Hidden connection between general relativity and Finsler geometry

Abstract

Modern formulation of Finsler geometry of a manifold M utilizes the equivalence between this geometry and the Riemannian geometry of VTM, the vertical bundle over the tangent bundle of M, treating TM as the base space. We argue that this approach is unsatisfactory when there is an indefinite metric on M because the corresponding Finsler fundamental function would not be differentiable over TM (even without its zero section) and therefore TM cannot serve as the base space. We then make the simple observation that for any differentiable Lorentzian metric on a smooth space-time, the corresponding Finsler fundamental function is differentiable exactly on a proper subbundle of TM. This subbundle is then used, in place of TM, to provide a satisfactory basis for modern Finsler geometry of manifolds with Lorentzian metrics. Interestingly, this Finslerian property of Lorentzian metrics does not seem to exist for general indefinite Finsler metrics and thus, Lorentzian metrics appear to be of special relevance to Finsler geometry. We note that, in contrast to the traditional formulation of Finsler geometry, having a Lorentzian metric in the modern setting does not imply reduction to pseudo- Riemannian geometry because metric and connection are entirely disentangled in the modern formulation and there is a new indispensable non-linear connection, necessary for construction of Finsler tensor bundles. It is concluded that general relativity--without any modification--has a close bearing on Finsler geometry and a modern Finsler formulation of the theory is an appealing idea...