To fully reconcile the application of the concepts presented in this paper with the Special Theory of Relativity, it is necessary to demonstrate that the Domain D₀ is equivalent to the space - time in which that theory applies, Pseudo - Euclidean Space - Time. It is therefore necessary to show that D₀ possesses the following additional characteristics to those already defined.

(i) The Temporal Co-ordinate X_o is related to the time t in D₀ by the expression, (after Minkowski ),

(ii) The magnitude of the maximum theoretically attainable spatial velocity in D₀, is equal to the Spatial Terminal Velocity, c.

(iii) When the spatial velocity of a moving point in D₀ is rectilinear and constant, measurements of time and distance related to axes associated with it, transform from those of D₀ according to the Lorentz Transformations of the Special Theory.

The Temporal Co-Ordinate X₀

In (2.7), if v is zero, i.e. a point is spatially at rest in D₀ , its Existence Velocity is reduced to:

which upon integration with respect to t gives:

where r_o, the constant of integration, is the constant spatial position in D₀ from some stationary reference. In this case the trajectory of motion is clearly, from (A.2), along the temporal axis X_o so that, in (A.3), the relationship of (A.1) is implicit.

The Maximum Theoretically Attainable Spatial Velocity in D_0.

Inserting (2.8) and (2.9) into (2.7) gives for Existence Velocity in spatial - temporal polar form

where s is a unit vector in the direction of v. Clearly the maximum theoretically attainable spatial velocity occurs when q = p/2, the Existence Velocity becoming simply:

At any other value of q, the spatial component of V must be less than c.

Transformation of The Axes

This is the more complex of the three criteria to prove. It is accomplished by derivation from first principles, of the relationship between the spatial and temporal axes of a point in D₀ moving with constant velocity, and those of D₀ itself. The temporal axis relationship is derived in the form of a time parameter in order to fully demonstrate agreement with the Lorentz Transformations. During this process a relationship for the proper time of the point is also derived.

Let B be a point in D₀ moving with a constant spatial rectilinear velocity v. Let R' be a space like" co-ordinate associated with B, let Q be any fixed point on R', and let t¢_q be the time along a "temporal" axis X'_q, associated with R' at the location of Q. Finally, let initial conditions be such that at some instant in D₀ designated t = 0, the position of B in D₀ is defined as a reference point, (Fig. A1 may be usefully referred to in following this derivation). Consider first the spatial axis R'. At time t the positions of B and Q in D₀ will be given by :

Differentiating (A.8) with respect to t

But with

and

and taking the magnitude of (A10)

but for Q to exist in D₀

so that in (A.13)

but r¢_q is constant, therefore dr'_q /dt must be zero. This gives in (A.15)

As v is constant (A.16) can be integrated immediately to give:

Substitution of (A.17) into (A.8) then yields:

Now as Q is any point on the R' axis, it can be co-incident with B to give r¢_q = 0. In this case (A.18) would reduce to:

and

The magnitude of (A.21) yielding:

As Q is any point on R', then (A.21) and (A.22) represent the relationship between the spatial axis associated with B and that of D₀, and (A.20) represents a measure of the proper time of Q in D₀ . The parameters r'_q and r_q may therefore, in (A.21) and (A.22), be replaced with the axis designators R', and R respectively,and the lengths r¢_q. and r_q therefore represent the relationship between their scales. Subsequent reference to (A.21) shows that R' possesses both spatial and temporal components and therefore a precise orientation in D_o . Substitution of (A.22), (2.8) and (2.9) into (A.18) gives:

Where s is a unit vector in the direction of v. This shows by comparison with (A.4) that R' is orthogonal to the Existence Velocity Vector and, therefore, the spatial - temporal trajectory of B in D₀ . Also, from (A.22) it is clear that units of length along R' (i.e. with t constant ) are greater than units of length along R, and that the increase is a direct result of the orientation of R' relative to R in D₀.

Now consider the temporal co-ordinate associated with R' at the location of Q. Firstly it is noted that since Q is fixed in relation to B, its only motion in axes associated with B is a temporal one. Therefore the primed temporal axis along which Q is in motion, X'_q, must lie along its trajectory in D₀ which must be parallel to that of B. As a consequence, this axis must be orthogonal to R'. Now (A.20), as stated above, is a measure of the proper time of Q in D₀ and by virtue of (A.7) is therefore directly proportional to its position on X_o from its initial position at t = 0. In like manner however, the proper time of Q on its primed temporal axis is directly proportional to its position on that axis from its initial position. Note however that, as Q possesses only temporal motion in the primed axes, its proper time along X'_q will be identical to that of X'_q itself. Therefore, to derive the relationship between time on the two temporal axes for any constant value of r_q, a one-to-one correspondence between incremental distances on them can be established as follows. If dX_o is an incremental distance along the temporal axis of D₀ , and dX'_q an incremental distance along the temporal axis associated with the point Q on R' such that they are temporally coincident in D₀ , then due to their relative orientation, they conform to the following expression:

As t¢_q is the time along X'_q, and since the primed temporal velocity of all points on R' is c, i.e. equal to |V|, then by (A.2), (A.24) may be rewritten thus:

Integrating (A.25)

The constant k is the initial condition that ensures temporal coincidence of the two incrementals within D₀, and k must therefore be such that when t¢_q is zero, the proper time of Q in D₀ is also zero. Thus, from (A.20) when t_q is zero, t is given by

which gives in (A.26) when t¢_q is zero

so that finally in (A.26)

This being the relationship between time on the X'_q and X_o axes.

Note that (A.25) shows that the units of time along X'_q are greater than those along X₀ by the same factor and, for the same reason that the units of length along the spatial axes differ. Clearly all such points on R', including B, must have associated with them a time, along a unique temporal axis, of the form of (A.29) in which the spatial term differs appropriately. The locus of the reference zero on these axes lies along the spatial axis of D₀ at t = 0. This together with the expansion of the units of time along these axes ensures the simultaneity of existence of each point on R' in both frames of reference. It is also noted that the mathematical relationship between t¢_q and t is the same as that between t_q and t. They cannot be equated however because of the difference in the magnitude of the units.

The above relationships, specifically associated with the point Q, can be diagrammatically represented as in Fig. (A1) below. The spatial motion of the point B, has, from the above argument, resulted in axes associated with it being expanded and rotated in the direction of motion in D₀ through the same spatial-temporal angle q as the Existence Velocity vector of B. This concurs with statements in the literature [3] that in Minkowski's `World' the Lorentz Transformations "correspond to a rotation of the co-ordinate system".

Clearly (A.22) and (A.29) are identical to the Lorentz Transformations of the Special Theory and together with the previous results of this Appendix, demonstrates the equivalence of the Domain D₀ with Pseudo - Euclidean Space - Time. The application of the concept of Existence Velocity within the latter is therefore a valid one.

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