A generalized Weyl structure with arbitrary non-metricity

Abstract

A Weyl structure is usually defined by an equivalence class of pairs (g, ω) related by Weyl transformations, which preserve the relation ∇g = ω ⊗ g, where g and ω denote the metric tensor and a 1-form field. An equivalent way of defining such a structure is as an equivalence class of conformally related metrics with a unique affine connection Γω, which is invariant under Weyl transformations. In a standard Weyl structure, this unique connection is assumed to be torsion-free and have vectorial non-metricity. This second view allows us to present two different generalizations of standard Weyl structures. The first one relies on conformal symmetry while allowing for a general non-metricity tensor, and the other comes from extending the symmetry to arbitrary (disformal) transformations of the metric.1 The usual definition of the non-metricity tensor is Q = ∇g.

Conformally invariant proper time with general non-metricity

“…We are here concerned with finding a definition of proper time in scale-invariant spacetimes with a general form of the non-metricity tensor Q, i.e. in generalized Weyl space-times in the sense of [40]. In order to do so, we will start with the same definition as presented by Perlick in [41], which can be generalized to an arbitrarily general space-time in a straightforward way:…”

“…In Ref. [40], it was found that in non-Riemannian manifolds, under a scale transformations of the metric, the scale invariance of the affine structure implies certain transformation properties of the non-metricity tensor under scale transformations. In fact, one can verify that the simultaneous transformations g = e φ g andQ = e φ ( Q + dφ ⊗ g) (8) leave invariant the affine connection (as scale transformations should), where φ is any arbitrary smooth scalar function.…”

“…This fact fostered the interest in Weyl geometries, since they provide a natural way of introducing scale transformations without changing the affine structure (which cannot be done in Riemannian geometries). However, although non-metricity is necessary for defining scale transformations that do not change the affine structure, the usual restriction on the non-metricity to be Weyl-like is unnecessary, and this can be achieved with general non-metricity [40]. In this case only the vectorial irreducible components of non-metricity transform as a gauge 1-form, while the tensorial irreducible components transform trivially by a conformal factor.…”

“…In this case only the vectorial irreducible components of non-metricity transform as a gauge 1-form, while the tensorial irreducible components transform trivially by a conformal factor. 2 Motivated by the discussion of Perlick [41] about how to construct a suitable Weyl-invariant notion of proper time in Weyl-invariant space-times that reduces to the standard definition of proper time in the Riemannian limit, and following [42] in its analysis of the physical role played by the Weyl 1-form, we will be concerned about finding a suitable definition of proper time that respects scale invariance in the presence of arbitrary non-metricity (or generalized Weyl invariance in the sense of [40]), and also with the physical consequences of having non-trivial non-metricity if physical time was described by this definition. To that end we will generalize the parametrization for generalized proper time found in [43] to the case of arbitrary non-metricity and find the existence of a conformally invariant second clock effect related to an arbitrary non-metricity tensor.…”

Conformally invariant proper time with general non-metricity

“…This fact fostered the interest in Weyl geometries, since they provide a natural way of introducing scale transformations without changing the affine structure (which cannot be done in Riemannian geometries). However, although non-metricity is necessary for defining scale transformations that do not change the affine structure, the usual restriction on the non-metricity to be Weyl-like is unnecessary, and this can be achieved with general non-metricity [41]. In this case only the vectorial irreducible components of non-metricity transform as a gauge 1-form, while the tensorial irreducible components transform trivially by a conformal factor 3 .…”