Germ of a synthesis: space–time is spinorial, extra dimensions are time-like

Abstract

A pressing issue for modern physics is the possibility of extra dimensions of space–time. Here, a novel approach to this question is put forward, with three facets: First, an integral transform is introduced into Einstein's general relativity that is non-local and spinorial. For Minkowskian space–time, the transform intertwines three spaces of six dimensions, which a priori are on an equal footing, linked by the octavic triality of Cartan. Two of these spaces are interpreted as null twistor spaces; the third may be regarded as giving space–time two extra time-like dimensions, for which the ordinary space–time is an axis of symmetry. Second, it is suggested that the extra dimensions perdure for a general space–time: the overall structure is controlled by a generalized Fefferman tensor. Accordingly, it is posited that the additional time-like dimensions arise naturally and constitute an aspect of space–time reality that ultimately will be amenable to experimental investigation. Conceivably, devices such as the Large Hadron Collider will uncover this reality. Third, it is argued that the structure hints at a synthesis of ideas deriving from general relativity, string theory, condensed matter physics, category theory and non-commutative geometry.