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

Source: [How the non-metricity of the connection arises naturally in the classical theory of gravity](<https://www.sota2.com/research/paper/68fa468f1e209131ce921601>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T05:53:03.867Z

Authors: B. B\\k{a}k, J. Kijowski

Published: 2022-09-06T18:42:40.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2209.05936>)

[PDF](<https://arxiv.org/pdf/2209.05936.pdf>)