Geodesic Killing orbits and bifurcate Killing horizons

Abstract Geodesic orbits of a one-dimensional group G of isometries of a semi-Riemannian manifold are classified into complete and incomplete orbits. It is shown that the latter (which are null), if extendable, define fixed points of G. A bifurcate Killing horizon N̂ in a four-dimensional Lorentz manifold is defined as the union of intersecting (smooth) Killing horizons (of the same group G). By means of an analysis of the action of G near a fixed point, the theorem is established that a Killing horizon N is contained, as a ‘branch’, in a bifurcate Killing horizon N̂ if and only if it contains an incomplete, extendable, null geodesic orbit. Examples in familiar relativistic space times are pointed out.