On the Theory of Killing Orbits in Space-Time

TL;DR

This paper provides a comprehensive theoretical analysis of Killing orbits in space-time, exploring their structure, stability, and relation to the Lie algebra of Killing vector fields, with illustrative examples and clarifications of existing theorems.

Contribution

It introduces a general decomposition of space-time based on orbit characteristics and discusses stability, linking orbit dimensions with the Killing algebra and isotropies.

Findings

Stable orbits are well-behaved and predictable.

Unstable orbits may exhibit non-independence of Killing vectors.

Examples demonstrate the theoretical concepts and clarify Fubini's theorem.

Abstract

This paper gives a theoretical discussion of the orbits and isotropies which arise in a space-time which admits a Lie algebra of Killing vector fields. The submanifold structure of the orbits is explored together with their induced Killing vector structure. A general decomposition of a space-time in terms of the nature and dimension of its orbits is given and the concept of stability and instability for orbits introduced. A general relation is shown linking the dimensions of the Killing algebra, the orbits and the isotropies. The well-behaved nature of "stable" orbits and the possible miss-behaviour of the "unstable" ones is pointed out and, in particular, the fact that independent Killing vector fields in space-time may not induce independent such vector fields on unstable orbits. Several examples are presented to exhibit these features. Finally, an appendix is given which revisits and attempts to clarify the well-known theorem of Fubini on the dimension of Killing orbits.