One cornerstone of special relativity is the Einstein clock postulate which links the length of distinguished spacetime curves to an observers measurement of time. Assuming a metric length measure this leads to general relativity. During this talk I consider a more general geometric length, not necessarily induced by a metric, which leads to non-metric geometry of space-time describing gravity, called Finsler geometry. I present a precise definition of such Finsler spacetimes which guarantees a clear notion of causality. Moreover I discuss the coupling of physical matter fields to Finsler spacetimes and how the matter field content determines the spacetime geometry dynamically. The latter yields a gravitational field equation which extends Einstein’s equations. As example I present a first order spherical symmetric solution of the extended equation and compare it to the linearised Schwarzschild solution of general relativity.