Hyperbolic Superluminal Scator Algebra

Abstract

An alternate time-space framework is proposed by means of a hyperbolic scator algebraic formalism where the deformed Lorentz transformations are an immediate consequence of the formal properties of the algebra. We survey the group properties of the hyperbolic product in the restricted, super-restricted and zero magnitude subsets as well as in the complete scator set. Some properties of the magnitude are derived together with a decomposition of 1 + n dimensional scators in n products of scator elements with scalar component and only one non-vanishing director component. These results are used to describe the scator properties in 1 + n dimensions. The super-restricted subset is associated with superluminal frames or tachyons. The various compositions of subluminal and superluminal events in the same direction follow the canonical superluminal Lorentz transformations interpretation. Composition of orthogonal components produce novel results. Products of scators with components within and outside the light pyramid yield sub or super luminal scators as well as scators belonging to a mixed state.

Geometric and Differential Features of Scators as Induced by Fundamental Embedding

“…Since zero divisors are excluded from considerations, we have It was shown in [13] that known subluminal particles described by time-like scators undergo the so-called restricted space conditions, as well as superluminal particles described with space-like scators would undergo so-called super-restricted space conditions. Restricted space conditions generate a closed subset under multiplication, but scators obeying super-restricted space conditions do not constitute a closed set under multiplication [13].…”

On the Geometry of the Hyperbolic Scator Space in 1+2 Dimensions

“…By the way, the scator appearing on the right-hand side of (1.5) will be referred to as dual to o c (see Definition 2.1 below). Many properties of the scator product (1.2) were widely investigated in many contexts [1,8,9], also physical [4]. To gain some insight in possible physical interpretation of the scator algebra, we recall here some basic terminology from [1] The classification proposed above is analogous to what is well known from special relativity.…”

“…Note that in both cases componentwise addition of scators or vectors does not preserve this classification. However, the norm of the product of two scators is just the product of their norms [9], which is very useful in this context. In this paper we present new results concerning the metric structure of the scator space, extending results of [4,9].…”

“…A closely related notion, "super-restricted space conditions", was introduced in [9] without a direct relation to the type of considered events. Superrestricted space conditions define either time-like events (in even-dimensional spaces) or space-like events (in odd-dimensional spaces).…”

“…From (1.6) it follows that the inverse of a scator (with respect to the scator product) is given by 9) and light-like scators are not invertible.…”