Conformal Lorentz geometry revisited
Authors: Teleman Kostake 
Published: July 2002

The group U(2,2) and its subgroup SU(2,2) were considered by Penrose in his study of the conformal compactification ℳ of the Minkowski space M [R. Penrose and W. Rindler, Spinors and Space-Time (Cambridge University, Cambridge, 1986) and R. O. Wells, Jr., Bull. Am. Math. Soc. I, 2 (1979)]. The standard representation of SU(2,2) in C4 and in ℳ are the corner stones of twistor theory, which was created by Penrose to the double purpose of obtaining new solutions of Einstein equations and new insights on gravitational radiation. We think that other representations of SU(2,2) or U(2,2) could also bring some information in relativity [see also, Barut O. Asjim, in Noncompact Lie Groups and some of their Applications, edited by E. A. Tanner and R. Wilson (Kluwer Academic, Dordrecht, 1994), p. 103] and, accordingly, we propose an extension of Penrose twistor program.
In this paper we deal with a new U(2,2)-space, which we denote by W. We show first that the SU(2,2)-space ℳ introduced by Penrose is isomorphic to U(2), endowed with an action of SU(2,2) given by non-Abelian homographic transformations. These transformations keep invariant the equation det(u−v)=0, characterizing the pairs (u,v)∈U(2)×U(2) such that ‘‘u lies on the light-cone of v.’’ By definition, our space W consists of all pairs (u,v)∈U(2)×U(2) satisfying the condition det(u−v)≠0. The starting point of this article is the observation that W carries an SU(2,2)-invariant pseudo-Riemannian metric L:=Tr[(u−v)−1u̇ ×(u−v)−1v̇], of signature (4,4). (W,L) is in fact an irreducible symmetric space in Cartan’s sense, which is isomorphic to the quotient SO(2,4)/S[O(1,1)×O(1,3)]. As an irreducible symmetric space, it is an 8-dimensional Einstein space, whose Ricci tensor is proportional to the metric tensor.
We study the geodesic paths of this space giving the general solutions in terms of initial data and studying the constants of motion. In particular we determine the geodesic paths which exhibit two periods. We also show that Mach’s principle on inertial motions receives an explanation in our theory by considering the particular geodesic paths, for which one of the partners of an interacting pair is fixed and sent to infinity. In fact we study a dynamical system (W,L) which presents some formal and topological similarities with a system of two particles interacting gravitationally. (W,L) is the only conformally invariant relativistic two-point dynamical system. At the end we show that W can be naturally regarded as the base of a principal GL(2,C)-bundle which comes with a natural connection. We study this bundle from differential geometric point of view. Physical interpretations will be discussed in a future paper. This text is an improvement of a previous version, which was submitted under the title ‘‘Hypertwistor Geometry.’’ [See, K. Teleman, ‘‘Hypertwistor Geometry (abstract),’’ 14th International Conference on General Relativity and Gravitation, Florence, Italy, 1995.] The change of the title and many other improvements are due to the valuable comments of the referee, who also suggested the author to avoid hazardous interpretations.