  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.
