Geometric Unification in Classical Physics

TL;DR

This paper demonstrates that Lorentz force naturally appears in Finsler geometry and unifies aspects of electromagnetism, dark matter, and gravity through torsion and curvature in a geometric framework.

Contribution

It introduces a geometric unification of classical physics using Finsler geometry, revealing the inescapable Lorentz force and deriving Maxwell and Einstein equations from torsion and curvature.

Findings

Lorentz force arises in Finsler geometry autoparallels.

Maxwell's equations emerge from Bianchi identities for torsion.

Einstein's equations are derived from curvature and torsion decomposition.

Abstract

We show that always present in the autoparallels, even in natural liftings to the Finsler bundle of arbitrary connections, the Lorentz force is inescapable in Finsler geometry. These liftings retain the form $R_{\,\nu \lambda }^{\mu }\omega ^{\nu }\wedge \omega ^{\lambda }$, but he soldering forms, $\omega ^{\mu }$, and the components have changed. Finslerian torsions, $\Omega ^{\mu }=d\omega ^{\mu }-\omega ^{\nu }\wedge \omega _{\nu }^{\mu }=R_{\,\nu \lambda }^{\mu }\omega ^{\nu }\wedge \omega ^{\lambda }+S_{\,\nu i}^{\mu }\omega ^{\nu }\wedge \omega _{0}^{i}$, $(\lambda =(0,l_{i})=0,1,2,3)$, span three sectors: (a) electrodynamic, defined by $\Omega ^{i}=0$ and $S_{\nu l}^{0}=0$, (b) "dark matter" (for lack of a better name), defined by $\Omega ^{i}=0,$ $S_{\nu l}^{i}\neq 0$ (It affects the equation of the autoparallels with additional terms, not only for the force but also for the momentum) and (c) the dark sector of the $\Omega ^{i}$, "dark" because it contributes to the energy equations but not to the equations of the motion. We then assume teleparallelism. The linearization of the first Bianchi identity for torsions of type (a) becomes the first pair of Maxwell's equations. And the vanishing of a curvature related vector-valued differential 3-form, and the splitting of the connection into contorsion and Levi-Civita differential 1-form yields Einstein's equations as a statement of annulment of a sum of three currents, metric dependent (gravitational), torsional (other fields and matter) and a third one that mixes torsion and metric and with cosmological flavor.