Weyl symmetry without the traceless condition

TL;DR

This work argues that Weyl (conformal) invariance in curved spacetime does not universally require the matter stress-energy trace to vanish. By adopting conformal general relativity with a dynamical scalar $\phi$ and an active Weyl rescaling framework, it shows that if timelike masses transform as $m=\kappa\,\phi$ (i.e., $m\to\hat{m}=\Omega^{-1}m$) and the fluid energy density scales as $\rho\to\hat{\rho}=\Omega^{-4}\rho$, then the matter action remains form-invariant and the $\phi$ equation of motion acquires a source term proportional to $T^{(m)}$, eliminating the need for $T^{(m)}=0$. The paper provides two derivations—one via standard variational procedures and another via Ward identities—that yield the same result, demonstrating that Weyl symmetry can be maintained with nonzero $T^{(m)}$. This reinterprets the role of the traceless condition, introduces a fifth-force coupling through $f_{\mu}=(\partial_{\mu}\phi)/\phi\,T^{(m)}$, and explores the cosmological consequences of an infinite set of Weyl-gravitational states, with potential implications for dark energy and dark matter. The findings invite a reconsideration of conformal invariance in gravitational theories and motivate further work on observational signatures and the Weyl anomaly in this framework.

Abstract

We show that the requirement that the trace of the stress-energy tensor of matter must vanish if invariance under Weyl rescalings is a symmetry of a given gravitational theory is not universal. This requirement holds wherever the masses of timelike fields are constant parameters that are not transformed by Weyl rescalings, or if the energy density of perfect fluids transforms as $ρ\rightarrowΩ^{-3}ρ$. In contrast, if the masses of timelike fields are point-dependent quantities transforming under the Weyl rescalings as $m\rightarrowΩ^{-1}m$, and the energy density of perfect fluids transforms as $ρ\rightarrowΩ^{-4}ρ$, the Weyl symmetry does not require the vanishing of the trace of the matter SET. This result is demonstrated in two different ways. In consequence, any matter fields, regardless of whether the trace of their stress-energy tensor vanishes or not, can be coupled to gravity. The phenomenological consequences of the novel result are drawn.