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Locally Inertial Reference Frames in Lorentzian and Riemann-Cartan Spacetimes

In this paper we scrutinize the concept of locally inertial reference frames (LIRF) in Lorentzian and Riemann-Cartan spacetime structures. We present rigorous mathematical definitions for those objects, something that needs preliminary a clear mathematical distinction between the concepts of observers, reference frames, naturally adapted coordinate functions to a given reference frame and which properties may characterize an inertial reference frame (if any) in the Lorentzian and Riemann-Cartan structures. We hope to have clarified some eventual obscure issues associated to the concept of LIRF appearing in the literature, in particular the relationship between LIRFs in Lorentzian and Riemann-Cartan spacetimes and Einstein's most happy though, i.e., the equivalence principle.