  We work on a 4-manifold equipped with Lorentzian metric $g$ and consider a
volume-preserving diffeomorphism which 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. Firstly, we show that for Ricci-flat manifolds our linearised field
equations are Maxwell's equations in the Lorenz gauge with exact current.
Secondly, for Minkowski space we construct explicit massless solutions of our
nonlinear field equations; these come in two distinct types, right-handed and
left-handed. Thirdly, for Minkowski space we construct explicit massive
solutions of our nonlinear field equations; 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 explicit solutions
of nonlinear field equations we resort to group-theoretic ideas: we identify
special 4-dimensional subgroups of the Poincare group and seek diffeomorphisms
compatible with their action, in a suitable sense.
