Kavli Affiliate: Anthony Lasenby

| First 5 Authors: Michael Hobson, Anthony Lasenby, , ,

| Summary:

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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