The conventional mathematical approach adopted here has, for the gravitational effects studied, produced results that are in agreement with those obtained from the General Theory of Relativity, and as observed for astronomical motion within the Solar System. However, the concepts from which this approach stems, e.g. that all matter must possess an Existence Velocity within a linear Relativistic Domain, as herein defined, differ from the concepts upon which the General Theory is based. That difference is in the manner in which the "gravitational field" of the source causes motion. In the General Theory this is defined as due to a "curvature" of the space-time continuum, proportional to, and in the direction of, the gravitational source. The "world line" of any gravitating mass is then said to be curved in the direction of the source.

In the development presented here, the space-time continuum of the Relativistic Domain D₁ is defined to be spatially and temporally linear, and gravitationally induced motion has been shown to be caused by an Acceleration Potential, generated by the gravitational source, proportional to its mass and inversely proportional to the distance from it. The effect of this Potential has in turn been shown to be augmented by the result of temporal dilatation produced by the source.

Because of this fundamental difference in which gravitational motion is caused, the concepts presented in this paper cannot be considered as merely a different mathematical formulation of gravitation, but should be considered as an alternative to that of the General Theory.

In both concepts the continuum of Pseudo-Euclidean Space-Time, is required to possess characteristics such that it interacts with matter energy to produce a new continuum, which in turn causes gravitationally induced motion. Also, both concepts incorporate temporal dilatation plus the small radial extension of distance from the source. Both of these latter effects are created within the body of the gravitational source and then extend beyond its geometric confines. In the General Theory temporal dilatation is treated as a consequence of gravitation while in the presentation here it is shown to contribute to the cause of it. Accordingly, in view of the great similarity of results in the two concepts, it is believed that the purported cause of gravitationally induced motion in the General Theory, the curvature of space-time, is a mis-interpretation, and this curvature is nothing more than the curvature of the trajectory of the gravitating mass, rather than of the space-time continuum in which it moves. If this is so, the consequence is that the continuum proper of the General Theory must therefore be identical to that of D₁. Some evidence for this is shown in Appendix D. Also, the derivation of Einstein's equation of planetary motion from the characteristics of D₁ further supports this opinion.

The primary cause of gravitational motion in the Domain D₁ is its Acceleration Potential. This Potential is in turn augmented by the spatially dependent temporal dilatation shown to exist in that Domain, and represented by the parameter u. The generation of these effects within a gravitational source in D₁ is therefore central to this theory, and a mechanism for it will be presented in a next paper.

The successful application of the concepts of Relativistic Domains in both a Pseudo-Euclidean Space-Time, (D₀, see [1]), and a Gravitational one (D₁), as well as representing a unification of mathematical analysis within them, has also established a unique link between them such that the variation of only one parameter, the temporal rate u, is sufficient to transform one into the other. In fact this can be completely generalised with the result that both the Pseudo-Euclidean and Gravitational space-times are merely particular cases of a potentially infinite number of hypothetical Relativistic Domains in which the parameter u can be of any form. Pseudo-Euclidean Space-Time would perhaps retain its special character insofar as its temporal rate possessed the unique value of unity. The Domain D₁ is of course equally if not more important because it describes the space time continuum of the gravitational effects within Solar System and beyond.

Throughout the text there are a number of results that can be taken further. Of particular interest are the questions of inertial mass, naturally generated kinetic energy and effects inside the geometric constraints of the gravitational source itself. The same applies when given a set of special conditions outside the source. Most particular in this latter respect is the situation, possibly hypothetical, where the geometrical radius of a gravitational source is of the same order of magnitude as twice the gravitational radius. At the point of equivalence the temporal rate of D₁ becomes zero, i.e. time stops.

Finally, it should be remembered that, for a single isolated gravitational source, despite the agreement of the results in this paper with those of the General Theory, and with observed planetary motion in the Solar System, all such results, strictly from the point of view of mathematical rigour, are approximations. This is so because in all cases the analysis assumes that the gravitational source is stationary, i.e. not affected by the gravitational influence of the gravitating mass. No matter how small the latter, there will always be an effect on the larger mass. Therefore the results here and in the General Theory are only approximately correct in the case where the gravitational source is much larger than the gravitating mass. Where sizes are comparable it is necessary to take account of the mutual gravitational attraction, which effectively results in a new Relativistic Domain, D₂.

The appendices to the main text provide additional evidence that gravitational motion within the Relativistic Domain D₁ is identical to that postulated in the General Theory.