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.

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 (3.17).

The former is given by

(4.1)

Where r is the distance to the centre of the gravitational source, m_g 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

(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 (3.17) by putting w to zero, viz.

(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 . This merely means that in Newton’s experimentation the effect of the velocity ratio , (or ), was too small to be observed. Thus equating the final approximation of (4.3) with (4.2) gives

(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

(4.5)

where

(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 =-½ which when inserted into (4.5) gives the function u thus

(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:

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.

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)

(4.8)

The equivalent expression in the co-ordinates of D₀ is

(4.9)

Consider the radial component of (4.8) first

(4.10)

temporally transforming this to the Domain D₀ using (2.16)

(4.11)

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

(4.12)

then upon integration

(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

(4.14)

differentiate this with respect to r.

(4.15)

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

(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

(4.17)

Thus in (4.12) this gives simply

(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 "extension" 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

(4.19)

Integrating

(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.

The temporal relationship between D₁ and D₀ can be established by integrating (2.9) and inserting (4.7). This gives

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

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)

(4.22)

so that for the left hand side of (4.2)

(4.23)

working this out gives

(4.24)

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

(4.25)

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

(4.26)

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

(4.27)

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

On to the next Section: Planetary Orbits, (Part 1)

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