UNIVERSAL REALITY

Sometimes a Little Knowledge

We have noted (in [1]) that, when space and time increments are measured, neither the object nor its observer should be accelerated- an impossible feat for aggregate entities whose macroscopic measurements are thereby confined to thought experiments. However, in the microscopic world elementary objects that are either at rest or at the speed of light (i.e. they are without mass), hence, (in order to attain either state) do not invoke the restriction [1] to which aggregate entities are inherently subject.

Spatial and temporal variables associated with aggregate entities that make up our macroscopic world are measured in the space-time of the observer's Lorentz reference frame. As a matter of course, the same approach is applied to measuring these same variables for elementary objects when they are observed individually. However, while taken for granted, it is not at all evident that the spatial extension of elementary objects is 'measured' in the observer's reference frame. In fact spatial information regarding an elementary object is conveyed to the observer's reference frame, e.g. in the form of electromagnetic radiation and thus obtained in unaltered form in the object's reference frame instead. In contrast, the temporal component is measured in the observer's stationary reference frame (by default) by clock(s) that are not normally related to the object. Consequently, not equation [2, 3 or 4] but equation [5] is pertinent to the observation of elementary entities. The primes are reversed in the right hand side of equation [5], which thus provides corresponding vantage conditions from the observed object.

Although it would seem that strictly the primed spatial variable should be changed to being unprimed and vice versa, such a switch has not been made herein. The reason is that we will demonstrate that while we have advanced that the conventional assignment of primes may be inappropriate when Lorentz geometry is applied to elementary objects, fortuitously it turns out to befit the geometry described in this discourse. Essentially, the primed and unprimed vectors are integral to a single frame, the universal reference frame, which has been rent into hyperbolic Lorentz geometry by our inability to observe the individual properties of objects that conform to the geometry of the ('in-situ' universal) space-time continuum (to which aggregate entities including the corporal embodiment of the observer are unable to conform). Moreover, a switch of primes would further complicate the text when alternating between Lorentz reference frames and the geometry of the equations presented herein. Accordingly, the text adheres to the conventional assignment of unprimed variables as originating in the observer's stationary Lorentz reference frame and of primed variables in a Lorentz reference frame with velocity β with respect to the former. In effect the locus of the unprimed vectors in hyperbolic Lorentz geometry spans both the observer and object, as does that of the primed vectors. The approach herein takes space-time data that defines two sets of complex two-space in a Lorentz continuum (of which both the primed and unprimed reference frames are subsets that intersect in the Einstein interval) and transforms their coordinates such that the real and imaginary parts are separately rendered in the universal reference frame in which both the observer and the object reside.

In order to project the 2-space defined by each of the two sets of vectors in equation [5] the Lorentz transformation will be applied to their scalar expression. We will encounter instances when projecting a vector that a radial vector emerges indicating that we are actually performing a reverse operation, i.e. we had started out with the projected component. Our first projection will be onto the unprimed part of the universal reference frame (either (δs')² or (δt')² may be transformed):

((δs - βδt)(1-β²)^-1/2)² + δt² = ((δt - βδs)(1-β²)^-1/2)² + δs² = (1 - β²)^-1(δs² - β2δsδt + δt²) [6]

where the velocity β between the primed and unprimed Lorentz frames is δs_r/δt_r = β, the hyperbolic tangent of angle θ (tanhθ); θ defines the angle between the temporal increment and the Einstein interval and is also called the velocity parameter.

Equation [6] shows that the projection is not a hyperbolic relationship since the squares of δs and δt are summed and their cross-product has a negative sign. The change in perspective brought about by the transformation of the hyperbolic relationships in equation [2] is indicative that trigonometric relationships of vectors whether they be real or imaginary are not necessarily preserved in a projection.

In fact a projection from an imaginary plane onto a real plane and vice versa will result in a loss of angular correlation: any vector on an imaginary plane regardless of its direction will appear as orthonormal with respect to real 3-space (the vector's magnitude cannot be determined since only its projected orthonormal component is perceived) and conversely a vector whose projection is orthonormal may have any direction in imaginary 3-space.

The Lorentz transformation coefficients (1 - β²)^-1 and β² (1 - β²)^-1 are said to be independent of “s and t”. The vectorial quality of vectors t_i and t'_i and acknowledgement of past failure to apply Relativity to elementary objects forces us to restate this notion as the coefficients’ independence from the values for the moduli of vectors s and t_i and their primed counterparts when the Lorentz reference frames are conceived to be independent of the object. The independence of the coefficients only applies to objects (whose gravity may be discounted) in Lorentz space-time because it is moot with respect to the attributes of elementary objects: the object subtends and is integral to the universal reference frame. In other words δs_r and δt_r associated with an elementary object are the same measurement as respectively δs and δt (of the object, not the macroscopic measurements of the Lorentz space-time in which it is situated). Implicitly, the two sets of measurements are separated by prior acceleration, the synonym for aggregation of objects, whose resultant velocity thus is necessarily sub-luminal (accordingly, luminal ‘velocity’ is not the result of acceleration but will be shown to be a geometric property manifested in the universal reference frame). Velocity between Lorentz frames is thus but a macroscopic measurement of a hyperbolic trigonometric function whose components are obtained in one Lorentz reference frame, as opposed to temporal and spatial components of an elementary object which originate in separate sections (each a Lorentz reference frame) of the universal reference frame

