It has been shown in Section 2 that the passage of time experienced by all material bodies within the three spatial dimensions of D₁, is due to the manner in which their velocity in the temporal direction is manifested in those dimensions, viz. from (2.2)

(3.1)

It was also shown in Section 2 that the temporal rate of D₁ with respect to D₀ is, from (2.7)

(3.2)

Because time in D₁, is by virtue of (3.1), derived from the temporal velocity of all spatial points within it, it is the actual or effective variation of this velocity, via the non-unity value of u, that results in the variation of the passage of time in the spatial dimensions. Clearly therefore, the time dilatation effect in D₁ is expressed by its temporal rate u.

For all spatial points external to the gravitational source, the value of u was shown, in [1] Eq.(4.7), to be

(3.3)

where, as in [1],

• a is the gravitational radius of the source.
• s is the radial distance from the centre of the source, to a point external to it.

Eq.(3.3) shows that the temporal rate in D₁ external to the source, is a non-dimensional inverse function of the radial distance s from its centre. Now, the generation of u can only take place within the physical constraints of the gravitational source, and it is therefore necessary that the mechanism responsible be shown to describe the equivalent of (3.3) within the source, as well as (3.3) itself outside the source. The equivalent of (3.3) within the source would be expected to exhibit the same form as (3.3) for reason of continuity at the boundary, the surface of the source.

To attempt an immediate discussion or mathematical development of this mechanism from first principles would require a number of somewhat speculative and complex assumptions concerning space, time and matter, assumptions which, at such an early stage in the procedure, would be impossible to adequately support. Fortunately however, it is possible to circumvent such an approach because the form for gravitational acceleration internal to a source, can be derived by using only one simple assumption. The result so obtained can then be used to derive a mathematical model for time dilatation in a more rigorous manner. The derivation of gravitational acceleration internal to a source is the subject of the next section.

To effect this short derivation, it is assumed that gravitational acceleration, both inside as well as outside a source, is of the form

(3.4)

With this assumption, it has been shown in [3], pp 9/10, that gravitational acceleration both internal and external to a thin spherical shell is given by

Internal

(3.5)

External

(3.6)

where

• m_sh is the total mass of the shell.
• l is the distance of the measuring point from its centre.

A solid spherical mass may be considered as a series of concentric spherical shells, and therefore, the gravitational acceleration inside such a mass at a distance s_i from its centre would be, from (3.5) and (3.6), simply

(3.7)

where

• m_i is the mass of that part of the source inside s_i.
• m_g is the total mass of the gravitational source.
• s_g is the physical radius of the source.

Here, and for the remainder of this paper, the subscript i denotes a parametric value inside the source at s_i.

The result (3.7) has also been derived in the reference.

Eq.(3.7) will be used later in developing a mathematical description of the mechanism causing time dilatation internal to the source. Prior to this however, the only three possible causes of this effect are reviewed for their likely applicability.

3.3.1 Discussions.

Consider again (3.1).The variability of this relationship can only come about in one of three possible ways.

(i) Temporal velocity, cu_i, within the gravitational source is directly reduced by some interaction between the source, and the space-time in which it exists.

(ii) Temporal distance, D x₀, is, via some interaction with the source, stretched such that an effective reduction in temporal velocity is simulated.

(iii) The source, via some interaction with the space-time in which it exists, generates a small spatial velocity along all radius vectors from the centre of the source. This spatial velocity, via the       criterion of existence within D₁, then results in a corresponding reduction in temporal velocity.

The lack of measurable interaction between the spatial and temporal dimensions of D₁ prevents the proof or disproof of any of these potential causes, it being only possible, via mechanical means, to measure the passage of time in units related to astronomical or atomic events. However, each of the above potential causes can be logically examined to enable an assessment to be made of which is the more likely.

Firstly, in the assessment of (i) above, for temporal velocity within the source to be the subject of a direct reduction, the interaction between the source and the fabric of the space-time continuum must be of the form of a resistance to motion in the temporal direction. The precedent for this is that it has been shown in both [1] and [4] that forces can exist in the temporal direction. The difference here is that the supposed forces causing the reduction in temporal velocity, would exist naturally in the temporal direction and not be due to, or associated with, any action or event in the spatial part of the Domain. This leads to a number of difficulties. The mathematical development for such a mechanism must lead to an expression similar to (3.3) for u inside the source. In the redefined temporal flow in Section 2, the future part of the Domain would represent the unity term in such an expression. Other terms constituting the reduction from unity of u inside the source, would then be the result of the resistance to temporal motion inherent in the material body of the source. However, it is stated here without proof, that a hypothetical temporal forcing function, applied to any mass in any Relativistic Domain, including that resulting from a resistance to temporal motion, would not result in a change in temporal velocity, but instead a change in mass. This has already been effectively demonstrated in [1] and [4], but will also be discussed in more detail in a future paper. This is the first difficulty. Irrespective of this however, a further problem exists concerning space outside of the source where time dilatation is also in effect. Outside of the source there is nothing to interact with a temporal resistive force to create a reduction of temporal velocity. How the effect is promulgated outside of the source in this scenario is therefore unclear.

