• Locally inertial reference frames;
  • Lorentzian spacetimes;
  • Riemann-Cartan spacetimes.


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.