Monochromatic plane-fronted waves in conformal gravity are pure gauge

TL;DR

This paper demonstrates that monochromatic plane-fronted gravitational waves in conformal gravity are pure gauge artifacts with no energy, challenging the assumption that such waves are physically meaningful in this theory.

Contribution

It shows that under certain gauge conditions, these waves are reducible to Minkowski space, indicating they are not genuine physical solutions but coordinate artifacts.

Findings

Plane-fronted waves are pure gauge and carry no energy.

Standard perturbative gravitational waves do not exist in this conformal gravity setting.

More general ansatz yields energy-carrying waves, unlike the simple plane waves.

Abstract

We consider plane-fronted, monochromatic gravitational waves on a Minkowski background, in a conformally invariant theory of general relativity. By this we mean waves of the form: $g_{\mu\nu}=\eta_{\mu\nu}+\epsilon_{\mu\nu}F(k\cdotx)$, where $\epsilon_{\mu\nu}$ is a constant polarization tensor, and $k_\mu$ is a lightlike vector. We also assume the coordinate gauge condition $|g|-1/4\partial_\tau(|g|1/4g^{\sigma\tau})=0$ which is the conformal analog of the harmonic gauge condition $g^{\mu\nu}\Gamma_{\mu\nu}^\sigma=-|g|-1/2\partial_\tau(|g|1/2g^{\sigma\tau})=0, where $\det[g_{\mu\nu}]\equivg$. Requiring additionally the conformal gauge condition $g=-1$ surprisingly implies that the waves are both transverse and traceless. Although the ansatz for the metric is eminently reasonable when considering perturbative gravitational waves, we show that the metric is reducible to the metric of Minkowski space-time via a sequence of coordinate transformations which respect the gauge conditions, without any perturbative approximation that \in{\mu}{\nu} be small. This implies that we have, in fact, exact plane-wave solutions; however, they are simply coordinate/conformal artifacts. As a consequence, they carry no energy. Our result does not imply that conformal gravity does not have gravitational wave phenomena. A different, more generalized ansatz for the deviation, taking into account the fourth-order nature of the field equation, which has the form $g_{\mu\nu}=\eta_{\mu\nu}+B_{\mu\nu}(n\cdotx)G(k\cdotx)$, indeed yields waves which carry energy and momentum [P.D. Mannheim, Gen. Relativ. Gravit. 43, 703 (2010)]. It is just surprising that transverse, traceless, plane-fronted gravitational waves, those that would be used in any standard, perturbative, quantum analysis of the theory, simply do not exist.