The tangent bundle exponential map and locally autoparallel coordinates for general connections with application to Finslerian geometries

Abstract

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas-Thomas normal coordinates introduced earlier. We expand the fundamental geometry function of a Finsler spacetime in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the non-linear curvature of the manifold. From physics these coordinates may be interpretation as the realisation of an Einstein elevator in Finslerian spacetime geometries.

29 pages, references and their discussion added, language optimised, Journal accepted version. Int. J. Geom. Methods Mod. Phys, 23 November 2015

References in corpus (4)

• General Very Special Relativity is Finsler Geometry
• Planck-scale modified dispersion relations and Finsler geometry
• Finsler geometric extension of Einstein gravity
• The connections of pseudo-Finsler spaces

Cited by in corpus (5)

• Curved spacetimes with local $\kappa$-Poincaré dispersion relation
• Deformed relativity symmetries and the local structure of spacetime
• Reaching the Planck scale with muon lifetime measurements
• Special coordinate systems in pseudo-Finsler geometry and the equivalence principle
• Normal coordinates based on curved tangent space