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36Solutions with special subspacesWhen studying exact solutions, subspaces of space-time occur naturallyand frequently. Trivially in any chosen coordinate system there are subspacesin which one or two of the coordinates are constant; these subspacesmay have some significance if the coordinates they are attached to have.More significantly, subspaces arise as group orbits of groups of motions.In this chapter we shall discuss a third idea, namely to look for (threedimensional)subspaces which admit intrinsic symmetries or have someother special properties which are not shared by the full space-time. Thisidea was formulated by Collins (1979) in an explicit way, but had beenimplicitly used earlier. It has been applied also to the space of Killingtrajectories of a timelike Killing vector; in particular the case where thisspace (which in general is not a subspace of space-time) is conformallyflat has been discussed, see e.g. Perjés et al. (1984) and §18.7.36.1 The basic formulaeWe parametrize the hypersurfaces we are interested in by the (spacelikeor timelike) coordinate x 4 , and denote their normal unit vector byn a =(0, 0, 0,εN), n a =(−N α /N, 1/N ), n a n a = ε = ±1,a, b, ... =1, ..., 4, α, β, ... =1, ..., 3.The space-time metric then readsand has(36.1)ds 2 = g αβ (dx α + N α dx 4 )(dx β + N β dx 4 )+ε(Ndx 4 ) 2 (36.2)g ab =(g 3αβ + εN α N β /N 2 −εN α /N 2 ). (36.3)−εN β /N 2 ε/N 2571