Bi-metric pseudo-Finslerian spacetimes

TL;DR

This paper explores the mathematical challenges of defining pseudo-Finsler metrics in bi-metric spacetimes with Lorentzian signature, highlighting issues with null vectors and metric construction.

Contribution

It provides foundational definitions and discusses the difficulties in constructing pseudo-Finsler metrics for bi-metric Lorentzian spacetimes, addressing a key mathematical problem.

Findings

Pseudo-Finsler structures are essential for modeling signal propagation in physics.

Constructing a well-defined pseudo-Finsler metric in Lorentzian signature is problematic on the null cone.

The paper clarifies the limitations of existing Finsler geometry approaches in bi-metric spacetimes.

Abstract

Finsler spacetimes have become increasingly popular within the theoretical physics community over the last two decades. Because physicists need to use pseudo-Finsler structures to describe propagation}of signals, there will be nonzero null vectors in both the tangent and cotangent spaces --- this causes significant problems in that many of the mathematical results normally obtained for "usual" (Euclidean signature) Finsler structures either do not apply, or require significant modifications to their formulation and/or proof. We shall first provide a few basic definitions, explicitly demonstrating the interpretation of bi-metric theories in terms of pseudo-Finsler norms. We shall then discuss the tricky issues that arise when trying to construct an appropriate pseudo-Finsler metric appropriate to bi-metric spacetimes. Whereas in Euclidian signature the construction of the Finsler metric typically fails at the zero vector, in Lorentzian signature the Finsler metric is typically ill-defined on the entire null cone.