Mannheim's linear potential in 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. Specializing 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 Schwarzschild form. Using Flanagan's method of obtaining a conformally invariant metric tensor we attempt to get a solution of Schwarzschild type. We find, however, that the 1/r terms disappear altogether. We conclude that a solution of Mannheim type cannot exist for these field equations.