Gauge Choice in Conformal Gravity
  In a recent paper (MNRAS 458, 4122 (2016)) K. Horne examined the effect of a
conformally coupled scalar field (referred to as Higgs field) on the
Mannheim-Kazanas metric $g_{\mu\nu}$, i.e. the static spherically symmetric
metric within the context of conformal gravity (CG), and studied its effect on
the rotation curves of galaxies. He showed that for a Higgs field of the form
$S(r) = S_0 a/(r + a)$, where $a$ is a radial length scale, the equivalent
Higgs-frame Mannheim-Kazanas metric $\tilde{g}_{\mu\nu} = \Omega^2 g_{\mu\nu}$,
with $\Omega = S(r)/S_{0}$, lacks the linear $\gamma r$ term, which has been
employed in the fitting of the galactic rotation curves without the need to
invoke dark matter. In this brief note we point out that the representation of
the Mannheim-Kazanas metric in a gauge where it lacks the linear term has
already been presented by others, including Mannheim and Kazanas themselves,
without the need to introduce a conformally coupled Higgs field. Furthermore,
Horne argues that the absence of the linear term resolves the issue of light
bending in the wrong direction, i.e. away from the gravitating mass, if $\gamma
r > 0$ in the Mannheim-Kazanas metric, a condition necessary to resolve the
galactic dynamics in the absence of dark matter. In this case we also point out
that the elimination of the linear term is not even required because the sign
of the $\gamma r$ term in the metric can be easily reversed by a simple gauge
transformation, and also that the effects of this term are indeed too small to
be observed.
