A one-parameter conformal invariance of Maxwell's equations allows the wavelengths of electromagnetic waves to change as they propagate, and do so even in otherwise field-free space. This produces an ambiguity in interpretations of stellar red shifts. Experiments that will determine the value of the group parameter, and thereby remove the ambiguity, are proposed. They are based on an analysis of the anomalous frequency shifts uncovered in the Pioneer 10 and 11 spacecraft studies, and physical interpretation of an isomorphism discovered by E. L. Hill. If the group parameter is found to be non-zero, Hubble's relations will have to be reinterpreted and space-time metrics will have to be altered. The cosmological consequences of the transformations are even more extensive because, though they change frequencies, they do not alter the energy and momentum conservations laws of classical and quantum-electrodynamical fields established by Cunningham and by Bialynicki-Birula.