Equivalent Gravities and Equivalence Principle: Foundations and experimental implications

TL;DR

This paper examines the geometric trinity of gravity theories, highlighting their equivalences and differences in relation to the Equivalence Principle, and discusses implications for understanding spacetime and fundamental physics.

Contribution

It clarifies the nuanced relationship between GR, TEGR, and STEGR, especially regarding the role and fundamentality of the Equivalence Principle in these theories.

Findings

GR, TEGR, and STEGR are dynamically equivalent in many aspects.

TEGR and STEGR recover the strong form of the EP without relying on it as a fundamental principle.

The EP may be an emergent rather than fundamental feature of gravity theories.

Abstract

The so-called Geometric Trinity of Gravity includes General Relativity (GR), based on spacetime curvature; the Teleparallel Equivalent of GR (TEGR), which relies on spacetime torsion; and the Symmetric Teleparallel Equivalent of GR (STEGR), grounded in nonmetricity. Recent studies demonstrate that GR, TEGR, and STEGR are dynamically equivalent, raising questions about the fundamental structure of spacetime, the under-determination of these theories, and whether empirical distinctions among them are possible. The aim of this work is to show that they are equivalent in many features but not exactly in everything. In particular, their relationship with the Equivalence Principle (EP) is different. The EP is a deeply theory-laden assumption, which is assumed as fundamental in constructing GR, with significant implications for our understanding of spacetime. However, it introduces unresolved conceptual issues, including its impact on the nature of the metric and connection, its meaning at the quantum level, tensions with other fundamental interactions and new physics, and its role in dark matter and dark energy problems. In contrast, TEGR and STEGR recover the EP, in particular in its strong formulation, but do not rely on it as a foundational principle. The fact that GR, TEGR, and STEGR are equivalent in non-trivial predictions, but the EP is not necessary for TEGR and STEGR, suggests that it may not be a fundamental feature but an emergent one, potentially marking differences in the empirical content of the three theories. Thus, the developments within the Geometric Trinity framework challenge traditional assumptions about spacetime and may help to better understand some of the unresolved foundational difficulties related to the EP.