Spin coefficients and gauge fixing in the Newman-Penrose formalism

TL;DR

This paper develops a method to express Newman-Penrose spin coefficients as functions of tetrad invariants, aiming to clarify gauge choices and physical degrees of freedom in the formalism.

Contribution

It provides the first general recipe for expressing spin coefficients in terms of tetrad invariants after choosing a tetrad.

Findings

Established a method to relate spin coefficients to tetrad invariants.

Provided a set of identities and quantities needed for the process.

Facilitated better gauge fixing and interpretation in Newman-Penrose formalism.

Abstract

Since its introduction in 1962, the Newman-Penrose formalism has been widely used in analytical and numerical studies of Einstein's equations, like for example for the Teukolsky master equation, or as a powerful wave extraction tool in numerical relativity. Despite the many applications, Einstein's equations in the Newman-Penrose formalism appear complicated and not easily applicable to general studies of spacetimes, mainly because physical and gauge degrees of freedom are mixed in a nontrivial way. In this paper we approach the whole formalism with the goal of expressing the spin coefficients as functions of tetrad invariants once a particular tetrad is chosen. We show that it is possible to do so, and give for the first time a general recipe for the task, as well as an indication of the quantities and identities that are required.