Conformal gravity does not predict flat galaxy rotation curves

We reconsider the widely held view that the Mannheim–Kazanas (MK) vacuum
solution for a static, spherically-symmetric system in conformal gravity (CG)
predicts flat rotation curves, such as those observed in galaxies, without the
need for dark matter. This prediction assumes that test particles have fixed
rest mass and follow timelike geodesics in the MK metric in the vacuum region
exterior to a spherically-symmetric representation of the galactic mass
distribution. Such geodesics are not conformally invariant, however, which
leads to an apparent discrepancy with the analogous calculation performed in
the conformally-equivalent Schwarzschild-de-Sitter (SdS) metric, where the
latter does not predict flat rotation curves. This difference arises since the
mass of particles in CG must instead be generated dynamically through
interaction with a scalar field. The energy-momentum of this required scalar
field means that, in a general conformal frame from the equivalence class of CG
solutions outside a static, spherically-symmetric matter distribution, the
spacetime is not given by the MK vacuum solution. A unique frame does exist,
however, for which the metric retains the MK form, since the scalar field
energy-momentum vanishes despite the field being non-zero and radially
dependent. Nonetheless, we show that in both this MK frame and the Einstein
frame, in which the scalar field is constant, massive particles follow timelike
geodesics of the SdS metric, thereby resolving the apparent frame dependence of
physical predictions and unambiguously yielding rotation curves with no flat
region. We also comment on how our analysis resolves the long-standing
uncertainty regarding gravitational lensing in the MK metric. (Abridged)
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