## 2.0  The Relativistic Space-Time Domain D1.

### 2.1  Definition.

The Relativistic Domain D0, as developed and shown in [1] to be equivalent to Pseudo-Euclidean Space-Time, is one in which gravitation exists in an artificially defined form only. A rigorously defined expression of gravitation, for a single gravitational source, requires that the Domain D0 is modified by the presence of the source, to produce a new Relativistic Domain, D1. The change is a simple one and the only differences between D0 and D1 are, firstly, a modification of the form of Existence Velocity, the central concept upon which such Domains are based, and secondly a consequential modification of the maximum theoretical spatial velocity attainable within the Domain. The new Domain is however still linear and does not exhibit any form of curvature.

Accordingly, D1 can be defined as a mutually orthogonal space-time of four linear dimensions, three of which Y1, Y2, and Y3, are spatial in nature, and the fourth, X0, is temporal and identical to the temporal dimension of D0. Time in D1 is represented by the parameter t and, as a consequence of the new Domain's modified Existence Velocity, is different from the time t in D0. The Domain is such that it possesses a preferred spatial origin, the centre of the gravitational source, from which the radius vector magnitude to any random point B is

 s = ( y12 + y22 + y32 )1/2
(2.1)

where y1 , y2 and y3 are each a distance along the respective spatial axes from the origin. s has been chosen to represent the radius vector magnitude in D1 to separately identify it from the same parameter, r, in D0.

All spatial-temporal points that exist within D1 must, at all times, possess a characteristic Existence Velocity, the magnitude of which, for the point B, is defined to be the resultant of all four velocities along the co-ordinate axes of D1 and, may therefore be expressed as

 V = æè × y 21 + × y 22 + × y 23 + × x 2p öø 1/2 = cu
(2.2)

where the

 × y #

are the spatial axial velocities of the point and
 × x p

is its temporal velocity.

The parameter c is a velocity constant numerically equal to the magnitude of Existence Velocity in D0 and u is initially defined to be an arbitrary dimensionless function of s.

Finally, the maximum theoretically attainable spatial velocity in D1, designated Spatial Terminal Velocity, is defined as follows. For motion purely along a radius vector, Spatial Terminal Velocity is defined to be equal to cu. For purely circular motion in any plane about the origin, it is defined to be equal to the velocity constant c. This difference exists because of the purely radial nature of gravitation.

### 2.2  Existence Within D1

The Spatial-Temporal Existence Velocity Vector V, for the random point B within the Domain is determined as follows.

The spatial-temporal position of the point B with respect to the spatial centre of the gravitational source and some chosen temporal reference will be

 S = iy1 + ly2 + ky3 + jxp
(2.3)

where the y# are each a distance along the three spatial axes Y1, Y2 and Y3 for which the i, l and k are normal unit vectors. The term xp is a distance along the temporal axis for which j is the unit vector with a magnitude of (-1)1/2.

From (2.1), Eq.(2.3) may be rewritten as

 S = sn + jxp
(2.4)

where n is a radial unit vector.

For planar motion, the velocity of this point is defined by differentiating (2.4) with respect to the time t thus

 V = × s n + uwst + j × x p
(2.5)

 where V = dS/dt and w = df dt and is the angular rate of the point B. Note that (2.5) involves the differential of the unit vector n thus

 dn dt = dn dj dj dt = uwt
(2.6)

within which

 dn dj = ut
(2.7)

and therefore similarly

 dt dj = - un
(2.8)

Relationships (2.7) and (2.8) occur because of the different Spatial Terminal Velocities in the radial, n, and radial normal, t, directions. Proof of the above relationships is presented in Appendix F.

Taking the magnitude of (2.5) gives, after invoking the characteristic of existence in D1 via the insertion of (2.2)

 | V | = V = cu = æè × s 2 + u2w2s2 + × x 2p öø 1/2
(2.9)

from which

×
x

p
= cu æ
ç
è
1 -
 × s 2

c2u2
-  w2s2

c2
ö
÷
ø
1/2

(2.10)

which when re-inserted into (2.5) yields

V =
×
s

n + uwst + jcu æ
ç
è
1 -
 × s 2

c2u2
-  w2s2

c2
ö
÷
ø
1/2

(2.11)

Eq.(2.11) is the Existence Velocity of the random point B in the Relativistic Space-Time Domain D1. This expression will be used in the next Section to develop the kinematics and kinetics of gravitational motion in D1 which will then be shown to be the natural state of existence in that Domain. Before that however, it is useful to note three other important characteristics of D1.

### 2.3  The Time t in D1

 The first concerns the time t in D1. From (2.11) when × s and w are both zero, motion exists only along the temporal axis of D1 so that the temporal velocity of a spatially stationary

point in D1 is

 dx0 dt = cu
(2.12)

and therefore an element of time in D1 may be defined by the relationship

 dt = dx0 cu
(2.13)

and is therefore a function of spatial position from the origin by virtue of the fact that u is a function of s.

### 2.4  The Proper Time in D1

The second point concerns the proper time of the point B in D1, i.e. the time measured by an observer located with the point B. Inserting (2.12) into (2.10) and re-arranging gives

dxp

dx0
=  æ
ç
è
1 -
 × s 2

c2u2
-  w2s2

c2
ö
÷
ø
1/2

(2.14)

using (2.13) to rewrite the LHS of (2.14) then gives

dtp

dt
=  æ
ç
è
1 -
 × s 2

c2u2
-  w2s2

c2
ö
÷
ø
1/2

(2.15)

where

 dtp = dxp cu
(2.16)

where dtp/dt is the temporal rate, and tp the proper time of the point B in D1.

### 2.5  Temporal Significance of the Function u.

Although u has been defined as a non-dimensional function of the spatial variable s, its appearance in the temporal components of the above relationships has a special significance in that it relates time in D1 to that in D0, (Pseudo-Euclidean Space-Time).

A spatially stationary point in D1, with a temporal velocity given by (2.12) would, in an element of time dt in D0 move an element of distance dx0 along the temporal axis, given by

 dx0 = cudt
(2.17)

Therefore in D0 , the proper time of such a point, i.e. the proper time of D1 would be

 dt = dx0 c = udt
(2.18)

so that

 dt dt = u
(2.19)

and u is therefore a measure of the temporal rate of D1 with respect to D0 and, for future reference, it is noted that it must therefore possess a positive sign. Also, it is clear that because u is a function of s, the temporal rate of D1 is a variable dependent upon radial distance from the centre of the gravitational source, i.e. D1 exhibits spatially dependent temporal dilatation, as is also evident from (2.13).

The relationship between the respective spatial axes of the two domains depends upon the characteristics of u and will be developed subsequent to the determination of the precise nature of this function in Section 4.

G1 Version 2.2.4
Ó P.G.Bass, November 2009

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