### Abstract

This paper gives a theoretical discussion of the orbits and isotropies which arise in a spacetime 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 spacetime 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 misbehaviour of the 'unstable' ones is pointed out and, in particular, the fact that independent Killing vector fields in spacetime may not induce independent Killing 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.

Original language | English |
---|---|

Pages (from-to) | 4067-4084 |

Number of pages | 17 |

Journal | Classical and Quantum Gravity |

Volume | 20 |

DOIs | |

Publication status | Published - 2003 |

### Keywords

- GENERAL-RELATIVITY
- VECTOR FIELDS
- FIXED-POINTS
- SYMMETRIES
- INTEGRABILITY
- CURVATURE

### Cite this

*Classical and Quantum Gravity*,

*20*, 4067-4084. https://doi.org/10.1088/0264-9381/20/18/313

**On the theory of Killing orbits in space time.** / Hall, Graham Stanley.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 20, pp. 4067-4084. https://doi.org/10.1088/0264-9381/20/18/313

}

TY - JOUR

T1 - On the theory of Killing orbits in space time

AU - Hall, Graham Stanley

PY - 2003

Y1 - 2003

N2 - This paper gives a theoretical discussion of the orbits and isotropies which arise in a spacetime 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 spacetime 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 misbehaviour of the 'unstable' ones is pointed out and, in particular, the fact that independent Killing vector fields in spacetime may not induce independent Killing 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.

AB - This paper gives a theoretical discussion of the orbits and isotropies which arise in a spacetime 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 spacetime 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 misbehaviour of the 'unstable' ones is pointed out and, in particular, the fact that independent Killing vector fields in spacetime may not induce independent Killing 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.

KW - GENERAL-RELATIVITY

KW - VECTOR FIELDS

KW - FIXED-POINTS

KW - SYMMETRIES

KW - INTEGRABILITY

KW - CURVATURE

U2 - 10.1088/0264-9381/20/18/313

DO - 10.1088/0264-9381/20/18/313

M3 - Article

VL - 20

SP - 4067

EP - 4084

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

ER -