Is there any Trinity of Gravity, to start with?

TL;DR

This paper critically examines the concept of a geometric trinity of gravity, arguing that alternative formulations like torsion and nonmetricity are just different languages for the same physics, not genuinely new theories.

Contribution

It clarifies that the so-called geometric trinity of gravity does not introduce new physics but merely rephrases the same gravitational phenomena in different geometric terms.

Findings

Standard approaches do not involve additional structures beyond the metric.

Alternative models like teleparallel gravity are physically equivalent to GR.

Rephrasing gravity in different geometric languages does not lead to new physics.

Abstract

In recent years, it has been rather fashionable to talk about geometric trinity of gravity. The main idea is that one can formally present the gravity equations in different terms, those of either torsion or nonmetricity instead of curvature. It starts from a very erroneous claim that the Levi-Civita connection, and therefore the (pseudo-)Riemannian geometry itself, are nothing but an arbitrary choice. The point is that, as long as we admit the need of having a metric for describing gravity, the standard approach does not involve any additional independent geometric structures on top of that. At the same time, any other metric-affine model does go for genuinely new stuff. In particular, the celebrated teleparallel framework introduces a notion of yet another parallel transport which is flat. It gives us curious new ways of modifying gravity, even though very often quite problematic. However, in GR-equivalent models, we only get a new language for describing the same physics, in terms of absolutely unobservable and unpredictable geometrical inventions. For sure, one can always safely create novel constructions which do not influence the physical equations of motion, but in itself it does not make much sense and blatantly goes against the Occam's razor.