3. Gravitational Planar Motion in D₁ .

It was shown in [1] that in the Relativistic Domain D₀, (Pseudo Euclidean Space-Time), a change in the Existence Momentum of a mass could only be effected by the application of an accelerative force. Indeed, for spatial motion of a mass to exist in any Relativistic Domain, including one exhibiting gravitation, it is firmly believed that it can only result from the application of such a force. To cause gravitational motion therefore, if an accelerative force is not artificially applied, then it must be generated within the mass itself as a result of interaction with the characteristics of the Domain. Assuming the gravitational effect in D₁ to be a purely spatial radial one, this internally generated gravitational force can be determined for any gravitating mass, by comparing the spatial variation of its total energy, with the temporal variation of its Existence Momentum. First, consider the variation of Existence Momentum with time. For planar motion it is derived as follows.

If m is the energy mass of a particle possessing free planar motion in D₁, then, from (2.11), its Existence Momentum will be:-

(3.1)

Differentiating (3.1) with respect to gives the time rate of change of M in D₁ as :-

(3.2)

where in taking the derivatives of the unit vectors n and t, the relationships of (2.7) and (2.8) have been inserted.

Equation (3.2) gives the reaction of the particle to changes in its Existence Momentum and, if the cause of gravitation is purely spatial, then the temporal component will be zero, so that

(3.3)

This naturally integrates immediately to give:-

(3.4)

Initial conditions may be chosen to correspond to an apse of the spatial trajectory so that when and giving

(3.4a)

Note that m₀ is not the rest mass but the energy mass at the apse.

Eq.(3.4a) inserted into (3.4) gives

(3.5)

Eq(3.5) represents the energy mass of the particle as a function of its velocity in D₁.

To eliminate the term in in (3.3), use is made of the fact that gravitation is a purely radial effect with respect to the origin so that the radial normal, , component of (3.2) must also be zero. Therefore this gives

(3.6)

so that

(3.7)

Note that this is identical to the statement that angular momentum is constant. Substitution of (3.7) into (3.3) then gives after reduction

(3.8)

so that with substitution of (3.8) into (3.2), both the radial normal, (), and the temporal components vanish and there is left after reduction

(3.9)

This represents the resultant reaction of the gravitating mass to changes in its Existence Momentum.

Next the variation of the total energy of the mass as a function of its radial position from the origin will be determined for comparison with (3.9).

The total energy of matter in D₀ was, in [1], shown to be the product of its energy mass and the square of the magnitude of its Existence Velocity. Extending this to the Domain D₁, the total energy of the mass here is given by

(3.10)

Differentiating this with respect to gives

(3.11)

Converting the differential of the mass to one involving the time then gives

(3.12)

(3.8) may now be substituted for to give after reduction

(3.13)

This represents the resultant spatial variation of total energy with radial distance.

Now comparing (3.13) with the magnitude of (3.9) it is clear that

(3.14)

However, for purely gravitational motion, there is no artificially applied force and therefore the total energy of such a free particle within D₁ will be constant, i.e.

(3.15)

Insertion of this into (3.14) then gives

(3.16)

This relationship shows that the cause of gravitational motion in D₁ is a reaction force generated within the particle proportional to it's energy mass. The term has the dimensions of acceleration and as can be seen is solely a function of the characteristics of the Domain. For this reason this term is now defined as the Gravitational Acceleration Potential of D₁

The equation of motion of the gravitating mass may now be obtained by the simple substitution of the magnitude of (3.9) into (3.16), the result being

(3.17)

This is also evident from (3.13) when (3.15) is inserted. Equation (3.17) is the equation of free planar motion of a mass within D₁. The term in is the centripetal acceleration resulting from the rotational nature of the motion about the origin. The term in is an acceleration caused by the radial velocity as the particle mass moves through the varying temporal field surrounding the gravitational source. Both this and the centripetal term act in opposition to the main gravitational term, the Acceleration Potential.

The nature of the gravitational motion is clearly determined by the sign of the gradient of u, and it will be shown later that this sign is positive for a Domain identical to the Solar System.

It was shown in [1] that in D₀ the cause of the variation of mass as a function of spatial velocity was the manner in which artificially induced kinetic energy was stored. In D₁, for the type of motion under consideration, by virtue of (3.15) artificially induced kinetic energy does not exist and consequently the variation of mass, as exhibited by (3.5) must have a different cause. To investigate this, (3.15) is substituted into (3.11) to give, after separation of variables

(3.18)

Solution of this simple equation gives

(3.19)

showing that in D₁ because is a function of , energy mass is solely a function of position on the radius vector from the origin. Now substitution of (3.19) back into (3.10) then gives

(3.20)

e.g. the constant value of the total energy of the gravitating mass which is seen to be that at the point taken for initial conditions.

As mentioned above, m by virtue of the function is solely a function of . However, because m is the mass equivalent of which, being constant for all , therefore indicates that in (3.19) it cannot be the amount of matter energy that is varying, but some other parameter associated with D₁. The only other parameter involved is and the mechanism behind the variation of mass derives from the fact that is a measure of the temporal rate of D₁ and if the motion of a mass involves movement along a radius vector from the origin, it therefore moves continuously through a varying temporal rate. Because the units of mass include the square of time, these, and consequently the value of mass must vary along according to the square of the function .

Now (3.15) and (3.16) indicate that the gravitationally accelerated condition of the mass is its natural state of existence in D₁. To change this state of existence, in either assisting or resisting the gravitational effect, energy must be provided. As an example of this consider the simple case in which gravitational motion of a particle mass is prevented at some distance s ₁ from the centre of the source. Thus in (3.14) putting

(3.21)

gives

(3.22)

But because there is no motion both sides of (3.22) must be constant and it can be written

(3.23)

where is the constant force applied to resist gravitation and is therefore a measure of the weight of the particle mass. Note that the weight of any particle mass is a variable dependent upon its radius vector position from the origin of the gravitational source. Therefore, if the particle were far enough away from the origin, u would become constant, its gradient zero and therefore the weight of the particle also zero. This is of course entirely in keeping with experience within the Solar System. Note also that if the sign of were negative, (3.23) shows that the weight of the particle mass in an anti-gravitational field would be negative.

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