Conformal gravity: light deflection revisited and the galactic rotation
  curve failure
  We show how Conformal Gravity (CG) has to satisfy a fine-tuning condition to
describe the rotation curves of disk galaxies without the aid of dark matter.
Interpreting CG as a gauge natural theory yields conservation laws and their
associated superpotentials without ambiguities. We consider the light
deflection of a point-like lens and impose that the two Schwarzschild-like
metrics with and without the lens are identical at infinite distances from the
lens. The energy conservation law implies that the parameter $\gamma$ in the
linear term of the metric has to vanish, otherwise the two metrics are
physically inaccessible from each other. This linear term is responsible to
mimic the role of dark matter in disk galaxies and gravitational lensing
systems. Our analysis shows that removing the need of dark matter with CG thus
relies on a fine-tuning condition on $\gamma$. 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 energy conservation law. Based on this
conservation law and by removing the fine-tuning condition on $\gamma$, i.e. by
setting $\gamma=0$, the energy 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 $\beta$ and $k$ of the Schwarzschild-like metric.
