Conformally-rescaled Schwarzschild metrics do not predict flat galaxy rotation curves

For conformally invariant gravity theories defined on Riemannian spacetime and having the Schwarzschild-de-Sitter (SdS) metric as a solution in the Einstein gauge, we consider whether one may conformally rescale this solution to obtain flat rotation curves, such as those observed in galaxies, without the need for dark matter. Contrary to recent claims in the literature, we show that if one works in terms of quantities that can be physically measured, then in any conformal frame the trajectories followed by 'ordinary' matter particles are merely the timelike geodesics of the SdS metric, as one might expect. This resolves the apparent frame dependence of physical predictions and unambiguously yields rotation curves with no flat region. We also 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 precluded, since this coefficient is most naturally interpreted as proportional to a global cosmological constant. We further extend our analysis beyond static, spherically-symmetric systems to show that the invariance of particle dynamics to the choice of conformal frame holds for arbitary metrics, again as expected. Moreover, we show that this conclusion remains valid for conformally invariant gravity theories defined on more general Weyl-Cartan spacetimes, which include Weyl, Riemann-Cartan and Riemannian spacetimes as special cases.

Automated summary

This study investigates conformally invariant gravity theories defined on Riemannian spacetime with the Schwarzschild-de-Sitter metric as a solution, exploring the possibility of obtaining flat rotation curves without the need for dark matter by conformally rescaling this solution. The research shows that trajectories of "ordinary" matter particles in any conformal frame are timelike geodesics of the SdS metric, which resolves the frame dependence issue and unambiguously yields non-flat rotation curves.