In this talk we develop a new mathematical model of elasticity in the Lorentzian setting. Working on a Lorentzian 4-manifold, we consider a diffeomorphism which is the unknown quantity of our mathematical model. We write down a functional of nonlinear elasticity and vary it subject to the volume preservation constraint. The analysis of our nonlinear field equations produces three main results. Firstly, we show that for Ricci-flat manifolds the linearised field equations are Maxwell's equations in the Lorenz gauge with exact current. Secondly, for Minkowski space we construct explicit massless solutions; these come in two distinct types, right-handed and left-handed. Thirdly, for Minkowski space we construct explicit massive solutions; these contain a positive parameter which has the geometric meaning of quantum mechanical mass and a real parameter which may be interpreted as electric charge. In constructing our solutions we resort to group-theoretic ideas: we identify special 4-dimensional subgroups of the Poincaré group and seek diffeomorphisms compatible with their action, in a suitable sense.

This is joint work with Matteo Capoferri, preprint arXiv:1805.01303.