3. An Analysis of Mass in D₁.

It was shown in [2] that in D₀ the force applied to accelerate a mass produced two spatial reaction terms similar to (i) and (iv) in Section 2 above. It was also shown that the combination of these terms resulted in the generation of inertial mass. Determination of whether the same effect is present in D₁ for purely gravitational motion, is most easily accomplished by initially evaluating the four terms derived in Section 2 in terms of the function u, and suitably chosen initial conditions. These conditins, for gravitationally induced rectilinear motion, are defined as the value of u at the location that motion starts, u₀, and the value of the energy mass, represented by m₀, at the same location.

First, the term (i) in Section 2 above. From [1] Eq. (4.3), for gravitational rectilinear motion, is given by

(3.1)

Also from [1] Eq.(B4), by putting w₀=0

(3.2)

Substitution of (3.2) into (3.1) then gives

(3.3)

From [1] Eq. (3.19), energy mass is given by

(3.4)

Thus from (3.3) and (3.4) for the first spatial reaction term (i) in Section 2 above

(3.5)

Next, for the second spatial reaction term, (iv), in Section 2, from (3.4)

(3.6)

which with insertion of (3.2) becomes

(3.7)

and thus the combination of (3.2) and (3.7) gives for the second spatial reaction term

(3.8)

Eqs.(3.5) and (3.8) may now be summed to give the total spatial reaction to gravitationally induced rectilinear motion thus

(3.9)

which from (3.4) finally reduces to

(3.10)

as derived in [1] Eq.(3.16).

Prior to a discussion of this result, the temporal terms (ii) and (iii) in Section 2 are compared to re-confirm the purely spatial nature of gravitation.

From (3.2) and (3.7)

(3.11)

and from (3.1), (3.2) and (3.4)

(3.12)

thereby confirming that these terms are equal in magnitude but opposite in sign and therefore cancel.

The above result, (3.10), shows that what in the literature has been termed "gravitational" mass, is equivalent to energy mass as defined in this series of papers. It also thereby shows that in the Relativistic Space-Time Domain D₁, this mass bears no relationship to the inertial mass of D₀, Pseudo-Euclidean Space-Time. i.e. compare (3.4) with [2] Eq.(3.9). However, this comparison is perhaps an unrealistic one in that it is across Domains. A more realistic comparison is that of the gravitational mass of D₁ with true inertial mass within the same Domain. To do this it is necessary to analyse the effect of the application of an artificially applied force to a mass in D₁. Such an analysis must however also take account of the gravitational effect that is still present.

Inertial Mass in D₁

Prior to conducting this analysis it is useful to simplify (2.1) as it will thereby in turn simplify the ensuing development. Because the gravitational effect is purely spatial, the temporal component of (2.1) is zero, so that from this component

(3.13)

which for an initially stationary mass integrates to

(3.14)

and which via (3.2) can be shown to be equal to (3.4).

Substitution of (3.13) into (2.1) then gives

(3.15)

This can also be obtained by putting w = 0 in [1] Eq.(3.9). To conduct the analysis of forced motion in D₁, assume now that an artificial force F is applied in opposition to the gravitational effect to a stationary, free mass point. As rectilinear motion only is being considered, the vector notation is dropped. The rate of change of spatial momentum of this mass will then be from (3.15),

(3.16)

Solving this for yields

(3.17)

which is clearly seen to be the total acceleration of gravitation augmented by an accelerative term due to the application of the artificial force F. The mass term associated with this force is the true inertial mass of the Domain D₁, which becomes, with the insertion of (3.14) for the energy mass

(3.18)

Here it is clear that (3.18) possesses the same form as the inertial mass of Pseudo-Euclidean Space-Time, [2], Eq.(3.9), but with the presence of the additional multiplicative term u₀/u. This extra term appears because the motion is taking place through the varying temporal rate generated by the gravitational source.

From the above results it is clear that a hitherto basic belief of gravitational theory, the equivalence of gravitational and inertial mass, does not apply in D₁. These two mass terms cannot be equated. Gravitational mass is, due to the nature of the generation of the accelerative force, equivalent to energy mass and for accelerative motion due solely to gravitation, inertial mass does not exist. This is an important result which will be discussed in detail later in this paper.

It is easy to derive the initial acceleration, (at t =0 and s =s ₀), from (3.17) by putting , i.e.

(3.19)

Also, it is noted that if in (3.17) F is such as to prevent gravitational motion, then both and may be put to zero and (3.17) reduces to

(3.20)

as derived in [1] Eq.(3.23), for the weight of the mass at some arbitrary distance s₁ from the centre of the gravitational source.

Finally, it can be seen from (3.17) that a further steady state solution exists, that for when . Inserting this condition into (3.17) and solving for yields

(3.21)

From (3.21) it can be seen that if is very much larger than the gravitational term, the velocity approaches the terminal velocity of the Domain. In this respect this condition is the same as in D₀. However, if F is very small compared to the gravitational term (3.21) reduces to

(3.22)

which shows that within D₁ the maximum velocity that can be achieved in solely gravitational induced motion, is very much less than the terminal velocity. The reason is that the positive acceleration, proportional to the square of the velocity of the motion through the reducing temporal rate, (see (3.1)), eventually reaches a level that exactly balances the negative acceleration produced by the Acceleration Potential of the Domain.

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