22 - A final miscellany

Summary

The authors are well aware that this book has only described a very limited number of the known exact solutions of Einstein's equations. We have concentrated on those we believe to be the most basic or that have particularly significant interpretations. However, many other interesting solutions have been analysed and have their own extensive published literatures. Indeed, a number of these are highly relevant – either because they represent some easily comprehended idealised physical situation, or because they demonstrate some important property. In this concluding chapter, we will therefore comment on a number of further space-times that fall into either of these categories.

After commenting on some aspects of the Bianchi and Kantowski—Sachs cosmological models, that have (at least) three spatial isometries corresponding to homogeneity, we will briefly review the main properties of some other space-times that possess two commuting Killing vectors which have not been described in previous chapters. In general, such space-times include those with cylindrical symmetry, stationary and axial symmetry, boost-rotation or boost-translation symmetry, as well as the interaction region of colliding plane waves, and the so-called Gowdy (or vacuum G₂) cosmologies. Remarkably, for space-times of all these types, which possess 2-surfaces that are orthogonal to the group orbits, the resulting vacuum or electrovacuum field equations happen to be integrable. In fact, the field equations in all these cases can be written as a combination of the Ernst equations (Ernst, 1968a,b) in their elliptic or hyperbolic forms and subsidiary quadratures forthe remaining metric function.