Why do we live in a Riemannian space time ?

TL;DR

This paper explores how a Riemannian spacetime naturally emerges from a more general metric affine gravity framework, linking the Einstein-Hilbert action to effective Levi-Civita gravity with broken Weyl symmetry.

Contribution

It demonstrates that breaking Weyl symmetry in metric affine gravity leads to a theory equivalent to general relativity, explaining why low-energy gravity is Riemannian.

Findings

Effective Levi-Civita gravity arises from metric affine gravity.

Weyl symmetry breaking yields a Riemannian spacetime.

Non-metricity and torsion can be removed, recovering Einstein gravity.

Abstract

We start from the pure Einstein-Hilbert action in Metric Affine Gravity, with the orthonormal metric. We get an effective Levi-Civita Dilaton gravity theory in which the Dilaton field is related to the scaling of the gravitational coupling. When the Weyl symmetry is broken the resulting Einstein-Hilbert term is equivalent to the Levi-Civita one, using the projective invariance of the model, the non-metricity and torsion may be removed, so that we get a theory perfectly equivalent to general relativity. This may explain why low energy gravity is described by a Riemannian connection.