This paper investigates the derivation of the mathematical formulation of the Relativistic Domain theory of gravitation, in the form of Maxwell's equations of electromagnetic theory.

During the 19^th Century, several attempts were made to express the mathematics of gravitation in the form of a field theory. The earliest is perhaps that of Michael Faraday who between the years 1832 to 1859, made a number of experimental searches for a "gravitoelectric" field. He was unsuccessful, but remained convinced until he died that such a field existed, [1].

Subsequently, circa 1860 to 1870, James Clerk Maxwell also considered the possibility of expressing gravitation as a field theory, along the lines of his own electromagnetic formulation. However, for like sources he had difficulty reconciling the reversed direction in which the respective forces operate, i.e. like charges repel electrostatically, whereas like masses attract gravitationally. He also had difficulty with the concept of äction at a distance" as implied in Sir Isaac Newton's then extant theory of gravitation. Consequently, he was unable to achieve a satisfactory result, [2].

In the 1890's, Oliver Heaviside also considered this problem. He published a paper, [3], reviewed in [14], in which he showed that the divergence of a static gravitational field obeys Poisson's equation, and for which the curl is zero. He then argued that by analogy to Maxwell's equations, a "gravitomagnetic" field should also exist which had a divergence of zero and a non-zero curl proportional to the velocity of the mass causing this field. Although evidence of such gravitomagnetic effects were not available, Heaviside nevertheless continued the derivation by including time dependent effects of the fields, to produce exact analogies of Maxwell's electromagnetic equations. By this means the propagation velocity of gravitational waves was postulated to be the velocity of light.

All of the above investigations were carried out with Newton's linear theory of gravity which infers äction at a distance", i.e. no intervening medium of transmission. With the advent of Einstein's General Theory of Relativity, the problem became considerably more complex in view of the non-linearity of same. To express this theory in terms of Maxwell type equations, it was necessary to "linearise" Einstein's theory which meant that only weak gravitational sources were being investigated. Nevertheless, the consensus is, [4], [5], [6], [7] and [8], that the existence of a gravitomagnetic field is predicted from this theory, although its effects are likely to be immeasurably small, [8].

In this paper, the gravitational theory to be formulated in the form of Maxwell's equations is that of the Relativistic Domain theory of gravitation as developed in [9], [10] and [11]. This task is much simpler than that for the General Theory because the Relativistic Domain theory of gravitation is a completely linear theory and is already a field theory. This was briefly demonstrated in [11].

It is very important that to fully appreciate the subject content of this paper, the references [9], [10] and [11] are read first.

Also note that in this paper, vectors and unit vectors are represented by emboldened characters.