An example of algebraically general para-K\"ahler Einstein space

TL;DR

This paper constructs a new example of algebraically general para-Kähler Einstein spaces with symmetries, showing that certain symmetry conditions reduce Einstein equations to a solvable differential equation.

Contribution

It provides the first explicit example of an algebraically general para-Kähler Einstein space and demonstrates reduction of Einstein equations under specific symmetry assumptions.

Findings

Reduction of Einstein equations to a first-order differential equation

Explicit solution in one case

First example of such a space with specified symmetries

Abstract

Algebraically general para-K\"ahler Einstein spaces equipped with 3D algebras of infinitesimal symmetries are considered. It is shown that if the algebra contains 2D trivial subalgebra then vacuum Einstein field equations with cosmological constant can be reduced to a single, first-order differential equation. One of the cases is solved explicitly. Hence, a first example of an algebraically general para-K\"ahler Einstein space is given.