UNIVERSAL REALITY

The Other Side of Pythagoras

We will again use the Lorentz transformation to render the projection of the relation of the two sets of vectors in equation [3] in the primed and unprimed part of the universal reference frame, starting with an unprimed section:

((δs - βδt)(1-β²)^-1/2)² - δs² = ((δt - βδs)(1-β²)^-1/2)² - δt² = β²(1 - β²)^-1(δs² - β^-12δsδt + δt²) [13]

We will note that the projection of the variables in equation [2] does not result in a hyperbolic function since equation [13] yields the sum of δs² and δt² and a negative cross-product. We will remember that while the Lorentz transformation coefficient β²(1 - β²)^-1 is a hyperbolic sine of angle θ, it is also the hyperbolic direction cosine of the complementary angle (π - θ). The hyperbolic tangent of the complementary angle is the inverse of β, i.e. β^-1. Thus, in the plane (defined by the right most bracketed expression in equation [13] β^-1 = cosσ_i, a cosine (again circular and not hyperbolic).

β²(1 - β²)-1(δs² - β^-12δsδt + δt²) ⇒ -csc²σ_i (δs² - 2δsδtcosσ_i + δt²) = -csc²σ_i δq² [14]

Once more we may recognise the ‘law of cosines’ in the term within the right most brackets in equation [14]. Vectors s_i and t_i are thus not necessarily orthogonal but form an angle σ_i that defines a third side ordinate vector (q_i in a circular relationship with t_i as abscissa and s_i as radius vector, i.e. q_i opposes angle σ_i and is orthogonal to t_i). Also note that that the hyperbolic sine, coefficient β²(1 - β²)^-1 is also a circular cosecant in equation [14] owing to the inversion of β. Our second imaginary plane is thus defined:

i√(δs² - 2δsδtcosσ_i + δt²) = + δiq, - δiq [15] ⇒

t_i + q_i = s_i [16] (projection)

We note that, while vector t_i has reemerged, vector s is no longer present in equation [16] after ‘convolution’ into i whence it becomes s_i in the plane defined with vector q_i which is its axis of rotation. However, since vector p derived in [9] is the axis of rotation of the real plane in which vector s lies, the former is the real component of a complex axial vector of which q_i is the imaginary component. It is important to differentiate the ‘convolution’ of a vector such as s into s_i and the ‘convolution’ of a component of a complex axial vector. (More so than the normal component vector s), it is radius vector t that is being ‘convolved’ following which operation it will be orthonormal regardless of the angle of vector t from the normal (i.e. with respect to component vector s). However, angle σ remains real after its projection as angle σ_i and will determine the angle that s_i will have with t_i. We had previously noted that after ‘convolution’ radius t becomes an abscissa t_i, while abscissa s becomes a radius s_i. Thus the ordinal component p> of t will again be an ordinate, q_i after ‘convolution’ and its modulus will remain unaffected (see hyperbolic-circular transformations) . Ironically because the vectors’ ‘convolution’ is not combined with a reflection, the sign of their imaginary counterparts will invert (equations [13] and [14] show their multiplication by (i)2). Thus, just like a real axial vector, a complex axial vector will as if rotate without change in orientation or modulus. In contrast, its components will be respectively real and imaginary with mutually opposite signs, as might be expected, since we now will recognise them as p and q_i defining their own hyperbolic vector space that intersects that of s and t_i in the complex axial vector which in both hyperbolic vector spaces is a hypotenuse (radius) and vector product

s ∧ p = r = t_i ∧ q_i

better known as the Einstein interval. The inequality

t_i ≠ p ∧ q_i

confirms that p and q_i lie in their own hyperbolic vector space where they are respectively spatial and temporal vectors (when observed in a third Lorentz frame).

We will recall that in Lorentz geometry the velocity of light is reached when δs_r and δt_r the increments associated with respectively (macroscopic) vector s and vector t_i have the same magnitude. We have also found that in the same geometry vector s of an elementary object will undergo Lorentz contraction, thus at the velocity of light we will expect instead that its vector s' and vector t_i will have the same magnitude.

However, while vector s' is a radial vector whose modulus is invariable, equation [15] shows that the modulus of t_i contracts to zero as it becomes orthogonal to s_i when cossi reaches zero. Consequently, not only do the roles of vectors s_i and t_i in [16] appear to be the converse of vectors s and t in equation [9] as may be expected from the inversion of β, but so would the relationship between vector s' and vector t_i seem to be the inverse from that between (macroscopic) s and t_i.

