On the definition and examples of cones and Finsler spacetimes

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 114; no. 1
• Main Authors: Javaloyes, Miguel Angel, Sánchez, Miguel
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Cones; Fields (mathematics); Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Spacetime; Theoretical; Time dependence; Cone structure; Finsler spacetime; Lorentz-Finsler metrics and norms; 53C50; 70B05; 83D05; Generalized Fermat principle; 53C60; Static and stationary spacetimes; Zermelo navigation problem
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-019-00736-y

A systematic study of (smooth, strong) cone structures C and Lorentz–Finsler metrics L is carried out. As a link between both notions, cone triples ( Ω , T , F ) , where Ω (resp. T ) is a 1-form (resp. vector field) with Ω ( T ) ≡ 1 and F , a Finsler metric on ker ( Ω ) , are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures C are bijectively associated with classes of anisotropically conformal metrics L , and the notion of cone geodesic is introduced consistently with both structures. As a non-relativistic application, the time-dependent Zermelo navigation problem is posed rigorously, and its general solution is provided.