On an Observer-Related Unequivalence Between Spatial Dimensions of a Generic Cremonian Universe

Given a generic Cremonian space-time, its three spatial dimensions are shown to exhibit an intriguing, “two-plus-one” partition with respect to standard observers. Such observers are found to form three distinct, disjoint groups based on which one out of the three dimensions stands away from the other two. These two subject-related properties have, to our knowledge, no analogue in any of the existing physical theories of space-time. When confronting a new theory, attention is always paid to those features that make the theory both unrivalled and subject to unambiguous falsifiability. The theory of Cremonian space-time(s) [1–4] can be no exception in this respect. As for the first aspect, this theory has already been able to shed a remarkably fresh light on such pressing issues of contemporary physics as the macroscopic dimensionality and signature of the Universe [1,2,4], its possible origin and/or evolution [5,7], as well as on a puzzling discrepancy between the physical and psychological/mental concepts of time [1,4,6]. The second aspect, its falsifiability, has so far been mentioned in passing only [8] and asks, therefore, for a closer inspection. At the current stage of its development, there are very few testable predictions of the theory going beyond the above-mentioned three domains. Yet, one of them, although being of a rather subtle nature, stands out as truly fascinating and enormously challenging, for, among other things, it seems to undermine the status of two currently most favoured paradigms of natural sciences, viz. reductionism and third-person perspective. The feature concerned is an unequal footing on which three “Cremonian” space dimensions stand with respect to the observer/subject. Rephrased in a more explicit way, introducing an observer into our Cremonian space-time breaks the original symmetry by inducing/generating a delicate, 2+1 “splitting-up” in the status of its three spatial dimensions. Mathematically, this fascinating property is intimately connected with the fact that the intrinsic geometry of a proper conic is identical with that of a projective line and a projectively invariant property of four distinct points of a projective line known as separation [see, e.g., Refs. 9,10]. To begin with, we shall recall that a generic Cremonian space-time [1,4] is an algebraic geometrical configuration that sits in a real three-dimensional projective space and comprises three pencils of lines (spatial dimensions) and a single pencil of conics (time). The pencils of lines, L̃α (α = 1, 2, 3), are located separately in three distinct planes sharing a line, L̂, and their respective vertices, B̂α, are assumed not to be collinear, none of them being incident with the line L̂. The pencil of conics, Q̃, is situated in the plane defined by B̂α and its base points are the three vertices and the point, L̂, at which the line L̂ meets the plane in question. In a suitably chosen system of homogeneous coordinates, z̆i (i = 1, 2, 3, 4), this configuration can analytically be described as follows: L̃1(θ1) : z̆2 = 0 = z̆3 + θ1z̆4, (1) L̃2(θ2) : z̆1 = 0 = z̆3 + θ2z̆4, (2) L̃3(θ3) : z̆1 − z̆2 = 0 = 2z̆3 − z̆2 − z̆1 + θ3z̆4, (3)