Rigidity of geodesic completeness in the Brinkmann class of gravitational wave spacetimes

TL;DR

This paper investigates how geodesic completeness constrains Brinkmann spacetimes, including gravitational wave models, and proves a rigidity result related to longstanding conjectures in the field.

Contribution

It establishes a restricted rigidity theorem for Brinkmann spacetimes, linking geodesic completeness to the Ehlers-Kundt conjecture and extending known results for Cahen-Wallach spaces.

Findings

Proved a restricted rigidity result for Brinkmann spacetimes.

Connected geodesic completeness with the Ehlers-Kundt conjecture.

Settled a special case of the conjecture for Cahen-Wallach spaces.

Abstract

We consider restrictions placed by geodesic completeness on spacetimes possessing a null parallel vector field, the so-called Brinkmann spacetimes. This class of spacetimes includes important idealized gravitational wave models in General Relativity, namely the plane-fronted waves with parallel rays, or pp-waves, which in turn have been intensely and fruitfully studied in the mathematical and physical literatures for over half a century. More concretely, we prove a restricted version of a conjectural analogue for Brinkmann spacetimes of a rigidity result obtained by M.T. Anderson for stationary spacetimes. We also highlight its relation with a long-standing 1962 conjecture by Ehlers and Kundt. Indeed, it turns out that the subclass of Brinkmann spacetimes we consider in our main theorem is enough to settle an important special case of the Ehlers-Kundt conjecture in terms of the well known class of Cahen-Wallach spaces.