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

TL;DR

This paper demonstrates that conformal rescaling of Schwarzschild--de-Sitter metrics cannot produce flat galaxy rotation curves without dark matter, and confirms that physical particle trajectories are invariant across conformal frames in various gravity theories.

Contribution

It clarifies that conformal transformations do not alter physical trajectories, ruling out flat rotation curves from such rescaling and extending the invariance to general Weyl--Cartan spacetimes.

Findings

Conformal rescaling does not produce flat galaxy rotation curves.

Particle trajectories are invariant under conformal transformations.

The quadratic term coefficient relates to a global cosmological constant.

Abstract

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 arbitrary 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.