Weyl covariance, second clock effect and proper time in theories of symmetric teleparallel gravity

TL;DR

This paper proposes a new way to define parallel transport in symmetric teleparallel gravity to eliminate the second clock effect, suggesting this geometry as a consistent foundation for alternative gravity theories respecting Lorentz and Weyl symmetries.

Contribution

It introduces a Weyl-invariant parallel transport method in symmetric teleparallel geometry, removing the second clock effect and supporting its use for alternative gravity models.

Findings

Second clock effect can be removed with a new invariant parallel transport.

Symmetric teleparallel geometry is suitable for consistent alternative gravity theories.

Proper time and test body dynamics are analyzed within this framework.

Abstract

Just after Weyl's paper (Weyl in Gravitation und Elektrizit\"at, Sitzungsber. Preuss. Akad., Berlin, 1918) Einstein claimed that a gravity model written in a spacetime geometry with non-metricity suffers from a phenomenon, the so-called second clock effect. We give a new prescription of parallel transport of a vector tangent to a curve which is invariant under both of local general coordinate and Weyl transformations in order to remove that effect. Thus since the length of tangent vector does not change during parallel transport along a closed curve in spacetimes with non-metricity, a second clock effect does not appear in general, not only for the integrable Weyl spacetime. We have specially motivated the problem from the point of view of symmetric teleparallel (or Minkowski-Weyl) geometry. We also conclude that if nature respects Lorentz symmetry and Weyl symmetry, then the simplest geometry in which one can develop consistently alternative gravity models is the symmetric teleparallel geometry; $Q_{\mu\nu}\neq 0, \; T^\mu=0, \; R^\mu{}_\nu=0$. Accordingly we discuss the proper time, the orbit equation of a spinless test body and the Lagrangian for symmetric teleparallel gravity.