ slices. We consider outgoing wave initial data (\ref{['out']}) with $(\sigma,\mu)=(0.1,-0.1)$. The horizon is at $r=1$. We can see that $\partial_r\phi$ becomes steeper near the horizon as $v$ increases. This implies that $\partial_r^2\phi$ blows up on the horizon at large $v$.
• Figure 4: Time dependence of $\phi|_{r=M}$ and $\partial_r^2\phi|_{r=M}$ for $l=0$ and $H_0\neq 0$. We choose parameters as $(\sigma,\mu)=(0.1,-0.1),(0.05,-0.1),(0.1,-0.05)$ in the outgoing wave initial data (\ref{['out']}). We can see that $\partial_r^2\phi|_{r=M}$ blows up as $\sim v$.
• Figure 5: Time dependence of $\delta$. We can see that $\delta$ is linear in $\log v$ at late time.