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 nonzero 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. Moreover, we show that attempts to model rising rotation curves by fitting the coefficient of the quadratic term in the SdS metric individually for each galaxy are also precluded, since the scalar field equation of motion introduces an additional constraint relative to the vacuum case, such that the coefficient of the quadratic term in the SdS metric is most naturally interpreted as proportional to a global cosmological constant. We also comment briefly on how our analysis resolves the long-standing uncertainty regarding gravitational lensing in the MK metric.

Automated summary

This study reconsiders important issues in conformal gravity, including the predictions of the Mannheim-Kazanas vacuum solution in static, spherically symmetric systems and the rotation curve scenarios in different frames of reference. The authors demonstrate the importance of dynamically generating particle mass through interaction with a scalar field, resolving the frame dependence of physical predictions.