(2) there exist equations of motion for the matter fields such that the Cauchy problem has a unique solution (see chapter 7);

(3) the Cauchy data on some spacelike three-surface is invariant under a group of diffeomorphisms of which is transitive on ¿f.

Since the intrinsic geometry of J? is invariant under a transitive group of diffeomorphisms, these are isometries and is complete, i.e. cannot have any boundary. It can be shown (see § 6.5) that if there is a non-spacelike curve which intersects more than once, then there exists a covering manifold in which each connected component of the image of will not intersect any non-spacelike curve more than once. We shall assume that Ji is timelike geodesically complete, and show that this is inconsistent with conditions (1), (2) and (3).

Let be a connected component of the image of ¿f in J?. By (3), the Cauchy data on J>r is homogeneous. Therefore by condition (2), the Cauchy development of any region of ir is isometric to the Cauchy development of any other similar region of Jr. This implies that the surfaces {« = constant} are homogeneous if they lie within the Cauchy

148 EXACT SOLUTIONS [5.4

Al A, development of Jf, where « is the distance from MP measured along the geodesic normals to . These surfaces must lie either entirely within or entirely outside the Cauchy development of./?', as otherwise there would be equivalent regions in Jr which had inequivalent Cauchy evolutions. The surfaces {« = constant} will lie in the Cauchy development of as long as they remain spacelike, because the boundary of the Cauchy development of M' (if it exists) must be null (§6.5).

The geodesies orthogonal to & will be orthogonal to the surfaces {« = constant}, as a vector representing the separation of points equal distances along neighbouring geodesies will remain orthogonal to the geodesies if it is so initially. As in §4.1, one can represent the spatial Reparation of neighbouring geodesies orthogonal to & by a matrix A which is the unit matrix on M". By homogeneity, it will be constant on the surfaces {« = constant} while these lie in the Cauchy development of Jr. While A is non-degenerate, the map from x to a surface (s = constant} defined by the normal geodesies will be of rank three and so the surfaces will be spacelike three-surfaces contained within the Cauchy development of Jf. The expansion

6 = (det A)"1 d (det A)/ds of these geodesies oboys Raychaudhuri's equation (4.26) with the vorticity and acceleration zero. By condition (1), Rab VaVb is positive for all timelike vectors Va. Thus 6 will become infinite and A will be degenerate for some finite positive or negative value s0 of «. The map from to the surface « = s0 can have at most rank two; there will therefore be at least one vector field Zon X such that AZ = 0. The

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