Cremona transformations and the conundrum of dimensionality and signature of macro-spacetime

Abstract

The issue of dimensionality and signature of the observed universe is analysed. Neither of the two properties follows from first principles of physics, save for a remarkably fruitful Cantorian fractal spacetime approach pursued by El Naschie, Nottale and Ord. In the present paper, the author's theory of pencil-generated spacetime(s) is invoked to provide a clue. This theory identifies spatial coordinates with pencils of lines and the time dimension with a specific pencil of conics. Already its primitive form, where all pencils lie in one and the same projective plane, implies an intricate connection between the observed multiplicity of spatial coordinates and the (very) existence of the arrow of time. A qualitatively new insight into the matter is acquired, if these pencils are not constrained to be coplanar and are identified with the pencils of fundamental elements of a Cremona transformation in a projective space. The correct dimensionality of space (3) and time (1) is found to be uniquely tied to the so-called quadro-cubic Cremona transformations – the simplest non-trivial, non-symmetrical Cremona transformations in a projective space of three dimensions. Moreover, these transformations also uniquely specify the type of a pencil of fundamental conics, i.e. the global structure of the time dimension. Some physical and psychological implications of these findings are mentioned, and a relationship with the Cantorian model is briefly discussed.