Integration in the GHP formalism II: An operator approach for spacetimes with killing vectors, with applications to twisting type n spaces

TL;DR

This paper develops an operator approach within the GHP formalism to analyze spacetimes with Killing vectors, simplifying the problem to differential equations and demonstrating its application to vacuum Type N spaces with symmetries.

Contribution

It introduces a new zero-weighted GHP operator and a systematic method to incorporate Killing vectors into the GHP formalism, enabling reduction to differential equations for symmetric spacetimes.

Findings

Reduction to a pair of ordinary differential operator master equations

Derivation of a third order differential equation for a complex function

Outline of how previous equations relate to the new master equations

Abstract

Held has proposed a coordinate- and gauge-free integration procedure within the GHP formalism built around four functionally independent zero-weighted scalars constructed from the spin coefficients and the Riemann tensor components. Unfortunately, a spacetime with Killing vectors will be unable to supply the full quota of four scalars of this type. However, for such a spacetime additional scalars are supplied by the components of the Killing vectors; by using these alongside the spin coefficients and the Riemann tensor components we have the possibility of constructing the full quota of four functionally independent zero-weighted scalars, and of exploiting Held's procedure. As an illustration we investigate the vacuum Type N spaces admitting a Killing vector and a homothetic Killing vector. In a direct manner, we reduce the problem to a pair of ordinary differential operator `master equations', making use of a new zero-weighted GHP operator. By first rewriting the master equations as a closed set of complex first order equations, we reduce the problem to one real third order operator differential equation for a complex function of a real variable --- but with still the freedom to choose explicitly our fourth coordinate. An alternative, more algorithmic approach, using a closed chain of real first order equations for real functions, reduces the problem to the same order, but in a more natural and much more concise form. It is also outlined how the various other third order differential equations, which have been derived previously by other workers on this problem, can be deduced from our master equations.