Mathematics

Geometry via plane wave limits

Document Type

Article

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.

Publication Title

Classical and Quantum Gravity

Publication Date

11-2025

Volume

44

Issue

22

ISSN

0264-9381

DOI

10.1088/1361-6382/ae1711

Keywords

Hawking-Penrose focusing theorem, plane wave limit, semi-Riemannian geometry

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