Sachs Equations and Plane Waves VI: Penrose Limits

TL;DR

This paper establishes the intrinsic nature of Penrose limits of Lorentzian metrics along null geodesics, revealing their dependence on weighted associated-graded models and residual gauge structures.

Contribution

It introduces a new intrinsic framework for Penrose limits using weighted models and null filtrations, clarifying gauge degeneracies and geometric structures involved.

Findings

Penrose limit is intrinsic on a weighted associated-graded model.

Residual gauge group is linked to null filtration splittings.

Canonical identification of Penrose limits with weighted associated graded of metrics.

Abstract

We prove that the Penrose limit of a Lorentzian metric along an affinely parametrized null geodesic is intrinsic, but intrinsic on a weighted associated-graded model determined by the null filtration rather than on a canonically identified spacetime neighborhood. Under the standard dilation scaling $(u,v,x)\mapsto (u,\lambda^2 v,\lambda x)$, admissible adapted coordinate changes collapse to their weighted homogeneous principal parts, so the large coordinate freedom of the classical construction degenerates to a small residual weighted gauge group, namely the group attached to the splittings of the null filtration. On the manifold of unparametrized null geodesics, the same weighted dilation is the grading derivation of a Heisenberg tangent model, and a $1$-jet of contact scale determines a realized degree-two direction without changing the underlying graded limit. These residual data assemble into an intrinsic unpolarized Penrose gauge bundle over the $1$-jet bundle of contact scales, with a polarized parabolic reduction after choosing a Lagrangian. Pulling the resulting model to the incidence space of spacetime points lying on null geodesics yields a canonical tautological soldering to the ambient weighted normal geometry, and fiberwise identifies the corresponding homogeneous plane-wave germ with the weighted associated graded of the ambient metric germ along the null geodesic. On the pullback of the incidence correspondence to the first jet bundle of contact scales, the tautological soldering canonically identifies the Penrose limit with an actual metric on the corresponding soldered spacetime neighborhood of the null geodesic.