Free fall in modified symmetric teleparallel gravity

TL;DR

This paper investigates the nature of free fall and geodesics in modified symmetric teleparallel gravity, revealing that despite differences from general relativity, test particles still follow metric geodesics due to the theory's structure.

Contribution

It demonstrates that in modified symmetric teleparallel gravity, test particles follow metric geodesics, clarifying the role of the equivalence principle and geodesic paths in this theory.

Findings

Test particles follow metric geodesics despite the theory's differences from GR.

The theory does not obey the equivalence principle in Weinberg's sense.

The geodesic equation involves the Levi-Civita connection.

Abstract

The status of the equivalence principle in modified symmetric teleparallel gravity is examined. In this theory, minimum length geodesics are distinct from autoparallel geodesics, that is, the ``shortest'' paths are not the ``straightest'' paths. We show that a standard argument that singles out metric geodesics in general relativity does not apply in modified symmetric teleparallel gravity. This is because the latter theory does not obey the equivalence principle in the sense of Weinberg. We argue, however, that the structure of the theory makes it inevitable that a freely falling test particle follows a shortest path, a geodesic of the metric. The geodesic equation that governs the motion of a freely falling test particle involves the Levi-Civita connection, not some other connection obtained by solving the connection field equations of the theory. This also has bearing on whether, under appropriate conditions, modified symmetric teleparallel gravity is fully equivalent to general relativity.