By Ehlers J. (ed.)
ISBN-10: 1751781801
ISBN-13: 9781751781806
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Additional resources for Relativity theory and astrophysics, vol.2: galactic structure
Sample text
7) cannot reverse the directions of the parameters u and v. Thus, even in region IV with M non-zero, f and g must be monotonically decreasing functions. 6) which apply globally. It is also possible to use further transformations to put c1 = 1 and c2 = 1. However, these particular constants describe a measure of the magnitudes of the approaching waves, and it is therefore often convenient to retain them without this further rescaling. For situations in which M = 0 in the inital regions I, II and III, the product c1 c2 has an invariant meaning as the amount of non-linearity in the interaction.
It thus excludes situations involving impulsive gravitational waves. It does not apply therefore to the solutions of Khan and Penrose (1971), Ferrari and Iba˜ nez (1987b), and the exceptional solution of Chandrasekhar and Xanthopoulos (1986b). Nor does it apply to the colliding electromagnetic wave solution of Bell and Szekeres (1974) in which impulsive gravitational waves are generated by the collision. In addition, it proves only geodesic incompleteness and says nothing about curvature singularities.
One of the more useful ways of attaching a boundary to a singular space-time is by a b(bundle)-boundary construction. The b-boundary 1 This brief review follows closely that of Konkowski and Helliwell, 1989. 1 Singularities 49 is the projection into a space-time of a natural boundary attached to a higher-dimensional Riemannian manifold. In the standard b-boundary construction, the Riemannian manifold is the bundle of frames over spacetime having a positive definite metric induced by the affine connection.