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

TL;DR

This paper analyzes Mannheim's conformal gravity equations in the weak field limit, revealing that solutions near compact sources like the Sun deviate from Schwarzschild form, challenging the existence of Mannheim-type solutions.

Contribution

It derives the Green function for the metric equation in Mannheim's model and demonstrates the non-existence of Mannheim-type solutions near compact sources.

Findings

Solutions differ from Schwarzschild near sources

Green function representation of the metric function

Mannheim-type solutions cannot exist for these equations

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. We conclude that a solution of Mannheim type cannot exist for these field equations.