In the present work is explored the profound symmetry between the theories of gravity and electromagnetism, with particular attention to the similarity between the fundamental forces of nature: the weight force and the electric Lorentz force, Newton's law and Coulomb's law, as well as Maxwell's equations and the equations of gravitational electromagnetism. There are papers that explain how to obtain the gravitational electromagnetism equations by applying the weak field approximation to the Einstein field equations. Precisely in honour of this symmetry, this work aims to obtain Einstein field equations valid for electromagnetism, i.e. field equations from which to derive, under weak field approximation, Maxwell's equations. This research suggests a parallel vision between electromagnetism and gravity, suggesting that perhaps there could be a curved electromagnetic field in 4 dimensions alongside the curved gravitational field (Einstein's space-time).