Is there weak birefringence of light in vacuo?

There are numerous reasons to study modifications of general relativity and the Standard Model of particle physics, ranging from modelling inflation to exploring galaxy rotation curves and the nature of dark matter. Here we study the most general linear theory of electromagnetism, which admits vacuum birefringence, and derive weak gravitational field equations for the underlying area metric spacetime geometry. We discuss the weak gravitational field sourced by a point mass in an area metric spacetime and find first order corrections to the linearized Schwarzschild metric of general relativity.