Abstract
A Weyl structure is usually defined by an equivalence class of pairs (mathbf{g}, {varvec{omega }}) related by Weyl transformations, which preserve the relation nabla mathbf{g}={varvec{omega }}otimes mathbf{g}, where mathbf{g} and {varvec{omega }} 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 Gamma _{({varvec{omega }})}, 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.
Highlights
General relativity and a major part of alternative theories of gravity are constructed on the assumption that spacetime is a differential manifold endowed with a metric and a Riemannian connection, i.e., a metric-compatible connection
This construction is fundamentally important in order to define scalars and a dynamics that is manifestly covariant under general coordinate transformations, which encompasses the main symmetry presented in gravitational theories in the past 100 years
As it follows from the above considerations, the particular form of the non-metricity condition given by the first axiom is conceptually independent of the invariance of the physics under conformal transformations of the metric
Summary
General relativity and a major part of alternative theories of gravity are constructed on the assumption that spacetime is a differential manifold endowed with a metric and a Riemannian connection, i.e., a metric-compatible connection. Weyl presents two axioms for this geometry, the first one states that parallel displacement of vectors defines what he calls a similarity map, i.e., a linear map that induces a conformal transformation in the inner product of any two vectors This axiom implies that for a given vector that is parallel displaced along any curve, the derivative of the vector’s norm is proportional to the norm itself, and the proportionality function is given by some 1-form calculated along the curve, which is equivalent to require vectorial non-metricity of the form Q = ω ⊗ g. It can be seen that a gauge transformation of this 1-form field can always compensate any conformal transformation of the metric As it follows from the above considerations, the particular form of the non-metricity condition given by the first axiom is conceptually independent of the invariance of the physics under conformal transformations of the metric.
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