Matching conditions for scattering solutions of scalar wave equations on extremal Reissner-Nordström spacetimes

Abstract

We study scattering solutions $φ$ of the linear wave equation on extremal Reissner-Nordström spacetimes, satisfying the following properties: i) $φ$ attains a prescribed radiation field $ψ_{\mathcal{I}}$ through future null infinity, which decays at an inverse polynomial rate; ii) $φ$ is regular in the exterior region up to and including the future event horizon, i.e. $φ\in C^N$, where $N\gg1$ is independent of the decay rate of $ψ_{\mathcal{I}}$. We prove that such solutions exist for arbitrary $N$, and that they are not unique. The proof consists of: 1) finding an approximate solution $φ_{\mathrm{app}}$ with fast decaying error; 2) the use of backwards energy estimates in order to correct $φ_{\mathrm{app}}$ to an exact solution. Extremality is used only in the second step. The methods of the linear case described above are then used to show the same results for semilinear equations where the nonlinearity satisfies the null condition, as well as to geometries describing the hyperbolic orbit of multiple extremal Reissner-Nordström black holes.

Figures (3)

• Figure 1: Depicted is the confromal compactification (Penrose diagram) of $\mathcal{M}_{\mathfrak{t}}$ together with the hypersurfaces: (1) the horizon $\mathcal{H}=\{r=M\}$, (2) partial Cauchy hypersurface $\Sigma_{\tau}=\{t_*=\tau\}$, (3) spacelike hypersurfaces $\mathcal{B}=(M-r)v=\mathfrak{t}$, (4) fictive asymptotic boundary called null infinity $\mathcal{I}$.
• Figure 2: Above we have drawn the analytic compactification of an extremal Reissner--Nordström spacetime denoting by $\mathcal{B}$ a spacelike hypersurface in the interior of the black hole region, by $\mathcal{H}$ the event horizon, by $\mathcal{I}^+$ future null infinity, and by $i^+$ timelike infinity, while $\mathcal{F}$ and $\mathcal{K}$ are defined above. Note that the local coordinates around $\mathcal{F},\mathcal{K},i^+$ are $\{(r-M)t_*,x/\left\lvert x\right\rvert,t_*^{-1}\}$, $\{x,t_*^{-1}\}$,and $\{\frac{x}{t},t_*^{-1}\}$ respectively.
• Figure 3: We drawn the compactification $\overline{\mathcal{M}} _{\mathrm{m},\mathfrak{t}}$ on which our scattering result takes place. We only indicated how the geometry looks around a single black hole, with $\mathcal{H}_{g_\mathrm{m}},\mathcal{H}_{g_{\mathrm{ERN}}}$ denoting the black hole horizon of $g_\mathrm{m}$ and the image of the horizon of $g_{\mathrm{ERN}}$ under the diffeomorphism $\Phi_z$. Part of $\mathcal{H}_{g_\mathrm{m}}$ is dotted as we do not compute the precise convergence rate of the two hypersurfaces and thus the asymptotic behaviour on the compactification $\overline{\mathcal{M}} _{\mathrm{m},\mathfrak{t}}$.

Theorems & Definitions (138)

• Theorem 1.1: Rough version
• Conjecture 1.1: dafermos_scattering_2024
• Theorem 1.2: Theorem 4.4 of angelopoulos_non-degenerate_2020
• Conjecture 1.2
• Theorem 1.3: Matching conditions for semilinear waves on $(\mathcal{M} , g_{\mathrm{ERN}})$
• Remark 1.1: Interior solution
• Remark 1.2: Conformal, conormal regularity
• Remark 1.3: Nonlinearity
• Remark 1.4: Finite regularity
• Remark 1.5: Polyhomogeneity