A remark about wave equations on the extreme Reissner–Nordström black hole exterior

We consider a massless scalar field propagating on the exterior of the extreme Reissner–Nordström black hole. Using a discrete conformal symmetry of this spacetime, we draw a one-to-one relationship between the behavior of the field near the future horizon and near future null infinity. In particular, we show that the polynomial growth of the second and higher transversal derivatives along the horizon, recently found by Aretakis, reflects well-known facts about the retarded time asymptotics at null infinity. We also observe that the analogous relationship holds true for an axially symmetric massless scalar field propagating on the extreme Kerr–Newman background. PACS number: 04.70.Bw Recently, Aretakis studied a massless scalar field on the exterior of the extreme Reissner– Nordström black hole and proved that second and higher transversal derivatives of the field grow polynomially along the horizon, provided that a certain conserved quantity on the horizon is nonzero [1, 2]. This fact was interpreted as indicating the instability of the extreme Reissner– Nordström black hole. The aim of this note is to point out that an important observation by Aretakis on the behavior of scalar fields on the horizon is a reflection of well-known results about the asymptotic behavior of scalar fields near null infinity. Moreover, we show that if Aretakis’ conserved quantity vanishes, then the second transversal derivative at the horizon is bounded but, generically, the third and higher ones grow. The exterior (or domain of outer communication) of the extreme Reissner–Nordström black hole is the globally hyperbolic spacetime with manifold M = {−∞ < t < ∞, 0 < r < ∞} × S2 and metric g = −A−2 dt + A (dr + r dω), A = 1 + m r , (1) where m is a positive constant and dω2 is the round metric on the unit 2-sphere. The metric g is the unique spherically symmetric solution of the Einstein–Maxwell equations with mass m and charge q = ±m. The Maxwell field is given by F = q/(r + m)2 dt ∧ dr. Note that we 0264-9381/13/065001+06$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1 Class. Quantum Grav. 30 (2013) 065001 P Bizoń and H Friedrich are using the isotropic radial coordinate r which is related to the areal radial coordinate R by r = R − m. Key to our discussion is the fact that the metric g admits a discrete conformal symmetry [8], namely, the spatial inversion ι : (t, r) → (t,m2/r) of M onto itself (suppressing the angular coordinates, they are unaffected by our considerations), which satisfies ι∗g = g with = m r . To see the action of this symmetry on M, it will be convenient to introduce the retarded and advanced time coordinates u = t − r∗, v = t + r∗ with r∗ = r + 2 m log(r/m) − m2/r. They satisfy du = dt − A2 dr, dv = dt + A2 dr and u ◦ ι = v. In the coordinates (v, r) the metric takes the form g = gv ≡ −A−2 dv + 2 dv dr + (rA) dω = − r 2 (m + r)2 dv 2 + 2 dv dr + (m + r) dω, which shows that the metric extends as a real analytic metric gv (in fact as a solution to the Einstein–Maxwell equations) onto the extension MH+ = {−∞ < v < ∞, −m < r < ∞} × S2 of M. The null hypersurface H+ = {−∞ < v < ∞, r = 0} × S2 represents the future event horizon for (M, g). In the coordinates (u, r) the metric takes the form g = gu ≡ −A−2 du − 2 du dr + (rA) dω. Expressed in source coordinates (u, r) and target coordinates (v, r) the inversion takes the form ι : (u, r) → (u,m2/r) and the relation above reads ι∗gv = 2 gu. (2) In the coordinates (u, ρ ≡ m2/r) the metric ĝu ≡ 2 gu and the conformal factor take the form ĝu = −A(ρ)−2 du + 2 du dρ + (ρA(ρ)) dω, = ρ/m, which shows that ĝu and extend as real analytic fields (solution to the conformal Einstein– Maxwell equations) onto the extension MJ+ = {−∞ < u < ∞, −m < ρ < ∞} × S2 of M. Because ĝu and gv are related on M by a diffeomorphism, the metric ĝu has vanishing Ricci scalar as well. The null hypersurface J + = {−∞ < u < ∞, ρ = 0} × S2 on which = 0, d = 0 represents future null infinity for (M, g). In terms of source coordinates (u, ρ) and target coordinates (v, r) the inversion takes on M the form ι : (u, ρ) → (v = u, r = ρ). It follows that ι extends to a real analytic isometry ι′ : (MJ+, ĝu) → (MH+ , gv ) which maps J + onto H+ and M onto itself. As a consequence, ‘conformally well-behaved’ fields on the extreme Reissner–Nordström background can hardly distinguish between J + and H+. This is quite clear for Maxwell or Yang–Mills fields. They are governed by equations which, in four dimensions, only depend on the conformal structure. We discuss here the slightly more subtle case of scalar fields satisfying the massless wave equation. On a four-dimensional spacetime (N , h) with vanishing Ricci-scalar (the situation considered here) the wave operator h ≡ ∇μ ∇μ is identical with the conformally covariant wave operator Lh = h −1/6 Rh, which satisfies for any conformal factor θ > 0 and any scalar field f ; Lθ2h(θ −1 f ) = θ−3Lh( f ). If φ : (N ′, h′) → (N , h) is a spacetime diffeomorphism with inverse ψ , then it holds Lh′ ( f ◦ φ) = Lψ∗h′ ( f ) ◦ φ.