Scalar waves on a naked-singularity background

TL;DR

This paper establishes global weighted-L^2 and L^4 estimates for scalar waves on a super-extremal Reissner-Nordstrom black hole background, addressing well-posedness and singular potential challenges.

Contribution

It introduces a method to derive Morawetz and Strichartz estimates for scalar fields on non-globally hyperbolic spacetimes with singular potentials.

Findings

Proves Morawetz and Strichartz estimates for scalar waves on super-extremal Reissner-Nordstrom backgrounds.

Shows the well-posedness of the Cauchy problem using Friedrichs extension.

Transforms the problem to wave equations with singular potentials satisfying necessary conditions.

Abstract

We obtain global space-time weighted-L^2 (Morawetz) and L^4 (Strichartz) estimates for a massless chargeless scalar field propagating on a super-extremal (overcharged) Reissner-Nordstrom background. We begin by discussing the question of well-posedness of the Cauchy problem for scalar fields on non-globally hyperbolic manifolds and the role played by the Friedrichs extension, go over the construction of the function spaces involved, show how to transform the problem to one about the wave equation on the Minkowski space with a singular potential, and finally prove that the potential thus obtained satisfies the various conditions needed in order for the estimates to hold.