The General Self-dual solution of the Einstein Equations

TL;DR

This paper derives the most general explicit (anti)self-dual solutions to Einstein's equations, showing they are characterized by harmonic functions and include Gibbons-Hawking multi-center metrics, with a reinterpretation as Maxwell equations.

Contribution

It provides a complete characterization of all explicit (anti)self-dual Einstein solutions using three free functions, including stationary solutions as harmonic functions, and links to Maxwell equations.

Findings

Most general explicit (anti)self-dual Einstein solutions identified.

Stationary solutions characterized by harmonic functions.

Gibbons-Hawking multi-center metrics are the most general stationary solutions.

Abstract

We obtain the most general explicit (anti)self-dual solution of the Einstein equations. We find that any (anti)self-dual solution can be characterised by three free functions of which one is harmonic. Any stationary (anti)self-dual solution can be characterised by a harmonic function. It turns out that the form of the Gibbons and Hawking multi-center metrics is the most general stationary (anti)self-dual solution. We further note that the stationary (anti)self-dual Einstein equations can be reinterpreted as the (anti)self-dual Maxwell equations on the Euclidean background metric.