On the geometry of pp-wave type spacetimes

TL;DR

This paper explores the geometric and causal properties of generalized pp-wave spacetimes, analyzing their completeness, connectedness, and conjugate points, and highlights the need to adapt Riemannian results to Lorentzian settings.

Contribution

It provides a comprehensive analysis of the geometry of pp-wave type spacetimes, extending classical results and emphasizing the adaptation of Riemannian tools to Lorentzian geometry.

Findings

No horizons exist in these spacetimes.

Certain conditions ensure geodesic completeness.

Results highlight the need to modify Riemannian theorems for Lorentzian contexts.

Abstract

Global geometric properties of product manifolds ${\cal M}= M \times \R^2$, endowed with a metric type $<\cdot, \cdot > = < \cdot, \cdot >_R + 2 dudv + H(x,u) du^2$ (where $<\cdot, \cdot >_R$ is a Riemannian metric on $M$ and $H:M \times \R \to \R$ a function), which generalize classical plane waves, are revisited. Our study covers causality (causal ladder, inexistence of horizons), geodesic completeness, geodesic connectedness and existence of conjugate points. Appropiate mathematical tools for each problem are emphasized and the necessity to improve several Riemannian (positive definite) results is claimed.