Twisting type-N vacuum fields with a group $H_2$

TL;DR

This paper derives equations for twisting type-N vacuum gravitational fields with specific symmetries, revealing a reduction to a third-order differential equation in a special case, advancing understanding of such spacetime structures.

Contribution

It extends previous methods to include cases with a homothetic Killing vector, deriving new equations and identifying a reduction to a third-order differential equation for particular parameters.

Findings

Reduction to a third-order ODE for the case  = -1

General equations for twisting type-N vacuum fields with one Killing and one homothetic Killing vector

Extension of previous approaches to more general symmetry conditions

Abstract

We derive the equations corresponding to twisting type-N vacuum gravitational fields with one Killing vector and one homothetic Killing vector by using the same approach as that developed by one of us in order to treat the case with two non-commuting Killing vectors. We study the case when the homothetic parameter $\phi$ takes the value -1, which is shown to admit a reduction to a third-order real ordinary differential equation for this problem, similar to that previously obtained by one of us when two Killing vectors are present.