Conformal gravity does not predict flat galaxy rotation curves

Abstract

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. The conformal equivalence of the MK and Schwarzschild-de-Sitter (SdS) metrics, where the latter does not predict flat rotation curves, raises concerns that the prediction in the MK frame may be a gauge artefact. This ambiguity arises from assuming that, in each frame, test particles have fixed rest mass and follow timelike geodesics, which are not conformally invariant. The mass of such particles must instead be generated dynamically through interaction with a scalar field, the energy-momentum of which means that the spacetime outside a static, spherically-symmetric matter distribution in CG is, in general, not given by the MK vacuum solution. A unique solution does exist, however, for which the scalar field energymomentum vanishes and the metric retains the MK form. Nonetheless, we show that in both the Einstein and MK frames of this solution, in which the scalar field is constant or radially-dependent, respectively, 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, 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, thereby also precluding the modelling of rising rotation curves by fitting this coefficient separately for each galaxy. We further find that the general form of the conformal transformation linking the Einstein and MK frames is unique in preserving the structure of any diagonal static, spherically-symmetric metric with a radial coefficient that is (minus) the reciprocal of its temporal one. We also comment briefly on how our analysis resolves the long-standing uncertainty regarding gravitational lensing in the MK metric.

Emergence of the Cotton tensor for describing gravity

“…In conformal gravity, the values of γ have been estimated [7][8][9]. It has been pointed out that the γr term can potentially explain the galactic rotation curve without the need of dark matter [5,7,10], but there is still controversy in astrophysics [11].…”