Geometry via Plane wave limits

TL;DR

This paper generalizes Penrose's plane wave limit to semi-Riemannian manifolds, constructing invariant limits along geodesics that reveal geometric and curvature properties, with applications to geodesic behavior and curvature vanishing conditions.

Contribution

It introduces a covariant construction of plane wave limits for semi-Riemannian metrics, extending Penrose's limit and linking it to tensorial geometry and geodesic deviation.

Findings

Constructed plane wave limits for any semi-Riemannian metric.

Extended Hawking-Penrose results to semi-Riemannian manifolds.

Proved a Morse Index Theorem for specific geodesics.

Abstract

Utilizing the covariant formulation of Penrose's plane wave limit by Blau et~al., we construct for any semi-Riemannian metric $g$ a family of "plane wave limits." These limits are taken along any geodesic of $g$, yield simpler metrics of Lorentzian signature, and are isometric invariants. We show that they generalize Penrose's limit to the semi-Riemannian regime and, in certain cases, encode $g$'s tensorial geometry and its geodesic deviation. As an application of the latter, we partially extend a well known result by Hawking & Penrose to the semi-Riemannian regime: On any semi-Riemannian manifold, if the Ricci curvature is nonnegative along any complete geodesic without conjugate points that is "causally independent" (in a sense we make precise), then the curvature tensor along that geodesic must vanish in all normal directions. A Morse Index Theorem is also proved for such geodesics.