Finding integrals and identities in the Newman Penrose formalism: a comment on Class. Quantum Grav. 26 (2009) 105022 and on Gen. Relativ. Gravit. (2014)46:1703

TL;DR

This paper demonstrates that the Newman-Penrose formalism alone can derive integral identities and classify Ricci flat, Petrov type D solutions to Einstein's equations, without computer algebra or tensorial methods, providing new insights.

Contribution

It shows that the NP formalism as an exterior differential system suffices for deriving identities and classifying solutions, challenging the reliance on GHP formalism and CAS tools.

Findings

Proves fundamental identities using NP formalism without CAS

Identifies a new integral of the exterior differential system

Provides a refined classification of Ricci flat, Petrov type D solutions

Abstract

In 1969 Kinnersley, using the NP formalism, found all the Petrov type D, Ricci flat, solutions to the Einstein's Field Equations. Yet, in doing so -as it seems- he neglected two fundamental identities (or constraints) on four NP variables and Cartan invariants as well, namely $\tau\bar{\tau}-\pi\bar{\pi}=0$ and $\rho\bar{\mu}-\mu\bar{\rho}=0$. Since then, these identities have been constantly either overlooked or proven under special circumstances (e.g. electrovac solutions). It was only until 2009 when Edgar et al. by making an extended use of the GHP formalism, and of a computer algebra system, succeeded in proving those identities in the general case. In that reference, it was -rather indirectly- implied that the results under consideration were provable only within the GHP formalism and thus the latter is the optimal tool towards the invariant classification and study of classes of solutions to the EFEs. In 2014 there was a kind of response to that paper by J.J. Ferrando & J.A. Saez. Using the tensorial algebra (of 2-forms), and without the aid of a CAS, the authors proved the desired result and they offer a much more refined and extended classification of the Petrov type D, Ricci flat, solutions. Never the less when someone reads that third work, although beautiful and conceptually simple, one has the feeling that the authors know in advance what they want to prove; something which is not always the case. The goal of the present short work is to prove, through the specific example (i.e., the class of Ricci flat, Petrov type D geometries), that the original NP formalism, seen as an exterior differential system suffices to provide the desired results -thus commenting on the second cited paper- not only without (in principle) the aid of a CAS, but also to obtain a new further (unknown until now) integral of the EDS -thus commenting on the third cited paper.