Cartographing Gravity: Operational Maps, the Conatus of Proper Time, and the Potentia of Matter

Abstract

Classical natural philosophy sought the natural law governing the motion of free bodies long before daring to construct the appropriate mathematical formalism. Following this longstanding methodological perspective, in the present work General Relativity is reconstructed from operational and dynamical principles, rather than taking spacetime geometry as its starting point. The conatus of a massive free particle is identified with its natural tendency to follow a worldline of stationary proper time, which, under the appropriate local conditions, corresponds to a maximum of its proper time, whereas gravitation is interpreted as the potentia of matter to modify the local rate of proper time. By requiring the exact validity of Galilean free fall, the weak-field static invariant is determined. The gravitational field of moving sources is subsequently obtained by Lorentz-transforming the elementary static solution and causally superposing the retarded contributions of infinitesimal source elements. The resulting linearized field equations coincide with those of General Relativity in the harmonic gauge. Finally, imposing general energy-momentum conservation together with the requirement of second-order field equations yields—within the class of theories satisfying the stated assumptions and by virtue of Lovelock’s theorem—the unique covariant completion represented by Einstein’s field equations. Within this construction, geometry is not postulated from the outset; instead, it emerges as the final mathematical synthesis of operational dynamics based on proper time, inertia, and free fall.