Axisymmetric Stationary Solutions as Harmonic Maps

TL;DR

This paper introduces a method to generate exact vacuum solutions to Einstein's equations using harmonic maps, enabling classification of axisymmetric stationary spacetimes based on group theory.

Contribution

It presents a novel approach that reformulates Einstein's equations as harmonic maps into SL(2,R), allowing systematic solution generation and classification.

Findings

Provides a harmonic map decomposition of Einstein equations

Classifies solutions into known mathematical classes

Offers a new framework for exact solution generation

Abstract

We present a method for generating exact solutions of Einstein equations in vacuum using harmonic maps, when the spacetime possesses two commutating Killing vectors. This method consists in writing the axisymmetric stationry Einstein equations in vacuum as a harmonic map which belongs to the group SL(2,R), and decomposing it in its harmonic "submaps". This method provides a natural classification of the solutions in classes (Weil's class, Lewis' class etc).