Another implication of the vectorial quality of time is that the transformation coefficients have a geometrical meaning: they do not merely have the value of respectively the hyperbolic Cosine and hyperbolic Sine but are the respective hyperbolic direction cosines of θ and its complementary angle (π - θ) that yield collinear projections on the unprimed frame of the spatial and temporal variables in the primed frame so that their associated vectors may be summed such as to yield either the spatial or temporal variable in the unprimed frame, and vice versa. Furthermore, these geometrical relations confirm that the Einstein interval is a hypotenuse, i.e. a (hyperbolic) radius vector in the vector relations of equation [4]. Accordingly β does not just have the value of a hyperbolic tangent (δs_r/δt_r = β, the imaginary ‘i’ has been omitted- here from δt_r, however, note that both δs_r and δt_r and their primed counterparts are macroscopic measurements in Lorentz geometry) but is one by association of the orthonormal ‘direction’ of vector s with regard to imaginary vector t_i. Therefore, δs_r and δt_r like the latter vectors are orthonormal in the surface defined by the unprimed set of vectors in equation [4]. However, the orthonormal relationship in equation [4] changes as a result of rotation by θ to the plane defined by the right most bracketed expression in equation [6], ‘convolution’ out of i of vector t_i therein (we will find that the latter is actually the imaginary projection of a real radial vector: convolution of imaginary vectors into a real plane and vice versa is further described in the paragraph that follows equation [16]) and its corresponding effect on δt_r (which also becomes real) and the latter’s relationship with δs_r, which is now a cosine (circular as both increments are now real and thus the function is not hyperbolic): β = cosσ.

(1 - β²)^-1(δs² - β2δsδt + δt²)⇒ csc²σ(δs² - 2δsδtcosσ + δt²) = csc²σδp² [7]

The term within the right most brackets in equation [7] will be recognised as having the form of the ‘law of cosines’. Vectors s and t (see [9] below) are thus not necessarily normal but form an angle σ that opposes a third side vector p. Because cosσ is at unity when β = 1 vector p is the ordinate thus normal and in a circular relationship with s as abscissa. Hence, vector p is also the axis of rotation for the reflection of vector s. However, another axial vector of the same apparent orientation as p (i.e. orthonormal to s) has brought about the ‘convolution’ of t_i out of i to t. Since t_i is imaginary, the axial vector that brought about the rotation of the plane in which it lies (this vector will be derived in equation [16]) must be the imaginary component of a complex axial vector of which vector p is the real component. Vector t as radius vector is invariant and forms a closed triangle with s and p (when the direction cosσ for t approaches unity, the modulus of p will contract toward zero until vectors s and t will have the same moduli and direction). Our first real plane is thus defined:

√(δs² - 2δsδtcosσ + δt²) = + δp₁ - δp [8]⇒

We will remember that an observer’s spatial measurement of an elementary object is vector s' and not s. In contrast, in the observer’s Lorentz reference frame vector s is only associated with an object and (its relation to) our macroscopic environment that in turn is the aggregate of each vector s' of its constituent elementary objects (the vectorial relationships of vector s' will be derived in equation [19]). In equation [9] the observer’s vantage point has as if turned on itself (or is as if one with the object) such that a point previously at infinite distance in the line-of-sight on the focal centre line of a Lorentz geometry hyperbola (i.e. the direction of vectors s' aggregated to macroscopic vector s), will appear axially as the centre of a circle with vector t as radius and rotation angle σ. Accordingly, vector t does not have a fixed angular relationship with vector s, in contrast t_i is orthonormal to s. However, the latter has returned with its sign changed, while vector t_i that aligns with our macroscopic time is not apparent. (Note that vector t in equation [9] is real, hence, vector p should not be confused with the Einstein interval. However, it is plain why equation [9] will yield the Einstein interval in macroscopic measurements when the value obtained for the temporal increment should erroneously be entered for t - the reason for this correspondence will become clear in the discussion following equation [16]).

We have noted that vector s' of an elementary can be directly measured in the observer’s Lorentz reference frame as opposed to a macroscopic object whose macroscopic vector s' can instead be calculated only if the differential velocity between the primed and unprimed frames is known. The modulus of vector s of the object will be zero when cosσ = 0 = β (circular), its contraction a consequence of being a component vector. Yet, in the observer’s Lorentz reference frame the object’s vector s will aggregate with the corresponding vectors of other objects to macroscopic vector s' and lie in the latter’s direction. Since the modulus of macroscopic vector s will be equal (δs_r/δt_r = 0) or smaller (0 < δs_r/δt_r ≤ 1) than its primed counterpart, both the macroscopic and object’s vector s will appear Lorentz contracted. Hence, the object will appear to be at the speed of light. Since the object’s macroscopic vector s is associated with its position in Lorentz geometry, a zero value for its increment δs_r means that the object may remain at a constant distance even as its local space is contracted to zero with respect to the observer. Consequently, an elementary object is spatially infinitesimal in Lorentz geometry and at macroscopic zero velocity (while at rest in its own universal reference frame). The condition when cosσ ≠ 0 will be discussed following equation [16].

Vector t is a radius that like p is not apparent in Lorentz space-time (in the direction of macroscopic vector s), but, nevertheless, may freely rotate as its angular relationship to macroscopic vector s is unknown, the latter’s direction being determined by the aggregate environment of which the observer is part. We will return to the significance of t the newly emerged real temporal vector, its freedom of rotation and its effect on vectors s and p as part of the discussion concerning equation [16] and again after having derived the expressions for the other three planes.

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