Linear Waves in the Interior of Extremal Black Holes I

Dejan Gajic

doi:10.1007/s00220-016-2800-y

Dejan Gajic

Open Access

https://doi.org/10.1007/s00220-016-2800-y

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Abstract

We consider solutions to the linear wave equation in the interior region of extremal Reissner–Nordström black holes. We show that, under suitable assumptions on the initial data, the solutions can be extended continuously beyond the Cauchy horizon and, moreover, that their local energy is finite. This result is in contrast with previously established results for subextremal Reissner–Nordström black holes, where the local energy was shown to generically blow up at the Cauchy horizon.

Highlights

We either take g = ga,M, the metric of a Kerr spacetime with mass M and angular momentum parameter a, with |a| ≤ M, or g = ge,M, the metric of a Reissner–Nordström spacetime with mass M and charge e, with |e| ≤ M
The finiteness of the local energy of φ at the Cauchy horizon stands in sharp contrast to the interior region of subextremal Reissner–Nordström, where the corresponding energy of φ has been shown to blow up for generic data [37]
We show in a subsequent paper [28] that under analogous assumptions for the decay in time of φ along the event horizon of extremal Kerr (|a| = M) and assuming axisymmetry of φ, we can extend φ continuously beyond the Cauchy horizon in the corresponding black hole interior

Summary

The Geometry of Extremal Reissner–Nordström

We will discuss some basic geometric properties of extremal Reissner–Nordström and construct regular double-null coordinates that cover the regions on both sides of either the event horizon or Cauchy horizon. We will consider Eddington–Finkelstein double-null coordinates, (u, v, θ, φ), with u, v ∈ R, in Mint, in which the metric is given by g = −4 2dudv + g/. In the notation on the left-hand side of (2.6), we integrate over spacetime with respect to the standard volume form, i.e. let f : M ∩ Du0,v0 → R be a suitably regular function and U an open subset of M, f := f (u, v, θ, φ) − det gdudvdθ dφ = f (u, v, θ, φ) 2 2r 2dμS2 dudv, where dμS2 is the standard volume form on the round sphere S2 of radius 1. Consider the following weighted vector field N in Eddington–Finkelstein double-null coordinates:.

Main Results

The Spherically Symmetric Mode φ0

H0 v log v

Energy Estimates

Pointwise Estimates