  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.
