Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry

TL;DR

The paper builds a unifying framework linking Zermelo navigation, Randers-Finsler geometry, and null geodesic flows in conformally stationary spacetimes, using Painlevé-Gullstrand form to encode Zermelo data and a stationary Randers metric for the Finsler side. It shows that Randers metrics of constant flag curvature correspond to conformally flat spacetimes, and it clarifies gauge equivalences among the three pictures, including magnetic-flow interpretations and analogue-model connections. A broad suite of explicit examples (Langevin’s rotating platform, Gödel-type, Störmer’s dipole, ESU/Anti-Mach, Kerr and extremal Kerr) illustrates how the three descriptions interlock, how breakdowns (ergosurfaces, VLS) arise, and how near-horizon geometries exhibit hyperbolic structure. The results provide concrete tools for translating geometric, dynamical, and physical insights across the spacetime, Finsler, and fluid-dynamical viewpoints, with potential applications to analogue models and classification schemes via conformal and Killing symmetries.

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.

Figures (6)

• Figure 1: A network of relations. Alongside the arrows are the equation numbers which provide the respective map. The dashed lines represent interpretation relations rather than different frameworks.
• Figure 2: Langevin's rotating platform in Minkowski space and the corresponding Randers and Zermelo pictures. We have chosen positive $\Omega$ to depict the tipping of the light-cones (spacetime picture) and the "tornado" direction (Zermelo picture). The direction of the magnetic field presented (Randers picture) is merely illustrative, since the embedding performed excludes the $z$ direction.
• Figure 3: Surfaces, in $\mathbb{R}^3$, where the Randers structure breaks down. Left: Heisenberg example of section \ref{['heisenbergsection']} for $a=10/3,10/6,10/9$ (smaller to larger cylinder); the Randers structure is valid inside the cylinder for each case. Right: Störmer example of section \ref{['stormersection']} for $\sqrt{\mu}=0.5,0.8,1$ (smaller to larger surface); the Randers structure is valid outside the surface for each case.
• Figure 4: Region of $\mathbb{H}^2$ covered by the Randers structure in the examples of section \ref{['godelsection']} and \ref{['cat']}. The surface on the left is the embedding diagram of $\mathbb{H}^2$ discussed in section \ref{['langevin']}. As we increase the constant magnetic field of section \ref{['godelsection']}, beyond the minimal value associated to $a=1$, the Randers structure breaks down outside a finite radius which decreases with increasing $a$ (top sequence). As we turn on the asymptotically decaying magnetic field of section \ref{['cat']}, the Randers structure breaks down inside a finite radius which increases with increasing $\lambda$ (bottom sequence).
• Figure 5: Foliation of the 3-sphere by Clifford tori and the Killing wind corresponding, in the spacetime picture, to a rigidly rotating coordinate system. Adapted from Gibbons:1999uv and inspired by the cover of Twistor Newsletter.