Symmetries and exact solutions of Einstein's equations

Abstract

The theorems in the last section which show that curvature collineations are almost always conformal motions are somewhat disappointing since they mean that the program of using curvature collineations (and indeed therefore many symmetries other than conformal motions) in the search for exact solution of Einstein's equations is not a very fruitful one. The relatively few spacetimes which have non-trivial curvature collineations are ones in general with lots of other symmetries or properties which mean that they are well known or probably easily found solutions. It is still interesting to see why such spacetimes have non-trivial curvature collineations and a discussion on this point will be published elsewhere.

In the cosmological case it has been shown that Eardley's suggested examination of models with homothetic motions is not as useful as may be hoped because of the restrictions on such motions as outlined in II. Thus, again, the program of looking at solutions with symmetries other than Killing ones is not quite as fruitful as may be hoped.

On the other hand, many authors when writing about symmetries only seem to think in terms of isometries, and so it must be stressed that it is important to think in wider terms and examine the role of other symmetries. There are still many useful and interesting results using such symmetries waiting to be found!