How the non-metricity of the connection arises naturally in the classical theory of gravity

TL;DR

This paper explores how non-metricity naturally arises in the classical theory of gravity and proposes a reformulation of Einstein's theory that simplifies its structure and potentially explains large-scale cosmic effects.

Contribution

It introduces a new formalism treating non-metricity as fundamental, simplifying Einstein's equations and offering a framework to explain dark matter and dark energy phenomena.

Findings

Reformulation of Einstein's theory with non-metric connection

Simpler mathematical structure of gravitational equations

Potential explanation for dark matter and dark energy effects

Abstract

Spacetime geometry is described by two -- {\em a priori} independent -- geometric structures: the symmetric connection $\Gamma$ and the metric tensor $g$. Metricity condition of $\Gamma$ (i.e. $\nabla g = 0$) is implied by the Palatini variational principle, but only when the matter fields belong to an exceptional class. In case of a generic matter field, Palatini implies non-metricity of $\Gamma$. Traditionally, instead of the (1st order) Palatini principle, we use in this case the (2nd order) Hilbert principle, assuming metricity condition {\em a priori}. Unfortunately, the resulting right-hand side of the Einstein equations does not coincide with the matter energy-momentum tensor. We propose to treat seriously the Palatini-implied non-metric connection. The conventional Einstein's theory, rewritten in terms of this object, acquires a much simpler and universal structure. This approach opens a room for the description of the large scale effects in General Relativity (dark matter?, dark energy?), without resorting to purely phenomenological terms in the Lagrangian of gravitational field. All theories discussed in this paper belong to the standard General Relativity Theory, the only non-standard element being their (much simpler) mathematical formulation. As a mathematical bonus, we propose a new formalism in the calculus of variations, because in case of hyperbolic field theories the standard approach leads to nonsense conclusions.