Matching van Stockum dust to Papapetrou vacuum

TL;DR

This paper demonstrates a method to match van Stockum dust solutions with a family of non-static Papapetrou vacuum metrics, revealing new insights into boundary conditions and transformations in general relativity.

Contribution

It establishes a novel correspondence between van Stockum dust solutions and Papapetrou vacuum metrics, including explicit examples and boundary characterization.

Findings

Every van Stockum dust can be matched to a 1-parametric family of Papapetrou vacuum metrics.

Boundary conditions are determined by the vanishing of a specific first-order invariant.

New vacuum exterior solutions to the Lanczos--van Stockum dust metric are constructed.

Abstract

Addressing a long-standing problem, we show that every van Stockum dust can be matched to a 1-parametric family of non-static Papapetrou vacuum metrics, and the converse. The boundary, if existing, is determined by vanishing of certain first-order invariant on the vacuum side. Moreover, we establish a relation to Ehlers and Kramer--Neugebauer transformations, which allows us to look for dust clouds with a prescribed boundary. Explicit examples include the Bonnor metric and a new vacuum exterior to the Lanczos--van Stockum dust metric, as well as dust clouds with nontrivial topology.