Rod Structures and Patching Matrices: a review

TL;DR

This review explores the twistor theory approach to constructing stationary, axisymmetric solutions of Einstein's vacuum equations, emphasizing the role of patching matrices and rod structures in classifying these solutions.

Contribution

It provides a comprehensive overview of the twistor construction method and investigates how rod structures and asymptotics determine the holomorphic patching matrix for these solutions.

Findings

Catalogue of examples of solutions

Analysis of how rod structures influence the patching matrix

Insights into the inverse problem of metric reconstruction

Abstract

I review the twistor theory construction of stationary and axisymmetric, Lorentzian signature solutions of the Einstein vacuum equations and the related toric Ricci-flat metrics of Riemannian signature, \cite{W,MW,F,FW}. The construction arises from the Ward construction \cite{W2} of anti-self-dual Yang-Mills fields as holomorphic vector bundles on twistor space, with the observation of Witten \cite{LW} that the Einstein equations for these metrics include the anti-self-dual Yang-Mills equations. The principal datum for a solution is the holomorphic patching matrix $P$ for a holomorphic vector bundle on a reduced twistor space, and $P$ is typically simpler than the corresponding metric to write down. I give a catalogue of examples, building on earlier collections \cite{F,AG}, and consider the inverse problem: how far does the rod structure of such a metric, together with its asymptotics, determine $P$?