UNIVERSAL REALITY

Imagine there is no...

We have noted that in our stationary Lorentz reference frame we would observe an object's length contract and an associated marked time period dilate at speeds successively closer to that of light. Yet an observer who has remained with the object sees its length and the time period unchanged but notes that in our receding reference frame the measuring stick that we are still holding as if it were along the length of the object, is progressively shrinking (in a sense as if it had been upright but is being tilted away to ultimately 90º when it would disappear in the line-of-sight). Obviously neither observer could consider the object's spatial extension measured by the other to be in the same three spatial dimensions of its respective Lorentz reference frame.

In Lorentz geometry the relation between the space and time increments in equation [2] is given a geometric interpretation since the variables compute as if they were perpendicular even though this is inconsistent with the mathematical classification of time as a scalar (scalars unlike vectors have no direction but do possess a sign and magnitude). Nevertheless, the reason for computational agreement with experimental observations can be explained and the anomaly resolved once we understand that both observers in their respective Lorentz reference frame perceive a projection being the real component (in the line-of-sight and direction of motion) of the receding object's length as it enters an imaginary dimension). At the speed of light the object's imaginary spatial component (mathematically and in the literal sense imaginary) will appear to have the same direction as time for reasons explained herein. Considering that its space increment vanishes as the temporal increment waxes and is all that remains of the object, it would seem we must accept that space and time are identical in nature as will be demonstrated herein (i.e. that the cause and effect of the difference in their experience stems from their observation in the context of our macroscopic aggregate surroundings). Furthermore, since space has three real dimensions we must conclude that each has a temporal counterpart, i.e. that time also has three-dimensions, which together with space form a symmetrical continuum. This conclusion in a way signals the end of time: it is neither the arrow of life's progression nor the mathematical scalar whose macroscopic manifestation is continual accrual while governing elementary interaction with positive increments (current interpretations of Quantum Mechanics and Relativity do not rule out negative time increments). Instead time is a vector (a composite vector in macroscopic time) in (mathematically) imaginary dimensions in a three-space, i.e. time has three dimensions: our reality comprises the real and imaginary dimensions of an extended vector space.

Actually this revelation should not startle us. An indication of the vectorial character of time is embedded in the relation for the Einstein interval: the variables in equation [2] must add up to the 'zero vector' (an event does not come to naught as a result of observation from separate Lorentz reference frames: both sides of equation [2] equate to the invariant interval vector; the zero vector results from subtracting one side of equation [2] from the other).

Accordingly, if we try (unsuccessfully) to recast equation [2] as a vector relationship we will find:

t.t - s - t'.t' + s' = 0 ⇒ s' - s =? ≠ t'.t' - t.t [2A]

is inconsistent since the difference between the squared temporal scalars on the right is another squared scalar that can no more than numerically equate to but never become the composite vector being the difference of the two spatial vectors to its left (in a sense the spatial vectors in equation [2] are treated as scalars to obscure their inadmissible multiplication with scalar time as a consequence of transformation). Modified vector equation [3] below removes the discordance:

s' - s = t'_i - t_i [3]

In equation [3] we note that, apart from the replacement of the temporal scalars by vectors, on the left side s' has traded places with δt² in equation [2] and on the right side of equation [3] that t_i has replaced s' in equation [2]. The distinction between signs of the variables and relations in equation [2], which can now be corrected as:

t_i - s = t'_i - s' [4],

on the one hand, and the vector signs in equation [3] on the other hand, in the context of its implication on the interpretation of [3] and subsequent equations, and derivative relations is discussed further below. First we will discuss the meaning of the other form in which equation [2] can be recast as a vector equation:

s' + t_i = s + t'_i [5]

It has been asserted above and shown in the discourses on class and vector theory (see above links to subtexts on these theories) and in the next sections that vectors t_i and t'_i in equation [4], [3] and [5] are imaginary, thus that the subscript i is correct. It will also become clear that if the other spatial coordinates (normal to s) that do not undergo relativistic effects (on account of also being normal to the direction of motion) would be added to the unprimed relation in equation [4], i.e. still ignoring the dimensionality of time, this does not turn it into a definition of relationships in 4-space. (The velocity factor of vectors that are normal to the direction of motion is zero thus the corresponding primed and unprimed vectors are equal and would cancel each other in equation [4]). Equation [4] defines hyperbolic relations each in its own complex vector space (explicitly, now that we recognise δt as the modulus of a temporal vector). Hence, the Einstein interval is a complex composite vector whose space-time component vectors are real and imaginary. In effect space-time is not defined in a 4-dimensional continuum but is an extended (complex) 3-space that includes coordinates that are imaginary in addition to those that are real (and positive and negative). We will find that the two Lorentz 2-spaces of equation [4] bisect in a (complex) sense the real and imaginary components of the extended 3-space. A 3-space can be constructed with two planes, each of which is rotated through an angle of 2π of a vector that lies in the other plane such that the axis of rotation is perpendicular to the other plane (rotation being a consequence of the observer's orientation which in the case of an elementary object may simultaneously change in any direction in the absence of constraints imposed by aggregation and Lorentz geometry- also see the sections entitled 'The Observer's Dual Modality' and 'Newton's Apple Restored' further below). Hence, our real and imaginary extended 3-space will require a definition of two such sets of planes. Assisted by perpendicular symmetry between the dimensions both aspects of this extended 3-space can be reconstructed by the simple act of projection, i.e. rotation (reflection) in either real or in imaginary vector space and what we will label 'convolution' ('rotation' would not be appropriate) from real to imaginary vector space and vice versa

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