Physical Finsler coordinates in spacetime

Abstract

In Finsler geometry, a Finsler coordinate is a coordinate in the tangent space manifold of a given base manifold. As such it has been given various definitions in the relativity and field theory literature and often even remains undefined physically. Physically meaningful coordinates of a point in the tangent bundle of spacetime are the spacetime and four-velocity coordinates of the measuring device. It is here emphasized that the four-velocity of the measuring device need not be the same as the four-velocity of the measured object, either classically or quantum mechanically. The four-velocity of a measured particle excitation of a Finslerian quantum field in the tangent space manifold of spacetime is not a suitable physical Finsler coordinate. The role of the Finsler coordinate is elaborated in a detailed example involving a Finslerian quantum field and associated microcausality.