Spacetime diffeomorphisms as matter fields
Abstract
We work on a 4manifold equipped with Lorentzian metric $g$ and consider a volumepreserving 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 Ricciflat 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, righthanded and lefthanded. 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 grouptheoretic ideas: we identify special 4dimensional subgroups of the Poincare group and seek diffeomorphisms compatible with their action, in a suitable sense.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.01303
 Bibcode:
 2018arXiv180501303C
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 53C50 (primary);
 22E43;
 35Q41;
 35Q61;
 74B20 (secondary)
 EPrint:
 Minor edits