Any spacetime has a Bianchi type I spacetime as a limit

TL;DR

This paper introduces an 'ultra-local limit' process showing that any spacetime can be locally approximated by a Bianchi type I spacetime near a timelike geodesic, extending the idea of Penrose limits.

Contribution

It establishes a new limiting process that generalizes the Penrose limit, allowing any spacetime to be approximated by a Bianchi type I model near a timelike geodesic.

Findings

Any spacetime has a Bianchi type I limit near a timelike geodesic.

The ultra-local limit generalizes the Penrose limit for null geodesics.

The process can produce non-diagonal Bianchi type I spacetimes.

Abstract

Pick an arbitrary timelike geodesic in an arbitrary spacetime. We demonstrate that there is a particular limiting process, an "ultra-local limit", in which the immediate neighborhood of the timelike geodesic can be "blown up" to yield a general (typically non-diagonal) Bianchi type I spacetime. This process shares some (but definitely not all) of the features of the Penrose limit, whereby the immediate neighborhood of an arbitrary null geodesic is "blown up" to yield a pp-wave as a limit.