Exact Anti-Self-Dual four-manifolds with a Killing symmetry by similarity transformations

TL;DR

This paper analyzes the symmetry properties of anti-self-dual null Kähler four-manifolds with a Killing vector, using Lie symmetry theory to find exact solutions and classify their geometric structures.

Contribution

It applies Lie symmetry methods to classify and construct exact solutions of anti-self-dual four-manifolds with symmetries, revealing an infinite symmetry structure.

Findings

Infinite symmetry group of the constraint equations

Construction of exact spacetime solutions via similarity transformations

Simplification of differential equations using symmetry methods

Abstract

We study the group properties and the similarity solutions for the constraint conditions of anti-self-dual null K\"{a}hler four-dimensional manifolds with at least a Killing symmetry vector. Specifically we apply the theory of Lie symmetries to determine all the infinitesimal generators of the one-parameter point transformations which leave the system invariant. We use these transformations to define invariant similarity transformations which are used to simplify the differential equations and find the exact form of the spacetime. We show that the constraint equations admit an infinite number of symmetries which can be used to construct an infinite number of similarity transformations.