Twistors and General Relativity

The basic motivations underlying twistor theory are aimed at finding an appropriate union between the principles of quantum mechanics and the space-time geometry notions of relativity physics. As twistor theory has developed, however, it has found many more applications in pure mathematics than in the areas of physics that directly relate to these initial basic aspirations of the theory. The main areas of pure-mathematical application have been differential geometry (e.g. construction of anti-self-dual 4-manifolds [2], of hypercomplex manifolds [19], Zoll manifolds [28], conformal geometry, classification of holonomy structures [30], representation theory [3], and integrable systems [29]). In this short article, however, space does not allow us to enter into any kind of serious discussion of this pure-mathematical work. We concentrate, instead, on some of the basic twistor-theoretic constructions that directly address the initial physical aims of the theory, in addition to underlying many of the subsequent pure mathematical developments. Most specifically, we are concerned with the programme of bringing Einstein’s general theory of relativity within the scope of the twistor formalism, as an intended prerequisite to finding the appropriate union of that theory with the principles of quantum mechanics.