Recovering Einstein’s mature view of gravitation: a dynamical reconstruction grounded in the equivalence principle

Abstract

The historical and conceptual foundations of General Relativity are revisited, putting the main focus on the physical meaning of the invariant ¿¿2 , the Equivalence Principle, and the precise interpretation of spacetime geometry. It is argued that Albert Einstein initially sought a dynamical formulation in which ¿¿2 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’s Principle to massive objects, the invariant ¿¿2 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’s second law in proper time and coincides with the weak-field limit of General Relativity in the harmonic gauge. This approach restores the operational meaning of Einstein’s theory: spacetime geometry represents dynamical relations between physical measurements, rather than the substance of spacetime itself. By deriving the gravitational modification of the invariant directly from the Weak Equivalence Principle, Fermat Principle and Lorentz invariance, this formulation clarifies the physical origin of the metric structure and resolves long-standing conceptual issues—such as the recurrent hole argument—while recovering all the empirical successes of General Relativity within a coherent and sound Machian framework.

"This research was founded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” grant number PID2021-123903NB-I00, by program Unidad de Excelencia María de Maeztu grant number CEX2020-001058-M and by AGAUR grant number project 2021-SGR-00171"

Peer Reviewed

Postprint (published version)