In the General Theory of Relativity, the cause of gravitational motion is purported to be a curvature of space-time, due to the presence of a gravitational source in the form of ponderable matter. The degree of curvature is stated to be proportional to the mass of the gravitational source, which then determines the strength of the gravitational effect. However, the precise mechanism that causes this space-time distortion by the source is not described. Clearly, the characteristics of the continuum would have to be such that they were capable of this distortion, but again these are not addressed. Only the geometry of the curvature is described by means of a mathematical concept - the geodesics. It is considered that for a full understanding of gravity, a complete physical interpretation should be available, as well as a mathematical one.

In this series of papers, the proposed gravitational theory of the Relativistic Space-Time Domain D₁, an alternative to that in the General Theory, the same requisites are of course just as applicable. In this Domain, D₁, space-time is not curved, but completely linear along all four axes, yet, as shown in [1], still exhibits identical gravitational characteristics to those in the General Theory. The cause of gravitational motion in D₁, as shown in [1], is the Acceleration Potential extant within the Domain. This, in turn, is augmented by the time dilatation effect produced by the gravitational source in concert with the velocity of the gravitating body. Therefore, these two parameters, the Acceleration Potential and time dilatation, are central to the gravitational theory of D₁, and a complete understanding of it can only be achieved if the manner in which these parameters are generated by the source is known.

It is the purpose of this paper to provide both a mathematical development for, and a physical interpretation of, the mechanism generating both time dilatation and the Acceleration Potential of D₁, and thereby attempt to enhance this understanding. Because both of these parameters are generated by the same mechanism, it is not significant which is used to develop the characteristics of it, and because it is a simpler process to use the time dilatation effect for this purpose, that is the approach adopted.

Prior to this however, it is necessary to inject a note of caution. The development that is to be presented here, involves a new idea concerning a direct relationship between space, time and matter. This relationship is one for which there does not exist, at the moment, any supporting observational nor experimental evidence. Nor does a means of conducting suitable observational or experimental research to verify the hypothesis seem clear. Consequently, it must be regarded as speculative, and may be the subject of considerable refinement.

In the interests of brevity, unless necessary for complete clarity, a parameter will only be defined in this paper, if it has not previously been so in either [1], [2] or [4], with which familiarity is assumed.

2  The Nature of Time in a Relativistic Domain.

2.1  Definitions.

As well as a preamble to the further developments presented herein, by providing a more in depth treatment on how time is generated in a Relativistic Domain, this section also augments certain definitions concerning time, given in [1] and [2].

Note that the symbol D has been introduced in this paper to represent a parametric elemental. This is to enable a distinction to be made between an element of time, and the symbology for temporal rates.

In any Relativistic Domain, time is generated as a result of motion along the temporal axis. In the Domain D₀, Pseudo-Euclidean Space-Time, with a temporal velocity of c, Fig 2.1 shows that motion on the temporal axis over an elemental distance Dx₀, creates an element of time Dt in the spatial part of the Domain.

Fig. 2.1 - Generation of Time in Pseudo-Euclidean Space-Time D₀.

So that in D₀.

Dt= Dx0 c

(2.1)

In the Domain D₁, the gravitational space-time, with a temporal velocity of cu, Fig. 2.2 shows that motion on the temporal axis over an elemental distance Dx₀, creates an element of time Dt in the spatial part of the Domain.

Fig. 2.2 - Generation of Time in Gravitational Space-Time, D₁.

So that in D₁.

Dt = Dx0 cu

(2.2)

and therefore from (2.1) and (2.2)

Dt Dt = 1 u

(2.3)

Eq.(2.3) is the ratio of an element of time in D₁ to that in D₀. Because u < 1, this shows that an element of time in D₁ is longer than an element of time in D₀. However, because both elements have the same value, time in D₁ passes more slowly than it does in D₀. Consequently, from (2.3)

1  sec  in  D0     º     u  sec  in  D1

(2.4)

Determination of the rate of passage of time in D₁ compared to D₀ is accomplished via the following simple exercise.

In moving from one specifically defined point to another on the temporal axis, a body in the spatial part of D₁ will experience a passage of time of

dt = u  sec

(2.5)

During the above event the amount of time that passes in D₀ is, by virtue of (2.4)

dt = 1  sec

(2.6)

Therefore the rate of passage of time in D₁, compared to D₀ is from (2.5) and (2.6)

dt dt = u

(2.7)

This was also derived in [1] via a slightly different process where u was thereby defined as the Temporal Rate of D₁.

Finally, in the gravitational space-time D₁, it is seen that because the passage of time is a function of the parameter u, which has, in [1], been shown to be a dimensionless function of the spatial radial distance s. The passage of time in D₁ is therefore also a function of this spatial distance.

2.2  Temporal Flow.

In previous papers [1] and [2], temporal velocity was always referred to as the velocity of a spatial point, or material object, on the temporal axis. However, because temporal velocities in D₁ are a function of the spatial variable s, if the above were the case, it would mean that different points in the spatial part of the Domain would exist at different locations on the temporal axis. Furthermore, as the temporal motion continued, different spatial points would move further apart on this axis. This is equivalent to 'travelling in time' which is obviously not within observed experience. Consequently, the above view of the nature of temporal velocity needs to be more precisely defined such that it satisfies both observed events and the requirements of gravitational theory. Consider Fig. 2.3.

Fig. 2.3 - The Nature of Temporal Flow

It is proposed that temporal flow is a motion of the temporal dimension from the 'future' to the 'past', relative to a three dimensional spatial plane of existence on which all material bodies co-exist. In the future part of the dimension, the velocity of temporal flow is constant throughout at the value c. As the flow passes the spatial plane of existence, if a material body is present, the flow is altered in the manner described later in this paper such that the velocity is reduced. Thus, the velocity of the flow in the past region of the temporal dimension is no longer constant but becomes, in accordance with the parameter u, a function of the spatial distance from the source in all radial directions.

In this way all material bodies and spatial points in the spatial part of the Domain, can remain on the same three dimensional spatial plane of existence, but still experience a spatial variability in the rate of the passage of time, i.e. time dilatation.

It is obvious that, in the absence of material bodies on the plane of existence, i.e. in D₀, temporal flow is at the constant velocity c throughout the entire Domain both future and past. It should also be remembered that should a material body gain a spatial velocity, then by virtue of the criterion of existence in the Domain, this will also result in a reduction in the rate of passage of time experienced by that body. This however, only affects the material body itself and not its internal or surrounding space.

The proposed nature of temporal flow presented above represents a more acceptable view in satisfying both the mathematical requirements of gravitation and observable facts. However, because the further mathematical analysis to be pursued is not compromised, and in the interests of consistency with previous papers, temporal velocity will continue in this paper to be referred to as that of material bodies and/or spatial points in the temporal direction.

The ensuing sections of this paper present a logical and mathematical description of a mechanism by which time dilatation, and subsequently the gravitational Acceleration Potential, is generated by a gravitational source.

G3 Version 1.2.3
Ó P.G.Bass, August 2009