2  The Space-Time Domain D0

2.1  Definition

The Domain D0 is defined as a space time of four mutually orthogonal linear dimensions, three of which, X1 , X2 and X3 are spatial, while the fourth, Xo , following Minkowski, is temporal. Xo is defined, and will be shown, (Appendix A), to be the product of the time t in D0 and a constant velocity parameter c, designated the Spatial Terminal Velocity of D0 .

The Domain D0 is further characterised in that for any spatial - temporal point to exist within it, that point must at all times possess a vector velocity, designated Existence Velocity, the magnitude of which has the same value as the Spatial Terminal Velocity. Thus, for any point to exist within D0, it is necessary for the magnitude of the vectorial sum of its velocities along the four co-ordinate axes, to be at all times, equal to c.

2.2  Existence within D0

The position of any random point B within D0, relative to some chosen reference can be expressed in spatial - temporal vector form as

S = ix1 + lx2 + kx3 + jxo
(2.1)

where x1, x2 and x3 are each a distance along the corresponding spatial axes of D0 , for which i, l and k are the appropriate unit vectors. xo is a distance along the temporal axis for which j is the unit

vector. i, l and k each have the usual magnitude of unity, while j has the magnitude of 
Ö
 
-1
 
.



As the temporal axis of D0 is the product of the constant c with the time t, so then will the distance xo be a product of c and some function of the time t in D0. Accordingly (2.1) may be rewritten as

S = r + jctp
(2.2)

where tp is a function of the time t in D0 and, also, where the spatial component of (2.1) has been replaced with its resultant spatial vector position on a polar spatial linear co-ordinate axis R.

The velocity of such a point in D0 is defined by differentiating (2.2) with respect to t thus:

V = v + jc dtp
dt
(2.3)

where V = dS/dt and v = dr/dt

Invoking the characteristic of existence in D0 , (2.3) must at all times conform to the following identity,

V = c
(2.4)

where V is the magnitude of V. Substitution of (2.4) into (2.3) gives, after taking the magnitude,

c2 = v2 + c2 æ
è 		dtp
dt
 ö
ø 		2

 
(2.5)
so that
dtp
dt
  =  æ
è 		1 -  v2
c2
 ö
ø 		1/2

 
(2.6)

where v is the magnitude of v. Substitution of (2.6) into (2.3) then gives

V = v + jc   æ
è 		1 -  v2
c2
 ö
ø 		1/2

 
(2.7)

and the following terms are defined thus:

V is the Existence Velocity of the the point B in D0

dtp /dt is the Temporal Rate of the the point B in D0

cdtp /dt is the Temporal Velocity of the the point B in D0

tp is the Proper Time of the point B in D0.

Thus tp is the time measured by any observer moving with a spatial velocity v in D0.

From (2.7) it is evident that V possesses a spatial - temporal orientation in D0 which is directly dependent upon the spatial velocity magnitude v. As v increases from zero, temporal velocity undergoes a proportional reduction such that V, relative to the temporal co-ordinate of D0 rotates through an angle in the Xo - R plane, related to v by the expression

Sin q =  v
c
(2.8)

Thus, for future reference it is noted that

Cosq =  æ
è 		1 -  v2
c2
 ö
ø 		1/2

 
 =  dtp
dt
(2.9)

This completes the definition and characterisation of D0 . Its equivalence with Pseudo-Euclidean Space-Time is demonstrated in Appendix A. The next section formulates the kinematics and kinetics of rectilinear motion within D0 for comparison with that of the Special Theory.



R1 Version 2.3.3
Ó P.G.Bass, February 2008
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