Classical Weyl Transverse Gravity

TL;DR

This paper explores the classical properties of Weyl transverse gravity, establishing its equivalence with other theories, analyzing the cosmological constant, and examining black hole and cosmological solutions.

Contribution

It clarifies the classical equivalence among conformally-invariant scalar tensor gravity, Einstein's gravity, and WTDiff gravity, and investigates their implications for cosmology and black hole solutions.

Findings

Cosmological constant is an integration constant in WTDiff gravity.

Weyl symmetry is a 'fake' symmetry with a vanishing Noether current.

Black hole solutions are valid only in Cartesian coordinates in WTDiff gravity.

Abstract

We study various classical aspects of the Weyl transverse (WTDiff) gravity in a general space-time dimension. First of all, we clarify a classical equivalence among three kinds of gravitational theories, those are, the conformally-invariant scalar tensor gravity, Einstein's general relativity and the WTDiff gravity via the gauge fixing procedure. Secondly, we show that in the WTDiff gravity the cosmological constant is a mere integration constant as in unimodular gravity, but it does not receive any radiative corrections unlike the unimodular gravity. A key point in this proof is to construct a covariantly conserved energy-momentum tensor, which is achieved on the basis of this equivalence relation. Thirdly, we demonstrate that the Noether current for the Weyl transformation is identically vanishing, thereby implying that the Weyl symmetry existing in both the conformally-invariant scalar tensor gravity and the WTDiff gravity is a "fake" symmetry. We find it possible to extend this proof to all matter fields, i.e. the Weyl invariant scalar, vector and spinor fields. Fourthly, it is explicitly shown that in the WTDiff gravity the Schwarzshild black hole metric and a charged black hole one are classical solutions to the equations of motion only when they are expressed in the Cartesian coordinate system. Finally, we consider the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology and provide some exact solutions.