On Cremonian Dimensions Qualitatively Different from Time and Space

We examine a particular kind of six-dimensional Cremonian universe featuring one dimension of space, three dimensions of time and other two dimensions that cannot be ranked as either time or space. One of these two, generated by a one-parametric aggregate of (straight-)lines lying on a quadratic cone, is more similar to the spatial dimension. The other, represented by a singly-parametrical set of singular space quartic curves situated on a proper ruled quadric surface, bears more resemblance to time. Yet, the two dimensions differ profoundly from both time and space because, although being macroscopic, they are not accessible to (detectable by) every Cremonian observer. This toy-model thus demonstrates that there might exist extra-dimensions that need not necessarily be compactified to remain unobservable. There are a number of features of the macroscopic physical world that still remain substantially beyond grasp of theoretical physics. Among them, the non-trivial structure of time and the observed dimensionality of the universe obviously represent a case in question. As we found [1,2] and have repeatedly stressed [3–5], the two properties seem to be intimately intertwined and ask, therefore, for a conceptually new approach to be properly understood. A (very promising) piece of such a formalism is undoubtedly the concept/theory of Cremonian space-times [6–14]. The Cremonian picture of space-time is indeed remarkable in several aspects. The first, and perhaps most notable, fact is that without employing any concept of metric (measure), this approach fundamentally distinguishes the time dimension(s) from spatial ones and, in its most trivial form, it straightforwardly leads to their observed number (4) and respective ratio (1+3) as well [6,7,9,10]. Second, it demonstrates that these dimensions are not primordial, but emerge from more fundamental algebraic geometrical structures [13]. Third, it indicates that the universe with the inverse signature might evolutionary be intimately connected with our universe [12]. And last, but not least, when the observer (subject) is concerned, it qualitatively reproduces our ordinary perception of time as well as a whole variety of altered/non-ordinary forms of mental space-times [10,15]; moreover, every observer in this basic Cremonian universe is found to face an intricate 2+1 break-up among the space dimensions themselves [14]. In this paper, we introduce and examine a particular kind of a more complex, six dimensional Cremonian universe whose spatio-temporal sector is still four dimensional, yet featuring three dimensions of time and just a single spatial coordinate. The character of other two dimensions is neither that of space nor time; in addition, these dimensions are only conditionally observable/accessible. This Cremonian universe sits in the 3-dimensional projective space over the fields of the real numbers R and is generated by the configuration of fundamental elements of a homaloidal web of cubic (i.e., third-order) surfaces that share a proper conic, Q̂, a (straight-)line, L̂, incident with the conic and not lying in its plane, and three different non-collinear points, B̂i (i=1,2,3), none of them incident with either L̂ or the plane of Q̂. The cubics of the web have also two double points, D1 and D2, in common; both the points lie on L̂, the former being the intersection of L̂ and Q̂ [16,17]. Selecting an allowable system of homogeneous coordinates z̆α (α=1,2,3,4) in such a way that L̂ : z̆1 = 0 = z̆2, (1) Q̂ : z̆4 = 0 = −2z̆1z̆2 + z̆1z̆3 + z̆2z̆3 ≡ C, (2) B̂1 : ̺z̆α = (0, a, b, c), a, c 6= 0, (3) B̂2 : ̺z̆α = (f, 0, g, h), f, h 6= 0, (4) B̂3 : ̺z̆α = (k, k, l,m), k, l,m 6= 0, (5)