On the Geometric Meaning of General Relativity and the Foundations of Newtonian Cosmology
Jaume de Haro, Emilio Elizalde
TL;DR
This work reframes general relativity as a gauge theory with active diffeomorphism invariance, arguing that physical content lies in relational structure rather than an absolute spacetime substrate. By analyzing the hole argument and Einstein's covariance, the authors adopt a Machian viewpoint in which diffeomorphically related metrics are physically equivalent, and coordinates gain meaning only through the metric's relational content. They then construct a fully geometric, Newtonian cosmology using differential forms, the Hodge operator, and Lie derivatives, and show that the Newtonian derivation of the Friedmann equations and the evolution of linear perturbations on an expanding background reproduce the corresponding relativistic results at linear order. This framework provides a bridge between Newtonian intuition and relativistic geometry, clarifying how gauge principles shape cosmological dynamics and offering a relational, gauge-informed perspective potentially relevant to quantum-inspired ideas about emergent spacetime.
Abstract
The geometric foundations of General Relativity are revisited, with particular attention to its gauge invariance, as a key to understanding the true nature of spacetime. Beyond the common image of spacetime as a deformable 'fabric' filling the Universe, curvature is interpreted as the dynamic interplay between matter and interacting fields; a view already emphasized by Einstein and Weyl, but sometimes overlooked in the literature. Building on these tools, a Newtonian framework is reconstructed that captures essential aspects of cosmology, showing how classical intuition can coexist with modern geometric insights. This perspective shifts the focus from substance to relationships, offering a fresh magnifying glass through which to reinterpret gravitational dynamics and the large-scale structure of the Universe. The similarities of this approach with other recent, more ambitious ones carried out at the quantum level are quite remarkable.