Vacuum spacetimes with a spacelike, hypersurface-orthogonal Killing vector: reduced equations in a canonical frame

TL;DR

This paper simplifies the Newman-Penrose equations for vacuum spacetimes with a spacelike, hypersurface-orthogonal Killing vector by reducing them to a minimal set in a canonical frame, facilitating analysis of their differential structure.

Contribution

It provides a reduced form of the Newman-Penrose equations in a canonical frame for spacetimes with a spacelike Killing vector, including explicit expressions and coordinate-independent metric choices.

Findings

Reduced equations to a minimal set in a canonical frame

Explicit expressions for frame vectors in arbitrary coordinates

Coordinate-independent metric function choices that simplify Ricci tensor components

Abstract

The Newman-Penrose equations for spacetimes having one spacelike Killing vector are reduced -- in a geometrically defined "canonical frame'' -- to a minimal set, and its differential structure is studied. Expressions for the frame vectors in an arbitrary coordinate basis are given, and coordinate-independent choices of the metric functions are suggested which make the components of the Ricci tensor in the direction of the Killing vector vanish.