Geometry via Plane wave limits

TL;DR

This work develops a covariant framework to produce plane wave limits of arbitrary semi-Riemannian metrics along any geodesic, extending Penrose's classical limit via Blau et al.'s formulation. The authors show these limits are Lorentzian plane waves whose curvature data $\mathrm{Rm}_{g^\gamma_L}(\partial_i,\partial_t,\partial_t,\partial_j) = -A^{\gamma}_{ij}(t)$ encode substantial geometric information from the original metric, including geodesic deviation in the causally independent case, and establish links to Penrose's original construction and to Fermi coordinates. They generalize the construction to semi-Riemannian settings with a causality-aware transverse dimension and derive a Bochner-type relation for vector fields, connecting transverse dynamics to the ambient curvature. As applications, they prove a semi-Riemannian Hawking–Penrose-type focusing result: if $\gamma$ is complete, causally independent, has no conjugate points, and $\operatorname{Ric}_g(\gamma',\gamma') \ge 0$, then $\mathrm{Rm}_g(\cdot,\gamma',\gamma',\cdot)|_{\gamma'^{\perp}}=0$, along with a finite conjugate-point statement via a Morse index argument, thus linking semi-Riemannian geometry with Lorentzian focusing theory.

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.

Theorems & Definitions (23)

• Theorem
• Theorem
• Theorem
• Definition 1: Riemannian plane wave limit
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1
• proof