Revisiting local-scale invariant gravitational theory

TL;DR

The paper shows that local-scale symmetry (LSS) can be an exact, classical symmetry of gravitational laws in the CCSG theory if LSS is incorporated into the variational principle. By treating the metric $g_{\mu\nu}$ and the conformal scalar $\Phi$ as interdependent under infinitesimal Weyl transformations, the KG-type equation for $\Phi$ becomes the trace of the Einstein equations, allowing consistent coupling of arbitrarily massive matter with a nonzero trace $T^{\text{mat}}$ without breaking conformal invariance. An active (genuine) gauge-choice framework is developed, yielding a physically meaningful metric $\mathfrak{g}_{\mu\nu}=\frac{\Phi}{M_{\text{pl}}^2} g_{\mu\nu}$ and revealing a fifth force acting on timelike matter via $f^{\mu}=\frac{1}{2}h^{\mu\lambda}\partial_{\lambda}\ln\Phi$, while massless fields remain unaffected. GR emerges as a particular CCSG gauge, with the GR gauge $\Phi= M_{\text{pl}}^2$ reproducing standard Einstein equations; in other gauges, CCSG can yield distinct phenomenology, potentially addressing cosmological issues such as accelerated expansion without dark energy and modifying dark matter interpretations through the dynamics of $\Phi$. The work thus strengthens the case for LSS as a classical symmetry with tangible observational consequences and motivates further exploration of its cosmological and astrophysical implications.

Abstract

We revisit the conformally coupled scalar gravitational theory. This is the simplest local-scale invariant theory of gravity which is linear in the curvature scalar. We demonstrate that, if incorporate local-scale symmetry into the variational procedure, it is not required that the trace of the stress-energy tensor of the matter fields vanished for this symmetry to be preserved. The relevance of this result for the understanding of local-scale symmetry along with its physical consequences, is discussed.

Figures (1)

• Figure 1: Drawing of the field-space manifold ${\cal M}_\text{fields}$, that corresponds to the oval area in the plane $\Phi-g_{\mu\nu}.$ Each point in ${\cal M}_\text{fields}$ represents a different (vacuum) gravitational state. In the left figure the active approach to LST \ref{['conf-t']} is illustrated. In this case the conformal transformations represent a "real motion," i. e., a real change of the gravitational state ${\cal S}_{\bf g}:(\Phi,g_{\mu\nu})$$\rightarrow{\bar{\cal S}}_{\bf g}:(\bar{\Phi},\bar{g}_{\mu\nu})$. The passive standpoint on the LST is illustrated in the right figure. According to this approach the conformal transformations can be thought of as "rotations" of the coordinate system $R:(\Phi,g_{\mu\nu})$ in the plane $\Phi-g_{\mu\nu},$ which leaves invariant the gravitational state ${\cal S}_\mathfrak{g}:(\mathfrak{g}_{\mu\nu})$, where the invariant metric is given by \ref{['g-passive']}. Here $\Omega^w$ represents the conformal factor while $w$ is the conformal weight, which is $+2$ for the metric and $-2$ for the scalar field $\Phi$.