Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry

Abstract

We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries with constant flag curvature always map to conformally flat spacetimes and that the Finsler condition maps to either a causality condition or it breaks down at an ergo-surface in the spacetime picture. The gauge equivalence in this network of relations is considered as well as the connection to analogue models and the viewpoint of magnetic flows. We provide a variety of examples.

37 pages, 6 figures

References in corpus (6)

• Applications of the Gauss-Bonnet theorem to gravitational lensing
• A gravito-electromagnetic analogy based on tidal tensors
• Universal properties of the near-horizon optical geometry
• Light-bending in Schwarzschild-de-Sitter: projective geometry of the optical metric
• Kaluza-Klein reduction of conformally flat spaces
• Analogue spacetimes: Toy models for "quantum gravity''

Cited by in corpus (47)

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• Isolated Surfaces and Symmetries of Gravity
• Classical-physics applications for Finsler $b$ space
• Weak Gravitational Lensing in Dark Matter and Plasma Mediums for Wormhole-like Static Aether Solution
• Solution to the quantum Zermelo navigation problem
• Classical kinematics and Finsler structures for nonminimal Lorentz-violating fermions
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• Dynamics in Varying vacuum Finsler-Randers Cosmology
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• Global embedding of the Kerr black hole event horizon into hyperbolic 3-space
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• Heavy-ion collisions, Gubser flow, and Carroll hydrodynamics
• Gravitational Faraday and Spin-Hall Effects of Light
• Violations of the Kerr and Reissner-Nordstrom bounds: horizon versus asymptotic quantities
• Optimal navigation strategies for microswimmers on curved manifolds
• Deriving Weak Deflection Angle by Black Holes or Wormholes using Gauss-Bonnet Theorem
• Applications of cone structures to the anisotropic rheonomic Huygens' principle
• Weak gravitational lensing by Einstein-non-linear-Maxwell-Yukawa black hole
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• The causal structure of spacetime is a parameterized Randers geometry
• Harmonic functions and gravity localization
• The Finsler spacetime condition for (α,β)-metrics and their isometries
• A Hamiltonian approach to the gradient-flow equations in information geometry
• Microlensing and event rate of static spherically symmetric wormhole
• Aspects of General Relativity: Pseudo-Finsler extensions, Quasi-normal frequencies and Multiplication of tensorial distributions
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• An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
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• Relativistic Fluids, Hydrodynamic Frames and their Galilean versus Carrollian Avatars
• Massive particle surfaces and black hole shadows from intrinsic curvature
• Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge
• The geometry of induced currents in two dimensional media
• Finslerian geometrodynamics