Bounds for scalar waves on self-similar naked-singularity backgrounds

TL;DR

This paper investigates the behavior of scalar waves on self-similar spacetimes with naked singularities, demonstrating boundedness of wave multipoles and energy densities at the Cauchy horizon, indicating stability features.

Contribution

It provides new bounds for scalar wave multipoles and energy densities on self-similar naked-singularity backgrounds, contributing to understanding their stability.

Findings

Scalar wave multipoles have finite $L^2$ norm at the Cauchy horizon.

Multipoles obey pointwise bounds at the horizon.

Energy densities of scalar waves are bounded at the horizon.

Abstract

The stability of naked singularities in self-similar collapse is probed using scalar waves. It is shown that the multipoles of a minimally coupled massless scalar field propagating on a spherically symmetric self-similar background spacetime admitting a naked singularity maintain finite $L^2$ norm as they impinge on the Cauchy horizon. It is also shown that each multipole obeys a pointwise bound at the horizon, as does its locally observed energy density. $L^2$ and pointwise bounds are also obtained for the multipoles of a minimally coupled massive scalar wave packet.