Hamiltonian formulation of the Finsler and Riemann geometries

Ludek Klimes


In the Riemann geometry, the metric tensor depends on coordinates. In the Finsler geometry, which is a generalization of the Riemann geometry, the metric tensor depends also on the direction of propagation. Light rays are the zero-length geodesics in Finsler space, determined by the light-propagation metric tensor.

We would like to demonstrate that the Hamiltonian formulation simplifies the equations of the Riemann geometry, and makes the Finsler geometry no more difficult than the Riemann geometry.

We shall refer to the case of a metric space, in which the length of all geodesics is positive, as the spatial case, and to the corresponding geodesics as spatial geodesics. We shall refer to the opposite case as the space-time case, and to the corresponding geodesics as space-time geodesics. In the space-time case, we shall distinguish between zero-length and time-like geodesics. In the spatial case, the Riemannian metric tensor is always positive-definite, but the Finslerian metric tensor may also be indefinite. In the space-time case, the metric tensor is always indefinite.

Sections 2 and 3 are the introductory sections only. In Section 2, we briefly explain why we should strictly distinguish the light-propagation metric tensor from the gravitational metric tensor of general relativity. Section 3 is devoted to a brief outline of the physical background behind the light-propagation Hamiltonian. Then we shall start to speak about geodesics and proper time in the Finsler geometry.

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Notes for Relativistic Coordinates, Reference and Positioning Systems, Salamanca, Spain, January 21-25, 2005.
In: Seismic Waves in Complex 3-D Structures, Report 15, pp. 207-216, Dep. Geophys., Charles Univ., Prague, 2005.
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