Recovering Einstein Mature View of Gravitation: A Dynamical Reconstruction Grounded in the Equivalence Principle
Jaume de Haro, Emilio Elizalde
TL;DR
This work revisits the foundations of General Relativity by arguing that spacetime curvature should be viewed as a mathematical encoding of dynamical relations, not a physical substrate. Building on the Equivalence Principle and Lorentz invariance, it reconstructs gravitation in the weak-field limit as a dynamical modification of the invariant $ds$, extending Fermat’s principle to massive bodies and deriving invariants that yield Newtonian dynamics and reproduce linearized GR in the harmonic gauge. It clarifies the historical path from the 1907–1912 ideas through the Entwurf and the adoption of curvature-based equations, and it discusses the hole argument and diffeomorphism invariance as resolutions that preserve a Machian, relational view of inertia. The paper further connects this dynamical interpretation to a geometric formalism where curvature is a language rather than an ontological reality and discusses the implications for theories like teleparallel gravity and Einstein’s ether, highlighting the continued relevance of a non-substantival spacetime perspective for understanding gravitation.
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.