On discrete features of the wave equation in singular pp-wave backgrounds

TL;DR

This paper investigates the behavior of the wave equation in specific pp-wave backgrounds with singularities, revealing a discrete dependence on profile normalization and establishing matching conditions across singularities.

Contribution

It uncovers the discrete features of wave solutions in singular pp-wave geometries and derives conditions linking geometries before and after singularities.

Findings

Existence of singular limits depends discretely on profile normalization.

Matching conditions relate pre- and post-singularity geometries.

Wave solutions exhibit scale-invariant behavior near singularities.

Abstract

We analyze the wave equation in families of pp-wave geometries developing strong localized scale-invariant singularities in certain limits. For both cases of well-localized pp-waves and the so-called null-cosmologies, we observe an intriguing discrete dependence of the existence of a singular limit on the normalization of the pp-wave profile. We also find restrictive matching conditions relating the geometries before and after the singularity (if a singular limit for the solutions of the wave equation with initial conditions specified away from the near-singular region is assumed to exist).