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
|
|
|
|
|
= 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
|
|
|
|
|
= |
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
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
|