Schwarzschild and linear potentials in Mannheim’s model of conformal gravity

Abstract

We study the equations of conformal gravity, as given by Mannheim, in the weak field limit, so that a linear approximation is adequate. Specialising to static fields with spherical symmetry, we obtain a second-order equation for one of the metric functions. We obtain the Green function for this equation, and represent the metric function in the form of integrals over the source. Near a compact source such as the Sun the solution no longer has a form that is compatible with observations. We conclude that a solution of Mannheim type (a Schwarzschild term plus a linear potential of galactic scale) cannot exist for these field equations.

Conformal gravity does not predict flat galaxy rotation curves

“…CG is most celebrated phenomenologically, however, for its fitting of flat galaxy rotation curves without the need for dark matter [50][51][52][53][54]. This is based on the trajectories of massive particles in the MK metric, with parameter values that are consistent with solar-system tests, although the requirement for matching the MK metric onto a static, spherically-symmetric matter source suggests that it may not be possible to set the parameters in a consistent manner [55][56][57]. More troubling, however, is that the galaxy rotation curve analyses assume simply that massive (test) particles follow timelike geodesics, which are affected by conformal transformations, as is well known.…”

Gravity Models with Nonlinear Symmetry Realization

“…Explicit non-linear realization of the conformal symmetry within AdS/CFT correspondence was constructed in [7]. It also should be mentioned that there are models with a linear realization of the conformal symmetry [8,9], but their applicability is debatable [10][11][12][13].…”