Conformal gravity: light deflection revisited and the galactic rotation curve failure

We show how conformal gravity (CG) has to satisfy a fine-tuning condition in order to be able to describe galactic rotation curves without the aid of dark matter, as suggested in the literature. If we interpret CG as a gauge natural theory, we can derive conservation laws and their associated superpotentials without ambiguities. We consider the light deflection of a point-like lens in CG and impose that the two Schwarzschild-like metrics with and without the lens at the origin of the reference frame are identical at infinite distances. The conservation law implies that the free parameter appearing in the linear term of the metric has to vanish, otherwise the two metrics are physically unaccessible from one other. This linear term is responsible for mimicking the role of dark matter in the standard model and it also appears in numerous previous investigations of gravitational lensing. Our result thus shows that the possibility of removing the presence of dark matter with CG relies on a fine-tuning condition on the parameter . We also illustrate why the results of previous investigations of gravitational lensing in CG largely disagree. These discrepancies derive from the erroneous use of the deflection angle definition adopted in general relativity, where the vacuum solution is asymptotically flat, unlike CG. In addition, the lens mass is identified with various combinations of the metric parameters. However, these identifications are arbitrary, because the mass is not a conformally invariant quantity, unlike the conserved charge associated to the conservation laws. Based on this conservation law and by removing the fine-tuning condition on , i.e. by setting , the difference between the metric with the point-like lens and the metric without it defines a conformally invariant quantity that can in principle be used for (1) a proper derivation of light deflection in CG, and (2) the identification of the lens mass with a function of the parameters and k of the Schwarzschild-like metric.