Vanishing Noether current in Weyl invariant gravities

Abstract

We revisit the issue that the Noether current associated with a local scale symmetry, or equivalently the Weyl symmetry, identically vanishes. Based on only the second Noether theorem for a local symmetry, we prove that the Noether current associated with the Weyl symmetry is in general vanishing in any Weyl invariant gravitational theories in four-dimensional Riemannian geometry. We also clarify the reason: The Weyl transformation is nondynamical in the sense that it does not contain the derivative term of the transformation parameter as opposed to the conventional gauge transformation.