Accordingly, while vectors s_i and t_i are coincident when cosσ_i is at unity, as soon as these vectors are not of the same direction an increase of the included angle will decrease the modulus of t_i to less than that of s_i which remains constant as behooves a radius. Hence, when s_i > t_i < s' one might ask whether the relation between vector s' and vector t_i should be interpreted as super-luminal in a Lorenz frame? In the context of an elementary object, component vectors merely relate its radial vectors to Lorentz geometry; the concept of Lorentz contraction is thus a geometrical phenomenon. Geometrically luminal velocity is the coincidence of a Lorentz reference frame with the universal reference frame when the magnitude of vector t_i is equal to those of both s and s' entailing that it also equals the magnitude of vector s_i. In other wordss when δs_r/δt_r = s'/ t_i the included angle between temporal and spatial vectors in circular 2-space is zero and thus its cosine at unity; a cosine cannot exceed unity thus neither can the speed of light. However, this discourse demonstrates that time is in all respects identical to space and would be experienced thus, if our reference frame had a different orientation in which space had become time-like and time space-like, i.e. in effect space and time would trade places (compare equations [9] and [16]). Hence, the velocity of light is an artifact of the plane in which we project our reality and no more than a horizon beyond which macroscopic observation cannot reach. In a sense equation [16] provides a view toward the horizon from the opposite side.

Equation [16] also demonstrates that after transformation the stereometric information in equation [3] has unprimed components that are orthonormal with respect to any real plane of observation so that the apparent angle between s_i and t_i will be zero (i.e. cosσ_i will appear to be at unity). Hence, rather than appearing to be superluminal, the plane in which the object’s vectors project (t_i) or originate (s_i) will have the velocity of light with respect to the observer (in a Lorentz reference frame). However, another question arises: in what manner are the ‘lost’ superluminal components expressed in Lorentz geometry? We will note that the (macroscopic) differential velocity β between the primed and unprimed Lorentz reference frames is zero in stereometric observation the increment δs_r equals its primed counterpart δs'_r (i.e. equality of the corresponding vectors’ moduli- of course, the parallax angle remains: equation [3] conveys the vectors’ angular relationship in its scalar expression).

Velocities according to Relativity are not additive so that the differential speed between any two Lorentz reference frames of which at least one is at the speed of light with respect to the observer (in a third frame), will still be the speed of light. The case being examined thus presents an apparent enigma: how should we conceive the transition in differential speed, being that of light, between two frames which (with respect to the observer) are both at the speed of light but not coincident- no matter how nearly coincident- to a differential speed of zero between two coincident frames? (Simply in terms of Lorentz geometry: provided δt_r = δs_r ≠ 0 then δs_r/δt_r = c except at luminal coincidence when δs_r = 0 ≤ δt_r and thus δs_r/δt_r = 0). We note that such coincidence can only happen if the frames concern the same elementary object as is the case in stereometric observation when β = 0 and σ_i = π/2. A transition to luminal coincidence will introduce a discontinuity in Lorentz space-time: radius vector s_i having had the same direction and modulus as vector t_i (when cosσ_i = 1) switches to the orthogonal direction of vector q_i. The switch in direction causes the modulus of vector q_i to become equal to that of s_i, while reducing the modulus of t_i to zero. However, t_i will no longer contribute to macroscopic time, hence, the reference of s_i and q_i to the macroscopic clock will be lost. An elementary object’s vector t_i represents its radial vector s_i in Lorentz geometry, while vector q_i being orthogonal to t_i has no extension in the latter’s direction, i.e. as if without temporal increment and, hence, neither q_i or s_i will be perceived temporally in hyperbolic Lorentz space-time (considering only their association with the relativistic line-of sight).

Vector t_i unlike vector t'_i is not a radius thus cannot freely rotate; instead its direction will align with that of the observer’s macroscopic clock thus providing a reference for its angle σ_i with s_i. However, invariant radius vector s_i is also orthonormal while it was established in connection with equation [9] that vector t can freely rotate, meaning angle σ and thus angle σ_i cannot be known. Hence, neither the direction of radius vectors s_i and t nor the modulus of vector t_i can be determined (the latter will be equal to or smaller than the modulus of s_i). The other component vector q_i conveys, in a sense, the ‘excess’ of s_i of over t_i, i.e. the part ‘lost to superluminosity’. Vector q_i in turn is a component vector of complex axial vector r, whose modulus is indeterminate. Hence, the latter in turn will modulate the modulus of real vector t whose component vectors s and p will be apparent (the latter normal to the line-of-sight for any σ_i ≠ 0) as a real two-space quantum black-hole whose boundaries from within are defined by vectors t_i and q_i; σ_i = 0 representing the luminal- black (Schwartzschild) boundary of a one-space infinitesimal in the direction of macroscopic s. A real object at the Schwartzschild radius of a black-hole in effect becomes imaginary, thus; conversely, an imaginary object that would emerge from the ‘other side of the Schwartzschild radius’ can be expected to become real as demonstrated above through the coupling between radius vectors s_i and t. (An elementary object without mass is not subject to macroscopic constraints on acceleration to the velocity of light. Nor would a quantum black-hole exhibit the ‘insatiable’ characteristics of a macroscopic black-hole since the associated requisite space-time curvature would be disturbed by a second elementary object: two elementary objects cannot occupy the same space-time in principle. However, an elementary black-hole does offer a geometrical account for particle interaction).

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