Revisiting local-scale invariant gravitational theory

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.