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

Abstract

Just after Weyl’s paper (Weyl in Gravitation und Elektrizität, 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; $$Q_{\mu \nu }\ne 0, \; T^\mu =0, \; R^\mu {}_\nu =0$$ Q μ ν ≠ 0 , T μ = 0 , R μ ν = 0 . Accordingly we discuss the proper time, the orbit equation of a spinless test body and the Lagrangian for symmetric teleparallel gravity.

The non-minimally coupled symmetric teleparallel gravity with electromagnetic field

“…Nevertheless, in the models of STPG the metric and the affine (full) connection could represent gravitational and gauge fields, respectively, [5]. (ii) While the Riemannian geometry in which GR has been constructed is not invariant, symmetric teleparallel geometry is under a scale transformation [6]. Therefore, STPG model was later constructed and its spherically symmetric solutions were founded in the papers [7][8][9].…”