3 Gravitational Planar Motion in D1 .3.1 The Accelerative Force of Gravitation.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:-
Differentiating (3.1) with respect to t gives the time rate of change
of M in D1 as :-
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
This naturally integrates immediately to give:-
Note that m0 is not the rest mass but the energy mass at the apse.
Eq.(3.5) inserted into (3.4) gives
Eq(3.6) represents the energy mass of the particle as a function of its velocity in D1.
this gives
so that
Note that this is identical to the statement that angular momentum is
constant. Substitution of (3.8) into (3.3) then gives after reduction
so that with substitution of (3.9) into (3.2), both the radial normal,
(t), and the temporal components vanish and there is left after reduction
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 (3.10).
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
Differentiating this with respect to s gives
Converting the differential of the mass to one involving the time then gives
This represents the resultant spatial variation of total energy with radial distance.
Now comparing (3.14) with the magnitude of (3.10) it is clear that
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.
Insertion of this into (3.15) then gives
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
3.2 The Equation of Motion.The equation of motion of the gravitating mass may now be obtained by the simple substitution of the magnitude of (3.10) into (3.17), the result being
This is also evident from (3.14) when (3.16) is inserted. Equation (3.18) is
the equation of free planar motion of a mass within D1. The term in
u2w2s is the centripetal acceleration resulting from 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.
3.3 Mass and Energy.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 (3.16) artificially induced kinetic energy does not exist and consequently the variation of mass, as exhibited by (3.6) must have a different cause. To investigate this, (3.16) is substituted into (3.12) to give, after separation of variables
Solution of this simple equation gives
showing that in D1 because u is a function of s, energy mass
is solely a function of position on the radius vector from the origin. Now
substitution of (3.20) back into (3.11) then gives
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 u, is solely a function of s. However, because m is the mass equivalent of E which, being constant for all s, therefore indicates that in (3.20) it cannot be the amount of matter energy that is varying, but some other parameter associated with D1. The only other parameter involved is u and the mechanism behind the variation of mass derives from the fact that u 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 s according to the square of the function u.
3.4 WeightNow (3.16) and (3.17) 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 s1 from the centre of the source. Thus in (3.15) putting
gives
But because there is no motion both sides of (3.23) must be constant and it
can be written
where Fg 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.
G1 Version 2.2.4
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P.G.Bass, November 2009
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