In this paper the concept of locally inertial reference frames (LIRFs) in Lorentzian and Riemann-Cartan spacetime structures is scrutinized. A rigorous mathematical definition of a LIRF in both structures is given, 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. Hopefully, the paper clarifies some obscure issues associated to the concept of a LIRF appearing in the literature, in particular the relationship between LIRFs in Lorentzian and Riemann-Cartan spacetimes and Einstein's most happy thought, i.e., the equivalence principle.
Definition 1We call the pentuple 〈M , g ,D, τ g , ↑〉 a Lorentzian spacetime and the pentuple 〈M , g , D, τ g , ↑〉 a Riemann-Cartan spacetime.
Remark 2 Minkowski spacetime structure is denoted bythe local charts (ϕ,U ) and (ψ,V ) and (χ,W ) with coordinate functions 〈ξ μ 〉, 〈x μ 〉, 〈x μ 〉, respectively. Recall to fix notation that, e.g., given p ∈ M and V ⊂R 4 we have