Spacetime diffeomorphisms as matter fields

We work on a 4-manifold equipped with Lorentzian metric g and consider a volume-preserving diffeomorphism that is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric h, the pullback of g. Motivated by elasticity theory, we introduce a Lagrangian expressed algebraically (without differentiations) via our pair of metrics. Analysis of the resulting nonlinear field equations produces three main results. First, we show that for Ricci-flat manifolds, our linearized field equations are Maxwell's equations in the Lorenz gauge with exact current. Second, for Minkowski space, we construct explicit massless solutions of our nonlinear field equations; these come in two distinct types, right-handed and left-handed. Third, for Minkowski space, we construct explicit massive solutions of our nonlinear field equations; these contain a positive parameter that has the geometric meaning of quantum mechanical mass and a real parameter that may be interpreted as electric charge. In constructing explicit solutions of nonlinear field equations, we resort to group-theoretic ideas: We identify special four-dimensional subgroups of the Poincaré group and seek diffeomorphisms compatible with their action in a suitable sense.