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group. Taub-NTJT space and its extensions are obtained from part of this space by idonfcifinnfcion of ¡loinJ.H iimlor 11 (linni'ln Htil>groii|i of tho inoiimtry group.

group. Taub-NTJT space and its extensions are obtained from part of this space by idonfcifinnfcion of ¡loinJ.H iimlor 11 (linni'ln Htil>groii|i of tho inoiimtry group.

One would also obtain a Hausdorff manifold by taking the quotient of (I+ + II+ + I_): this corresponds to extending like (J2"', gF') at the surface't = t+ but extending like &v") at the surface t = t_. By taking the quotient of the whole space & minus the points and p_ one obtains a non-Hausdorff manifold; and taking the quotient of & one obtains a non-Hausdorff non-manifold in a way analogous to that in the example above. As in that example, one can take the quotient of the bundle of linear frames over iF and obtain a Hausdorff manifold.

By combining these extensions of the (i, i/r) plane with the coordinates \8,<j>) one can obtain corresponding extensions of the four-dimensional space g). In particular, the two extensions and (J*"", gy") give rise to two different locally inextendible analytic extensions of [J?, g), and both are geodesically incomplete. Consider one of these extensions, say (ui", g'). Tho throo-nphoron which nro (ho nurfiioon of l.rannil ivil.y of Mm inomolry group uxe espuoo-

like surfaces in the region t_<t <t+ and are timelike for t > t+ and t < t_. The two surfaces of transitivity t = t_ and t = i+ are null surfaces and they form the Cauchy horizon of any spacelike surface contained in the region < t < t+, because there are timelike curves in the regions t < t_ and t > t+ which do not cross t = t_ and t = t+ respectively (for example, closed timelike curves exist in the regions t < t_ and t > t+). The region of space-time t_ < t < t+ is compact yet there are timelike and null geodesies which remain within it and are incomplete. This kind of behaviour will be considered further in chapter 8.

Further details of Taub-NUT space may be found in Misner and Taub (1060), Misncr (1063).

5.9 Further exact solutions

We have examined in this chapter a number of exact solutions and used them to give examples of the various global properties which we shall wish to discuss more generally later. Although a large number of exact solutions are known locally, relatively few have been examined globally. To complete this chapter, we shall mention briefly two other interesting families of exact solutions whose global properties are known.

The first of these are the plane wave solutions of the empty space field equations. These are homeomorphic to R*, and global coordinates (y, z, u, v), which range from — oo to + oo, can be chosen so that the metric takes the form ds2 = 2 dw di> + dy*+dz2 + H(y, z, u) dw2, where H = (y2—z2)f{u) — 2yzg(u);

f(u) and g(u) are arbitrary C2 functions determining the amplitude and polarization of the wave. These spaces are invariant under a five-parameter group of isometries multiply transitive on the null surfaces {u = constant}; a special subclass, in which/(w) = cos 2u,g(u) = sin 2u, admit an extra Killing vector field, and are homogeneous space-times invariant under a six-parameter group of isometries. These spapes do not contain any closed timelike or null curves; however they admit no Cauchy surfaces (Penrose (1965a)). Local properties of these spaces have been studied in detail by Bondi, Pirani and Robinson (1959), and global properties by Penrose (1965a); Oszv&th and Schiicking (1962) have studied global properties of the higher symmetry space. The way in which two impulsive plane waves scatter each other and give rise to a singularity has been studied by Khan and Penrose (1971).

The other is the five-parameter family of exact solutions of the source-free Einstein-Maxwell equations found by Carter (19686) (see also Demianski and Newman (1966)). These include the Schwarzschild, Reissner-Nordstrom, Kerr, charged Kerr, Taub-NUT, de Sitter and anti-de Sitter solutions as special cases. A description of some of their global properties is given in Carter (1967). Some cases closely related to this family have been examined by Ehlers and Kundt (1962) and Kinnerfiley and Walker (1970).

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