Causal structure and electrodynamics on Finsler spacetimes

TL;DR

This paper introduces a new definition of Finsler spacetimes that generalizes Lorentzian manifolds, enabling consistent modeling of causality and electrodynamics, including light propagation along Finsler null geodesics.

Contribution

It provides a rigorous mathematical framework for Finsler spacetimes, extending geometric tools and formulating electrodynamics within this context.

Findings

Causal structure characterized by timelike and null curves.

Timelike directions form an open convex cone with null boundary.

Electrodynamics propagates light along Finsler null geodesics.

Abstract

We present a concise new definition of Finsler spacetimes that generalize Lorentzian metric manifolds and provide consistent backgrounds for physics. Extending standard mathematical constructions known from Finsler spaces we show that geometric objects like the Cartan non-linear connection and its curvature are well-defined almost everywhere on Finsler spacetimes, also on their null structure. This allows us to describe the complete causal structure in terms of timelike and null curves; these are essential to model physical observers and the propagation of light. We prove that the timelike directions form an open convex cone with null boundary as is the case in Lorentzian geometry. Moreover, we develop action integrals for physical field theories on Finsler spacetimes, and tools to deduce the corresponding equations of motion. These are applied to construct a theory of electrodynamics that confirms the claimed propagation of light along Finsler null geodesics.