The inverse square law of gravitation—II

1—The general theory of relativity was largely developed owing to the failure of attempts to bring the law of gravitation within the scope of Lorentz transformations. Many such attempts were made in the early days of relativity. A notable one was contained in Poincare’s paper of 1906; the astronomical consequences of this were developed by de Sitter in 1911. Other attempts are associated with the names of Abraham (1912), Nordstrom (1912, 1913a ,b ), Behacker (1913) and others. These attempts did not incorporate the phenomenon of the expanding universe, then undiscovered. Consequently they did not give recognition to the circumstance that, in accordance with Mach’s principle, moving particles must be described in frames of reference associated with the actual distribution of matter-in-motion in the universe. By suitable graduation of the clocks carried by the fundamental observers associated with the frames defined by the extra-galactic nebular nuclei, these frames may be taken to be in uniform relative motion. The present paper then obtains a completely relativistic formulation of the law of gravitation by purely kinematic methods. The results are expressed in the first instance in terms oft -measures, and in accordance with previous work they yield an inverse square law of gravitation with a “ gravitational constant”y proportional to the timet reckoned from the natural origin of time. When they are transformed tor -measures,||y becomes a constanty 0, and the formulae are appropriate to classical orr -dynamics. Int -measures we use the observers’ flat private spaces; there proves to be no need to introduce any local curvature of space in the neighbourhood of a massive particle. Inr -measures we use a public hyperbolic static space in which the nebular nuclei appear at rest; the inverse square law of attraction then appears as an approximation whose accurate expression is the hyperbolic functiony 0 m 1 m 2 (ct 0 )2 sinh2 (λ /ct 0 ).