4  Correlation Between the Domain D1 and the Solar System

To this point the analysis has been somewhat generalised because the function u, and therefore D1, has only been partially defined. u is the most important parameter associated with D1 dictating its inherent characteristics and those of existence within it. u could be specified to be any arbitrary function of s, the resulting hypothetical domains thereby exhibiting gravitational and other characteristics of various types and degrees. To determine D1 such that it possesses the gravitational characteristics of ponderable matter in the Solar System, requires therefore the determination of the appropriate function for u.

4.1  Determination of the Function u.

Because gravitation is a purely radial effect, the function u can most easily, and without any loss of generality, be determined by establishing a correlation between Newton's gravitational equation for free rectilinear motion, and the appropriate approximated form of (3.18).

The former is given by

d2r
dt2
 = - g mg
r2
(4.1)

Where r is the distance to the centre of the gravitational source, mg is the generating mass of the source and g is Newton's constant of proportionality. Now (4.1) is expressed in terms of the spatial and temporal axes of D0, Pseudo-Euclidean Space-Time. However, this relationship was empirically derived within primarily the gravitational influence of the Earth, and consequently the most accurate form of it would be obtained by its expression in terms of the spatial and temporal axes of a Domain representing the Earth's gravitational field. Equation (4.1) would then be the classical approximation to such an equation. If D1 is to represent this Domain then Newton's rectilinear gravitational equation expressed in the spatial and temporal axes of D1 would be

××
s
 
 = - g mg
s2
(4.2)

and s must differ from r and, over short periods, t from t by incremental amounts not discernible in Newton's experimentation. These conditions will be proved later.

The equation of free rectilinear motion in D1 is obtained from (3.18) by putting w to zero, viz.

××
s
 
 = - c2u du
ds
 + 2
×
s
 
 2
 
u
 du
ds
(4.3)


Comparing (4.2) with (4.3), as the former is independent of a velocity term, the appropriate approximation of (4.3) is obtained by ignoring the term involving  
×
s
. This merely means that in Newton's



experimentation the effect of the velocity ratio  
×
s
cu
, (or   dr/dt
c
) , was too small to be observed. Thus equating the final approximation of (4.3) with (4.2) gives

g  mg
s2
 = c2u du
ds
(4.4)

and where there now appears on the right hand side the Gravitational Acceleration Potential of D1. Equation (4.4) may be integrated to give

- a
s
 = u2
2
 + k
(4.5)

where

a = g  mg
c2
(4.6)

is the so called gravitational radius of the source. The constant of integration is obtained by introducing the boundary condition that as s® ¥, u ® 1, i.e. as s® ¥, D1 ® D0. This gives k = - 1/ 2 which when inserted into (4.5) gives the function u thus

u = æ
è 		1 - 2a
s
 ö
ø 		1/2

 
(4.7)

From this it is easily seen that with u being positive, the gradient of u along all radius vectors from the source is also positive. The consequence is that the gravitational effect within D1 is to result in motion towards the origin, or gravitational source, as it does in the Solar System.

To demonstrate that D1 as characterised herein is truly representative of the gravitational effects of concentrated matter in the Solar System, it is necessary to provide proofs to the following three statements:


(i) That s differs from r, and, over short periods, t from t by incremental amounts not discernible in mechanical experimentation.

(ii) That the relationship between D1 and D0 is such that (4.1) is indeed the classical approximation of (4.2).

(iii) That free motion within D1 is identical to that observed in the Solar System.


The first two of these statements can be proven via establishment of the relationship between the respective polar axes, and the time, in D1 and D0.

The third proof will be demonstrated in the next section by the derivation of the equation of a central orbit in D1 and its comparison with that obtained from the General Theory of Relativity and with observable planetary motion in the Solar System. A rigorous solution to this equation is also presented.

4.2  Relationship Between the Polar Co-ordinate Axes of D1 and D0.

This relationship can be established by the temporal transformation of a spatial velocity in D1 to the Domain D0, and comparing the resulting co-ordinate terms with the equivalent parameters in D0.

