Sachs Equations and Plane Waves VI: Penrose Limits

Abstract

Let $(M,g)$ be a Lorentzian manifold whose local space $\mathcal{N}$ of unparametrized null geodesics is smooth. We show that its Penrose limits assemble into a smooth bundle of plane-wave germs with a canonical gauge-theoretic soldering to spacetime. The incidence correspondence $\mathcal{U}=\{(x,[\gamma ]):x\in \gamma \}$ identifies the contact space modulo the tangent space to the sky of $x$ with the screen space at $x$, and identifies the contact line with the weight-two null quotient. These evaluation maps solder the associated-graded Penrose model. A first jet of contact scale fixes the affine scale and a Reeb representative of the weight-two direction. Above the resulting pullback of $J^1\mathcal{S}$, full neighborhood realizations form a torsor under the group ${\mathcal{G}}_{>}$ of weighted diffeomorphism germs with identity principal part. We also identify the sky-parabolic reduction and its further transverse-Lagrangian reduction to Rosen gauges. The reduction fails precisely where the chosen Lagrangian meets the moving sky, which is the Rosen caustic locus. The trace of the Grassmannian Schwarzian of the sky curve defines a conformally natural projective structure on each null geodesic, and its trace-free part is the Weyl tidal profile of the Penrose germ.

v3: Corrected a name in a reference