Einstein-Weyl from Kaluza-Klein

TL;DR

This paper explores how Kaluza-Klein reduction of four-dimensional spaces with self-dual or anti-self-dual Weyl tensors leads to Einstein-Weyl equations in three dimensions, providing new insights into gravitational instantons.

Contribution

It demonstrates the emergence of Einstein-Weyl equations from Kaluza-Klein reduction of special four-dimensional geometries, offering a new perspective on gravitational instantons.

Findings

Derivation of Einstein-Weyl equations from Kaluza-Klein reduction.

Expression for gravitational instanton density in terms of Kaluza-Klein functions.

Connection between self-dual Weyl tensors and Einstein-Weyl structures.

Abstract

We discuss the Kaluza-Klein reduction of spaces with (anti-)self-dual Weyl tensor and point out the emergence of the Einstein-Weyl equations for the reduction from four to three dimensions. As a byproduct we get a simple expression for the gravitational instanton density in terms of the Kaluza-Klein functions.