A unified, purely affine theory of gravitation and electromagnetism

In the purely affine formulation of gravity, the gravitational field is represented by the symmetric part of the Ricci tensor of the affine connection. The classical electromagnetic field can be represented in this formulation by the second Ricci tensor of the connection. Such a construction is dynamically equivalent to the sourceless Einstein-Maxwell equations. We generalize this construction to the case with sources, represented by the derivative of the affine Lagrangian density with respect to the connection. We show that the Maxwell equations with sources emerge for the simplest affine Lagrangian for matter, while the Einstein and Lorentz equations arise if mass has electromagnetic origin. We also show that the Maxwell equations replace the unphysical constraint imposed by the projective invariance of purely affine Lagrangians that depend explicitly on the connection.