Cremonian Space-Time(s) as an Emergent Phenomenon

It is shown that the notion of fundamental elements can be extended to any, i.e. not necessarily homaloidal, web of rational surfaces in a three-dimensional projective space. A Cremonian space-time can then be viewed as an emergent phenomenon when the condition of “homaloidity” of the corresponding web is satisfied. The point is illustrated by a couple of particular types of “almost-homaloidal” webs of quadratic surfaces. In the first case, the quadrics have a line and two distinct points in common and the corresponding pseudo-Cremonian manifold is endowed with just two spatial dimensions. In the second case, the quadrics share six distinct points, no three of them collinear, that lie in quadruples in three different planes, and the corresponding pseudo-Cremonian configuration features three time dimensions. In both the cases, the limiting process of the emergence of generic Cremonian space-times is explicitly demonstrated. Although the theory of Cremonian space-time, first introduced in [1], is relatively new, it has already proven to be remarkably fertile and attracted attention of both the lay public [2] and specialists [3] alike. The concept not only offers us a feasible explantion for why our Universe is, at the macroscopic scale, endowed with three spatial and one time dimensions [1,4–7], but also indicates unsuspected intricacies of a coupling between the two [6]. Moreover, it gives us important qualitative hints as for a possible underlying algebraic geometrical structure of a large variety of non-ordinary forms of psychological time (and space) [6,8]. These fascinating properties alone are enough to realize that the theory deserves further serious exploration. One of the most natural and fruitful ways of getting a deeper insight into a(ny) theory is to relax one (or several) assumptions that the theory is based on and see what structural and conceptual changes such a step involves. In order to pursue this strategy in our case one has to recall the underlying geometrical principle behind our concept of Cremonian space-time: the existence of a homaloidal web of (quadric) surfaces in a three-dimensonial projective space, P3 [1,4,5]. An aggregate of surfaces (of any order, not necessarily quadratic) in P3 is homaloidal [1,9]: if a) it is linear and of freedom three (i.e. contains a triple infinity of surfaces), b) all its surfaces are rational, and c) any three distinct members of the set have only one free (variable) intersection. The generalized theory outlined in what follows is based on abandoning the last assumption, i.e. on allowing any three distinct surfaces of the web to have two (or more) points in common. Of the couple of different types of homaloidal web of quadrics we have so far dealt with we shall first consider that whose corresponding Cremonian space-time was found to mimic best the basic observed macroscopic properties of the Universe, viz. a web whose base configuration comprises a (straight-)line and three distinct, non-collinear points, none incident with the line in question [1,6]. So, in this case, any web of quadrics that features a base line and where the number of isolated base points falls short of three may serve our purpose. We shall, naturally, focus on the case where there are just two base points as this case represents, obviously, the smallest possible deviation from the “homaloidity” condition. In order to facilitate our reasoning, we shall choose a system of homogeneous coordinates z̆i, i = 1, 2, 3, 4, in which the equation of base line, L̂, reads L̂ : z̆1 = 0 = z̆2 (1) and the two isolated base points, B̂1 and B̂2, coincide, respectively, with the vertices V1 and V2 of the coordinate tetrahedron, i.e. B̂1 : ̺z̆i = (1, 0, 0, 0), (2) B̂2 : ̺z̆i = (0, 1, 0, 0), (3)