On some examples of para-Hermite and para-K\"{a}hler Einstein spaces with $\Lambda \ne 0$

TL;DR

This paper explores specific Einstein spaces with null string congruences, reducing Einstein equations to a nonlinear PDE, and presents new explicit metrics for para-Hermite and para-Kähler Einstein spaces with nonzero cosmological constant.

Contribution

It introduces new explicit metrics for para-Hermite and para-Kähler Einstein spaces with  e 0, derived from a reduction of Einstein equations to a second-order nonlinear PDE.

Findings

Derived a second-order nonlinear PDE from Einstein equations.

Presented several new explicit metrics for the studied spaces.

Discussed connections to modern mechanical approaches.

Abstract

Spaces equipped with two complementary (distinct) congruences of self-dual null strings and at least one congruence of anti-self-dual null strings are considered. It is shown that if such spaces are Einsteinian then the vacuum Einstein field equations can be reduced to a single nonlinear partial differential equation of the second order. Different forms of these equations are analyzed. Finally, several new explicit metrics of the para-Hermite and para-K\"{a}hler Einstein spaces with $\Lambda \ne 0$ are presented. Some relation of that metrics to a modern approach to mechanical issues is discussed.