(Conformal) Killing vectors and their associated bivectors

TL;DR

This paper extends a formalism for analyzing spacetime symmetries using bivectors to include conformal and homothetic Killing vectors in arbitrary spacetimes, enhancing methods to identify such vectors.

Contribution

It generalizes existing bivector-based formalisms to encompass conformal and homothetic Killing vectors without requiring tetrad alignment.

Findings

Formalism applicable to arbitrary spacetimes with conformal symmetries

Efficient method for finding conformal Killing vectors using preferred tetrads

Illustrative example with Kimura's metric demonstrates practicality

Abstract

Fayos and Sopuerta have recently set up a formalism for studying vacuum spacetimes with an isometry, a formalism that is centred around the bivector corresponding to the Killing vector and that adapts the tetrad to the bivector. Steele has generalized their approach to include the homothetic case. Here, we generalize this formalism to arbitrary spacetimes and to homothetic and conformal Killing vectors but do not insist on aligning the tetrad with the bivector. The most efficient way to use the formalism to find conformal Killing vectors (proper or not) of a given spacetime is to combine it with the notion of a preferred tetrad. A metric by Kimura is used as an illustrative example.