Spacetime diffeomorphisms as matter fields

TL;DR

This paper explores a novel geometric framework where spacetime diffeomorphisms are modeled as matter fields, leading to solutions resembling Maxwell's equations and particles with mass and charge in Minkowski space.

Contribution

It introduces a new algebraic Lagrangian based on a pair of metrics and derives explicit solutions, connecting spacetime symmetries with matter field models.

Findings

Linearized equations reduce to Maxwell's equations in Ricci-flat manifolds.

Explicit massless solutions with chiral properties are constructed in Minkowski space.

Explicit massive solutions with parameters interpreted as mass and charge are found.

Abstract

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.