Recovering Einstein Mature View of Gravitation: A Dynamical Reconstruction Grounded in the Equivalence Principle

TL;DR

This paper revisits Einstein's original dynamical view of gravitation, reconstructing the invariant ds from the Equivalence Principle and extending it to dynamic, moving matter scenarios, aligning with weak-field General Relativity.

Contribution

It offers a novel reconstruction of gravitational dynamics grounded in Einstein's mature view, emphasizing the physical meaning of ds and the Equivalence Principle without relying on spacetime curvature.

Findings

Invariant ds derived from the Equivalence Principle.

Reproduces relativistic Newton's second law in proper time.

Matches weak field limit of General Relativity in harmonic gauge.

Abstract

The historical and conceptual foundations of General Relativity are revisited, putting the main focus on the physical meaning of the invariant ds, the Equivalence Principle, and the precise interpretation of spacetime geometry. It is argued that Albert Einstein initially sought a dynamical formulation in which ds encoded the gravitational effects, without invoking curvature as a physical entity. The now more familiar geometrical interpretation (identifying gravitation with spacetime curvature) gradually emerged through his collaboration with Marcel Grossmann and the adoption of the Ricci tensor in 1915. Anyhow, in his 1920 Leiden lecture, Einstein explicitly reinterpreted spacetime geometry as the state of a physical medium (an ether endowed with metrical properties but devoid of mechanical substance) thereby actually rejecting geometry as an independent ontological reality. Building upon this mature view, gravitation is reconstructed from the Weak Equivalence Principle, understood as the exact compensation between inertial and gravitational forces acting on a body under a uniform gravitational field. From this fundamental principle, together with an extension of Fermat Principle to massive objects, the invariant ds is obtained, first in the static case, where the gravitational potential modifies the flow of proper time. Then, by applying the Lorentz transformation to this static invariant, its general form is derived for the case of matter in motion. The resulting invariant reproduces the relativistic form of Newton second law in proper time and coincides with the weak field limit of General Relativity in the harmonic gauge.