1 | See, e.g., [89], p. 239 or [151], p. 92 for the proof. | |

2 | This theorem was first proven by Schoen and Yau [288, 289] and somewhat later, using spinor techniques, by Witten [325] (compare [265]). See [12] for a version relevant to the uniqueness problem, which allows degenerate components of the event horizon. | |

3 | Non-existence of certain static -body configurations (possibly, but not necessarily, black holes) was established in [21, 20]). These results rely on the positive energy theorem and exclude, in particular, suitably regular configurations with a reflection symmetry across a noncompact surface, which is disjoint from the matter regions. | |

4 | Here we are interested in stationary multi–black-hole configurations; nonexistence of some suitably regular stationary -body configurations was established, under different symmetry conditions, in [20, 21]. | |

5 | Studies of regularity and causal structure of black rings and Saturns can be found in [82, 67, 68, 305, 71]. | |

6 | It should be noted that, although formulated for 4-dimensional spacetimes, the results in [84] remain valid without changes in higher-dimensional spacetimes. | |

7 | An early apparent success rested on a sign error [46]. Carter’s amended version of the proof was subject to a certain inequality between the electric and the gravitational potential [50]. | |

8 | As already mentioned in Section 5.4, these black holes present counter-examples to the naive generalization of the staticity theorem; they are nice illustrations of the correct non-Abelian version of the theorem [302, 303]. | |

9 | The symmetry condition (6.13) translates into and , which can be used to reduce the EYM equations in the presence of a Killing symmetry in a gauge-invariant manner [158, 159]. | |

10 | In addition to the actual scalar fields, the effective action comprises two gravitational scalars (the norm and the generalized twist potential) and two scalars for each stationary Abelian vector field (electric and magnetic potentials). | |

11 | A workable formula for is provided by (4.3) (compare [228, eq. 4.14]). |

Living Rev. Relativity 15, (2012), 7
http://www.livingreviews.org/lrr-2012-7 |
This work is licensed under a Creative Commons License. E-mail us: |