Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes (Q6605408)

This memoir is a good contribution to the interplay between Finsler geometry, navigation problems, and spacetime causality. By unifying these diverse areas under the conceptual framework of wind Finslerian structures, it shows both mathematical rigor and broad applicability.\N\N\textbf{Strengths by Topics and Subtopics}\N\N\textbf{Wind Finslerian Structures:} The authors introduce and rigorously develop \textit{wind Finslerian structures} as a natural generalization of traditional Finsler metrics. A key strength lies in the clear explanation of how these structures model physical scenarios involving ``wind'' or vector field effects, which lead to indicatrices that no longer contain the zero vector. \N\NThe use of conic pseudo-Finsler metrics and the distinction between regions of mild, critical, and strong wind is elegant. This classification provides a solid mathematical foundation for analyzing diverse physical phenomena, including constrained motion under external forces. The transversality condition imposed on the structure ensures smoothness, preserving analytical tractability. This is crucial for extending classical Finslerian results to these new settings.\N\N\textbf{Applications to Zermelo's Navigation Problem:} The extension of Zermelo's navigation problem to scenarios involving arbitrary time-independent winds is one of the most compelling applications presented in this work.\N\NThe authors solve the navigation problem not only for minimal time trajectories but also explore maximizing trajectories and closed geodesics. This is a notable advancement over classical approaches that often restrict themselves to mild winds. By connecting navigation trajectories with the geodesics of the wind Finslerian structure, the work provides a geometric interpretation that transcends conventional optimal control techniques.\N\N\textbf{Causality in Spacetime Geometry:} The application of wind Finslerian structures to the causal structure of spacetimes endowed with a Killing vector field is a remarkable bridge between geometry and relativistic physics. The authors demonstrate how wind Riemannian structures naturally describe the causal properties of SSTK (standard with space-transverse Killing) spacetimes. This interpretation provides powerful tools to analyze phenomena such as horizons, Cauchy developments, and regions of no escape. The connection between the Zermelo navigation problem and spacetime causality via Fermat's principle highlights the deep interplay between non-relativistic and relativistic geometric problems.\N\N\textbf{Fermat's Principle and Variational Calculus:} The generalization of Fermat's principle is a cornerstone of the work, offering a variational interpretation of geodesics in spacetimes and generalized Zermelo navigation.\N\NThe extension to arrival curves that are not necessarily timelike is highly original and relevant for both theoretical and applied contexts. It refines prior results in mathematical relativity and variational calculus. The analysis of minimizing and maximizing geodesics using signature-changing tensors demonstrates technical sophistication and a mastery of variational methods.\N\N\textbf{Mathematical Framework:} The mathematical framework is rigorous, with careful attention to the theoretical underpinnings of wind Finslerian structures. The introduction of tools like c-balls and w-convexity enriches the geometric analysis. The careful construction of conic pseudo-Finsler metrics and the smooth embedding of their indicatrices ensure the robustness of the results. These techniques may inspire future research in generalized geometry.\N\N\textbf{Conclusions and Future Directions:} The concluding chapter offers an insightful summary of the results while outlining paths for further research. The authors discuss applications to wave propagation, spacetime splitting, and other geometric constructs. The discussion of potential extensions, such as the use of these structures in non-relativistic wave propagation and mathematical relativity, opens avenues for interdisciplinary research.\N\N\textbf{Overall Impression:} This memoir combines mathematical rigor with a deep understanding of its physical implications, creating a bridge between abstract geometry and practical applications. Each chapter is carefully constructed to build on the preceding material, ensuring a logical progression from foundational principles to complex applications. The primary strength of this work lies in its ability to generalize and unify existing concepts while extending them to novel contexts.