Matching conditions for scattering solutions of scalar wave equations on extremal Reissner-Nordstr\"{o}m spacetimes

Abstract

We study scattering solutions $\phi$ of the linear wave equation on extremal Reissner-Nordstr\"{o}m spacetimes, satisfying the following properties: i) $\phi$ attains a prescribed radiation field $\psi_{\mathcal{I}}$ through future null infinity, which decays at an inverse polynomial rate; ii) $\phi$ is regular in the exterior region up to and including the future event horizon, i.e. $\phi\in C^N$, where $N\gg1$ is independent of the decay rate of $\psi_{\mathcal{I}}$. We prove that such solutions exist for arbitrary $N$, and that they are not unique. The proof consists of: 1) finding an approximate solution $\phi_{\mathrm{app}}$ with fast decaying error; 2) the use of backwards energy estimates in order to correct $\phi_{\mathrm{app}}$ 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-Nordstr\"{o}m black holes.