Some Thoughts on Geometries and on the Nature of the Gravitational Field

TL;DR

This paper explores various geometric representations of gravitational fields, demonstrating their equivalence and suggesting that the geometric structure of spacetime is conventional, which prompts further investigation into the physical origin of gravity.

Contribution

It introduces a unified framework showing gravitational fields can be represented by different geometries or as a field in Minkowski spacetime, with explicit Lagrangian and Maxwell-like equations.

Findings

Different geometric structures can represent the same gravitational field.

The field equations are equivalent to Einstein's equations.

The geometric structure of spacetime is conventional and not unique.

Abstract

In this paper we show how a gravitational field generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel and non null nonmetricity spacetimes) or that we even can dispense all those geometrical structures and simply represent the gravitational field as a field in the Faraday's sense living in Minkowski spacetime. The explicit Lagrangian density for this theory is given and the field equations (which are Maxwell's like equations) are shown to be equivalent to Einstein's equations. Some examples are worked in detail in order to convince the reader that the geometrical structure of a manifold (modulus some topological constraints) is conventional as already emphasized by Poincare long ago, and thus the realization that there are disctints geometrical representations (and a physical model related to a deformation of the continuum supporting Minkowski spacetime) for any realistic gravitational field strongly suggests that we must investigate the origin of its physical nature. We hope that this paper will convince readers that this is indeed the case.