Mannheim's linear potential in conformal gravity

TL;DR

This paper investigates Mannheim's conformal gravity equations in the weak field limit, revealing that solutions analogous to Schwarzschild are not possible, challenging the viability of Mannheim's model for describing gravitational fields around compact objects.

Contribution

The study derives the Green function for Mannheim's conformal gravity equations and demonstrates the non-existence of Schwarzschild-like solutions within this framework.

Findings

No Schwarzschild form near compact sources

1/r terms vanish in the solutions

Mannheim's 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. 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.