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Killing horizons and orthogonally transitive groups in space-time

1969
11 pages
Published in:
  • J.Math.Phys. 10 (1969) 70-81
DOI:
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Abstract: (AIP)
Some concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null hypersurface whose null tangent vector can be normalized to coincide with a Killing vector field) and of the related concepts of invertibility and orthogonal transitivity of an isometry group in an n‐dimensional pseudo‐Riemannian manifold (a group is said to be orthogonally transitive if its surfaces of transitivity, being of dimension p, say, are orthogonal to a family of surfaces of conjugate dimension n ‐ p). The relationships between these concepts are described and it is shown (in Theorem 1) that, if an isometry group is orthogonally transitive then a local isometry horizon occurs wherever its surfaces of transitivity are null, and that it is a Killing horizon if the group is Abelian. In the case of (n ‐ 2)‐parameter Abelian groups it is shown (in Theorem 2) that, under suitable conditions (e.g., when a symmetry axis is present), the invertibility of the Ricci tensor is sufficient to imply orthogonal transitivity; definitions are given of convection and of the flux vector of an isometry group, and it is shown that the group is orthogonally transitive in a neighborhood if and only if the circulation of convective flux about the neighborhood vanishes. The purpose of this work is to obtain results which have physical significance in ordinary space‐time (n = 4), the main application being to stationary axisymmetric systems; illustrative examples are given at each stage; in particular it is shown that, when the source‐free Maxwell‐Einsteinequations are satisfied, the Ricci tensor must be invertible, so that Theorem 2 always applies (giving a generalization of the theorem of Papapetrou which applies to the pure‐vaccuum case).
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