On twisting type $[\textrm{N}] \otimes [\textrm{N}]$ Ricci flat complex spacetimes with two homothetic symmetries

TL;DR

This paper investigates twisting type [N] ⊗ [N] Ricci-flat complex spacetimes with two homothetic symmetries, deriving explicit solutions and reducing complex PDE systems to simpler ODEs.

Contribution

It introduces new coordinates that simplify the PDE system to an ODE and explicitly solves the special case, advancing understanding of these complex spacetimes.

Findings

Derived the general form of homothetic vector fields.

Reduced PDE system to a single ODE using new coordinates.

Explicitly solved the special case of the ODE.

Abstract

$\mathcal{HH}$ spaces of type $[\textrm{N}] \otimes [\textrm{N}]$ with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic symmetries. The general form of the homothetic vector fields are found. New coordinates are introduced which enable us to reduce the $\mathcal{HH}$ system of PDEs to one ODE on one holomorphic function. In a special case this is a second-order ODE and its general solution is explicitly given. In the generic case one gets rather involved fifth-order ODE.