A new approach to the Newman-Penrose formalism

TL;DR

This paper introduces a simplified and invariant approach to the Newman-Penrose formalism using transverse tetrads, enhancing understanding and potential generalization to higher dimensions.

Contribution

It presents a new compact formulation of the Newman-Penrose equations in transverse tetrads, with invariant expressions for Weyl scalars and spin coefficients, facilitating higher-dimensional extensions.

Findings

Equations can be recast in a more compact form

Weyl scalars expressed as functions of curvature invariants

Spin coefficients determined invariantly

Abstract

The Newman-Penrose formalism in transverse tetrads, namely those tetrads where \Psi_1=\Psi_3=0, is studied. In particular it is shown that the equations governing the dynamics within this formalism can be recast in a particularly compact way, leading to a better understanding of the formalism itself. The particular choice of tetrad allows not only to obtain the expression of Weyl scalars as simple functions of curvature invariants, but also the spin coefficients can be partly determined in an invariant way, by means of a new expression for the Bianchi identities that shows the various degrees of freedom in a more intuitive and direct way. We expect this approach to be very promising for a better understanding of all the equations governing the Newman-Penrose formalism. Such a new insight to the equations can also turn out to be useful for a generalization of the Newman- Penrose formalism to higher dimensions, thus allowing a better analysis of the various degrees of freedom, in view of extending to this case results already known in four dimensions.