From Universal Gravitation to Relativity: How Newton's Mutual Attraction of All Bodies Necessitated the Special and General Theories

This paper traces the specific incompatibility that forced the reformulation of gravitational theory in the early twentieth century, and identifies precisely why two successive corrections were required rather than one. Newton's law of universal gravitation asserts that every body attracts every other body by a single force law, and it was formulated within Galilean relativity, whose linear coordinate transformation holds space and time to be separate and absolute, entails an unbounded velocity-addition rule, and places no limit whatsoever on how quickly an influence may propagate between bodies. Newton's law inherits that absence of a limit as an implicit assumption of instantaneous action at a distance. When Maxwell's electromagnetic theory established a finite speed of light, and the Michelson-Morley experiment established that this speed is the same for every observer, the assumption became untenable. Einstein's 1905 special theory of relativity resolved the kinematic half of the problem by replacing the Galilean transformation with the Lorentz transformation, but special relativity is restricted to inertial frames, says nothing about gravitation, and is in fact incompatible with an instantaneous gravitational force. A second correction was therefore required, and Einstein's 1915 general theory supplied it by replacing the instantaneous force with spacetime curvature sourced by mass-energy and propagating at exactly the speed of light, while recovering Newton's law in the weak-field limit. This account was confirmed observationally in 2017, when gravitational and electromagnetic signals from the neutron-star merger GW170817 arrived 1.7 seconds apart after travelling 130 million light years. The paper then traces the development of the tensor calculus this completion required, from Riemann through Ricci-Curbastro and Levi-Civita to Einstein's collaboration with Grossmann, and evaluates even-handedly the claim advanced elsewhere in this series that three-dimensional vector-calculus alternatives are better suited to at least some purposes.