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We consider plane-fronted, monochromatic gravitational waves on a Minkowskibackground, in a conformally invariant theory of general relativity. By this wemean 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 alightlike vector. We also assume the coordinate gauge condition$|g|-1/4\partial_\tau(|g|1/4g^{\sigma\tau})=0$ which is the conformal analog ofthe 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 gaugecondition $g=-1$ surprisingly implies that the waves are both transverse andtraceless. Although the ansatz for the metric is eminently reasonable whenconsidering perturbative gravitational waves, we show that the metric isreducible to the metric of Minkowski space-time via a sequence of coordinatetransformations which respect the gauge conditions, without any perturbativeapproximation that \in{\mu}{\nu} be small. This implies that we have, in fact,exact plane-wave solutions; however, they are simply coordinate/conformalartifacts. As a consequence, they carry no energy. Our result does not implythat conformal gravity does not have gravitational wave phenomena. A different,more generalized ansatz for the deviation, taking into account the fourth-ordernature of the field equation, which has the form$g_{\mu\nu}=\eta_{\mu\nu}+B_{\mu\nu}(n\cdotx)G(k\cdotx)$, indeed yields waveswhich carry energy and momentum [P.D. Mannheim, Gen. Relativ. Gravit. 43, 703(2010)]. It is just surprising that transverse, traceless, plane-frontedgravitational waves, those that would be used in any standard, perturbative,quantum analysis of the theory, simply do not exist.

Monochromatic plane-fronted waves in conformal gravity are pure gauge