From the above discussion, this concept for the generation of time dilatation in D₁ would therefore be a speculative, complex mechanism exhibiting a number of fundamental difficulties, which could not be adequately nor completely explained. It is therefore considered to be an unlikely mechanism for the generation of this effect.

In the assessment of (ii) above it is noted that there appears to be a precedent in [1] Eq.(4.18), where, compared with the Domain D₀, it is shown that an apparent small extension of spatial distance exists along all radius vectors from the centre of a gravitational source. This effect is discussed further in Appendix A to this paper. If the same effect were to occur on the temporal axis, due to the presence of the source, then (3.1) should really be written

(3.8)

and the term u would be associated with the incremental distance D x₀, and not the velocity constant c. Consequently, temporal velocity would remain at the constant value c throughout the temporal dimension as it is in D₀. Once again, there are difficulties with this approach. Firstly, the results of all previous analyses in [1] and [4] have been obtained by associating the parameter u with the velocity constant c, thus inferring that there is a real reduction in temporal velocity. The results of these previous analyses could not be fully realised if (3.8) was the true relationship for the passage of time in D₁. Also, because the time dilatation effect appears outside the source, the stretching of temporal distance would also have to occur in this region. It is similarly unclear as to how this would be effected.

Consequently, it is also considered unlikely that this mechanism is the origin of time dilatation.

Elimination of the above two possibilities leaves this concept as the potential cause of time dilatation in D₁. Consequently, it is necessary that it be capable of adequate logical description, possessing none of the problems inherent in the previous two ideas. In particular, this concept must contain a clear mechanism to promulgate both the time dilatation effect, and the Acceleration Potential, outside the source.

The only precedent for this hypothesis is the mathematical form for u external to the source. This is shown as (3.3). Assume now that this relationship can also be expressed in the same form as [2] Eq(2.5), viz.

(3.9)

where v_s is a small spatial velocity generated by the gravitational source. It is proposed that for reasons of continuity a similar relationship would also hold inside the source. If v_s was a real velocity, and was to lead to the generation of time dilatation and the Acceleration Potential of D₁, it would, because of the nature of gravitation, have to exist along all radius vectors from the centre of the source. Therefore, it could not be a velocity that was associated with the material body of the gravitational source itself, and the only possible alternative is that v_s would therefore have to be a velocity of the very fabric of space, both within and surrounding the gravitational source. This is a new concept concerning the inter-relationship of space, time and matter, for which no supporting observational or experimental evidence exists. Nevertheless, it is proposed that v_s is indeed a real velocity as described above, and which will be shown herein to be capable of causing both time dilatation and the Acceleration Potential of D₁. It is further proposed that the generation of this "velocity of space", is effected by an expansion of space from within the source via a transition from the temporal dimension to the spatial, as the source motion along the temporal axis proceeds. The expansion takes place as an infinitely close series of spherical spatial wavefronts emanating from the centre of the source, and v_s is the wavefront linear expansion velocity along all radius vectors outside the source. The equivalent inside the source is designated v_is. Both v_s and v_is by virtue of the criterion of existence in D₁, result in a corresponding reduction in source temporal velocity thereby causing the time dilatation effect. Furthermore, in order for this spatial expansion velocity to be capable of producing the Acceleration Potential of D₁, it is proposed, and will be shown, that v_s and v_is both exhibit a radial spatial gradient. Also, because gravitational acceleration is a function of the mass of the source, so therefore must v_s and v_is be, together with the spatial expansion and time dilatation effects they produce. One important consequence of this concept is that the resulting expansion of space, Acceleration Potential and time dilatation effect, both internal and external to the source, will be shown to occur with no discontinuity at the boundary.

To complete the discussion of this proposal, it is necessary to consider whether there is an energy transference or dissipation occurring during the process. As far as is currently known, space itself is not constructed of any form of distributed energy field, and the only energy that exists within it is particulate matter energy, and electromagnetic energy that is transmitted through it. It is not known to react directly with any form of energy in the space-time continuum currently envisaged. This suggests that the expansion of space by a gravitational source is not fuelled by any form of energy conversion from the source itself. Also supporting this view is that it has not been possible to construct a mathematical derivation, in which the time dilatation effect is produced by an energy conversion process within the source. It is therefore considered that this process is one in which the energy contained within the source is not involved, and the process, while proportional to the source mass, is governed only by its temporal motion.

While the above discussion, accepting the proposals made therein, provide a logical dissertation on a mechanism for the generation of time dilatation and the Acceleration Potential of D₁, to ensure a degree of scientific credibility, it is necessary to provide proper mathematical support. Such support must enable a rigorous derivation of v_s and v_is from a basic consideration of the fundamental parameters involved. This is the subject of the next section.

Prior to its derivation proper, it is first noted that the magnitude of the linear expansion velocity, after it has promulgated to the outside of the source, can easily be obtained by equating (3.3) and (3.9) and solving for v_s thus

(3.10)

It will also be obvious that the differential of (3.9) with respect to time in D₁, equates to the Acceleration Potential of that Domain outside the source. This will be confirmed in Section 3.3.3.

