APPENDIX A

Inter-Relationship Between the Spatial and Temporal Expansion

Velocities and Their Acceleration Potentials.


For simplicity, in this Appendix only vector magnitudes of all parameters are considered.

In [11], Section 3.3.2, relationships for the velocities of spatial expansion, internal and external, emanating from a gravitational source, and the corresponding temporal velocities were developed. These are repeated below for convenience.

(i) Spatial, external.


us = æ
è 		2gM
s
 ö
ø 		1/2

 
 = c( 1 - u2 )1/2
(A.1)

(ii) Spatial, internal.


uis = æ
è 		3gM
sg
- gMsi2
sg3
ö
ø 		1/2

 
 = c( 1 - ui2 )1/2
(A.2)

(iii) Temporal, external.


ut = c æ
è 		1 - 2gM
c2s
 ö
ø 		1/2

 
 = cu
(A.3)

(iv) Temporal, internal.


uit = c æ
è 		1 - 3gM
c2sg
+ gMsi2
c2sg3
ö
ø 		1/2

 
 = cui
(A.4)

From these relationships it is clear that


V = ( us 2 + ut 2 )1/2 = c    and    Vi = ( uis2 + uit2 )1/2 = c
(A.5)

and all terms are defined thus

us    the spatial expansion velocity of the source external to itself.

uis    the spatial expansion velocity of the source internal to itself.

ut    the temporal velocity of all spatial points external to the source.

uit    the temporal velocity of all spatial points internal to the source.

g    Newton's constant of proportionality.

M     the mass of the gravitational source.

s    the distance of a point external to the gravitational source from its centre.

u     the temporal rate at s.

sg     the external radius of the gravitational source.

si     the distance of a point internal to the gravitational source from its centre.

ui     the temporal rate at si.

V     the spatial-temporal velocity magnitude of existence external to the source.

Vi     the spatial-temporal velocity magnitude of existence internal to the source.

It was also shown in [11] that


u = æ
è 		1 - 2gM
c2s
 ö
ø 		1/2

 
(A.6)

and


ui = æ
è 		1 - 3gM
c2sg
+ gMsi2
c2sg3
ö
ø 		1/2

 
(A.7)

Finally, it was also shown in [11] that


As = - gM
s2
 = - c2u du
ds
(A.8)

and


Ais = - gMsi
sg3
= - c2ui dui
dsi
(A.9)

This Appendix will now derive the associated acceleration parameters in the temporal direction and establish their relationship with their spatial counterparts.

From (A.3) above, allowing s to vary


At = dut
dt
 = c d
ds
 æ
è 		1 - 2gM
c2s
 ö
ø 		1/2

 
 ds
dt
(A.10)

where ds/dt is given by (A.1). Eq.(A.10) works out to be


At = 
æ
è 		2gM
s
 ö
ø 		1/2

  		 gM
s2
c æ
è 		1 -  2gM
c2s
 ö
ø 		1/2

 

    = c2( 1 - u2 )1/2  du
ds
  = -  ( 1 - u2 ) 1/2
u
 As
(A.11)

and similarly from (A.4)


Ait = 
æ
è 		3gM
sg
 -  gMsi2
sg3
ö
ø 		1/2

 
  gMsi
sg3
c æ
è 		1 -  3gM
c2sg
 +  gMsi2
c2sg3
ö
ø 		1/2

 

    = c2( 1 - ui2 )1/2  dui
dsi
 = -  ( 1 - ui2 ) 1/2
ui
Ais
(A.12)

Consequently, the spatial-temporal Acceleration Potential vector magnitude is, from (A.5) and (A.8), given by


A = ( As 2 + At 2 )1/2 =  
gM
s2 æ
è 		1 -  2gM
c2s
 ö
ø 		1/2

 

    =  As
u
  =  At
( 1 - u2 ) 1/2
  = c2 du
ds
(A.13)

and from (A.6) and (A.9)


Ai = ( Ais2 + Ait2 )1/2 =  gM
si2 ( 1 - [( 3gM)/( c2sg )] + [( gMsi )/( c2sg3 )] )1/2

    =  Ais
ui
 =  Ait
( 1 - ui2 )1/2
  = c2 dui
dsi
(A.14)

The four dimensional geometrical representation of these parameters can then be shown as


Picture 4


Fig. A.1 Geometrical Representation of Existence Velocities and Acceleration Potentials at Some Point External to the Source.

A similar representation exists internal to the source. Note that the spatial-temporal ängle of existence" is given by


cosq =  ut
c
  =  æ
è 		1 - us 2
c2
 ö
ø 		1/2

 
  =  As
A
 = u
(A.15)

and is the same as that derived in [15] for the physical velocity of a mass in Pseudo - Euclidean Space Time.



G4 Version 1.0.0
Ó P.G.Bass, August 2009

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