Thus taking the spatial component of (2.11)

-
u
 
 =
×
s
 
 n + uwst
(4.8)

The equivalent expression in the co-ordinates of D0 is

v = dr
dt
 n +  dj
dt
 r t
(4.9)

Consider the radial component of (4.8) first

×
s
 
 = ds
dt
(4.10)

temporally transforming this to the Domain D0 using (2.16)

u
×
s
 
 = ds
dt
(4.11)

Comparing this with the radial component of (4.9), if

ds
dt
 = dr
dt
(4.12)

then upon integration

s = r + k
(4.13)

The constant of integration relates to the boundary conditions of the two Domains. The lower boundary condition is not known because D1 is not homogeneous in this region. However, the other boundary at which both domains are homogeneous can be utilised as follows. From (4.7) write

u2s2 = s2 - 2as
(4.14)

differentiate this with respect to r.

d
dr
 ( u2s2 ) = 2 ds
dr
 ( s - a )
(4.15)

Inserting (4.12) and (4.13) into the right hand side of (4.15) then gives

d
dr
 ( u2s2 ) - 2r = 2( k - a )
(4.16)

and this expression must conform to the boundary condition that as r® ¥, s® r and u® 1. i.e. D1 ® D0. At this boundary the left-hand side of (4.16) vanishes leaving

k = a
(4.17)

Thus in (4.12) this gives simply

s = r + a
(4.18)

as the relationship between the radial axes of D1 and D0. This expression, together with (4.6), shows that s differs from r by an incremental amount not discernible in mechanical experimentation. i.e. proof of the spatial part of statement (i) above. It should be noted that the above process is essentially the same as that in [2] where the same result is obtained from the requirement that the two sets of co-ordinates be "harmonic". It should also be noted that (4.7) and (4.18) are only valid in homogeneous regions of D1 i.e. outside the generating mass of the gravitational source. This ëxtension" of radial distance occurs inside the gravitational source and (4.18) is the resultant effect outside the geometric dimensions of the source.

Now consider the radial normal terms in (4.8) and (4.9). Extracting and equating the angular rates, noting that the angular rate in D1 is already a function of time in D0


u w = df
dt


= dj
dt
(4.19)

Integrating

j = f
(4.20)

Where the constant of integration may be made zero by assuming a common reference radial in both domains. Thus angles in D1 and D0 are identical. This is to be expected as gravitation is a purely radial effect.

4.3  The Temporal Relationship Between D1 and D0

The temporal relationship between D1 and D0 can be established by integrating (2.19) and inserting (4.7). This gives

t = ut = t æ
è 		r + a
r - a
 ö
ø 		1/2

 
(4.21)


where the constant of integration has been put to zero by taking a common artificial temporal origin in both D1 and D0. Thus, where, as in Newton's experiments, the inequality  s >> 2 gmg
c2
 (or,  r >> gmg
c2
),



is valid, over short periods of time t » t. This constitutes proof of the temporal part of Statement (i) above.

4.4  The Classical Approximation to the Equation of Free Rectilinear Motion in D1

Utilising the above results it is now possible to show that Newton's gravitational equation, (4.1), is indeed the classical approximation of (4.2).

From (2.16) and (4.18)

ds
dt
 = 1
u
 dr
dt
(4.22)

so that for the left hand side of (4.2)

d2s
dt2
 = 1
u
 d
dt
 æ
è 		1
u
 dr
dt
 ö
ø
(4.23)

working this out gives

d2s
dt2
 = 1
u2
 d2r
dt2
  - 1
u3
 æ
è 		dr
dt
 ö
ø 		2

 
 du
ds
(4.24)

Substitution for u from (4.7) and for its spatial gradient, and then for s from (4.18) gives

d2s
dt2
 = æ
è 		r + a
r - a
 ö
ø 		d2r
dt2
 + a
( r - 		a )2
 æ
è 		dr
dt
 ö
ø 		2

 
(4.25)

Once again if r >> a, then (4.25), for the left hand side of (4.2) approximates to

d2s
dt2
 » d2r
dt2
(4.26)

For the right hand side of (4.2), substitution from (4.18) and (4.6) and taking the approximation yields

- gmg
s2
 » - gmg
r2
(4.27)

Thus completing the exercise and providing proof of Statement (ii) above.



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