While (3.10) gives the correct form for v_s, the derivation just summarised does not provide any insight into the manner in which it is generated. To accomplish this, it is necessary to start from a more fundamental position.

Thus to start this development proper, it is necessary to determine the criterion that controls the whole process. To do this it is first noted that the parameter g has units of

or

per unit mass

Thus any term of the form g(mass), will have units (length)³/(time)², i.e. a second order rate of change of volume. Therefore, because this product, g(mass), is predominant throughout gravitational theory, it is proposed that the generation of the spatial expansion within the source, can be described by some function of its second order rate of change of generated volume.

At this point, to ease the development process, the situation outside of the source will be considered first.

• Generation of Time Dilatation Outside the Gravitational Source.

Firstly, because it has been established earlier that the source does not dissipate energy in this process, any variation in the mass of the source for other reasons will be independent of it, and therefore a mathematical development to describe it as it appears outside the source, can treat the source mass as effectively constant. For this reason therefore, the second order rate of change of generated volume outside the source will also be constant, and this fact may be used to set up the following describing equation

(3.11)

where

• W is the spatial volume generated by the gravitational source external to itself.
• K₀ is a dimensionless constant of proportionality.

And the solution of (3.11) must result in (3.10) as the linear velocity of expansion.

First, expand the second order differential in terms of its first and second order radial linear derivatives. e.g. with

(3.12)

this differentiates to

(3.13)

so that substitution into (3.11) gives

(3.14)

To solve this for the first order linear derivative, first insert, from [1] Eq(4.2) for the second order linear derivative, (see also Section 3.3.3).

(3.15)

and solve for the first order derivative of s thus

(3.16)

Now differentiate (3.16) with respect to time to give

(3.17)

Again inserting (3.15) and solving for K₀ gives

(3.18)

Substituting this back into (3.16) then finally gives

(3.19)

The desired result. This is the linear spatial expansion rate emanating from the gravitational source and which exhibits a spatial variation such that its first derivative with respect to time results in (3.15), the Acceleration Potential of D₁.

This small spatial velocity, by virtue of the criterion of existence in D₁, causes a corresponding reduction of the temporal velocity of all spatial points outside of the source thus, via effectively [2] Eq(2.5)

(3.20)

and therefore the time dilatation effect as shown by the final term of (3.20) in conjunction with (3.1).

The situation inside the source will now be considered.

• Generation of Time Dilatation Inside the Gravitational Source.

Eq(3.14) may also be used as the describing equation inside the source except that now, the right hand side is no longer a constant. The right hand side is therefore unknown in this situation and consequently (3.14) must be modified to accommodate this. Therefore initially write

(3.21)

where f() is some function involving g, s_i, the position of measurement, and m_g the mass of the source. A solution for the first linear derivative can still be found by using the method adopted above. First insert (3.7) for the second order derivative and solve for the first order to give

(3.22)

Differentiating with respect to time then gives

(3.23)

Inserting (3.7) again for the second order derivative permits a simple differential equation in f() to be set up thus

(3.24)

this equation is linear and can be solved with an integrating factor of 1/s_i. The result is

(3.25)

k is the constant of integration and can be determined by inserting (3.25) back into (3.21) and repeating the above process as follows, (or inserting (3.25) directly into (3.22)).

(3.26)

Inserting (3.7) and solving for the first order derivative gives

(3.27)

From the previous analysis for the situation outside the source, it is known that at the lower boundary, the surface of the source, when s = s_g

(3.28)

and for reason of continuity this also applies at the upper boundary for s_i. Therefore in (3.27) putting s_i = s_g and inserting (3.28) gives the value for k as

(3.29)

So that in (3.27) this finally gives

(3.30)

for the linear rate of spatial expansion inside a gravitational source. The derivative of (3.30) with respect to time then gives (3.7) for the gravitational acceleration within the source.

The relationship for the temporal rate and therefore time dilatation inside the source, is then obtained in the same manner as in the previous section, viz.

(3.31)

3.3.3. The Acceleration Potential of D₁.

In [1] the Acceleration Potential of D₁ was primarily quoted in terms of its temporal rate and therefore, that form will also be derived here. For the situation outside the source, from (3.20)

(3.32)

Differentiating (3.32) with respect to s gives

(3.33)

so that

(3.34)

The negative signs here have been inserted for directional correctness. Clearly (3.34) is identical with (3.23) and the analytical results of [1] Section 4.

A similar process applied inside the gravitational source from (3.31) yields for the Acceleration Potential, (3.6) thus,

(3.35)

It is important to note that the Acceleration Potential of D₁, both inside and outside the source is not in the true sense a time dependent variable. It exists as a spatial field due to the spatial variabilities of both v_is and v_s

Note that in (3.31) when s_i = 0, the value of u_i at the very centre of the source is given by

(3.36)

which has implications regarding a limiting value for s_g. This together with other "close proximity" effects will form the subject of a future paper.

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