General heavenly equation governs anti-self-dual gravity

TL;DR

This paper demonstrates that the general heavenly equation describes anti-self-dual gravity, deriving related metrics, presenting algebraic solutions, and providing a real solution for neutral signature, thus advancing understanding of ASD gravitational structures.

Contribution

It establishes the governing role of the general heavenly equation in ASD gravity and derives explicit metrics and solutions, including a real neutral signature case.

Findings

Derived ASD Ricci-flat metrics from the general heavenly equation

Presented algebraic exact solutions as zeros of homogeneous polynomials

Obtained a real solution for neutral signature case

Abstract

We show that the general heavenly equation, suggested recently by Doubrov and Ferapontov \cite{fer}, governs anti-self-dual (ASD) gravity. We derive ASD Ricci-flat vacuum metric governed by the general heavenly equation, null tetrad and basis of 1-forms for this metric. We present algebraic exact solutions of the general heavenly equation as a set of zeros of homogeneous polynomials in independent and dependent variables. A real solution is obtained for the case of neutral signature.