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.Comment: In this version a new reference has been added, some misprints and
typos have been corrected and some few sentences in two remarks and in the
conclusions have been changed for better intelligibilit
