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

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.

Linear Waves in the Interior of Extremal Black Holes I

“…In [4] he showed moreover that, under the assumption of pointwise decay of solutions and their tangential derivatives along the event horizon in affine time v, second-order transversal derivatives even blow up as v → ∞. This blow-up phenomenon has been dubbed the "Aretakis instability" in the literature [33].…”

“…Heuristics and numerics regarding latetime tails for extremal Reissner-Nordström in [33,44,47] suggest an extremal variant of "Price's law" that in particular predicts:…”

“…Similarly, the following limit in ingoing Eddington-Finkelstein coordinates (v, r, θ, ϕ) is independent of v: [33]. In this case, the following late-time tails are suggested: 5) where the constants in (1.5) are non-vanishing for generic initial data on with…”

“…The numerical analysis in [33] suggests that the leading-order terms for the spherically symmetric mode φ 0 along H + , arising from compactly supported initial data on , are given by…”

“…By restricting to spherically symmetric solutions, we moreover discover that φ can in fact be extended as a C 2 function beyond the Cauchy horizon, if (and only if) we assume precise asymptotics of φ that are suggested by the numerics in [33].…”