  In this paper we scrutinize the so called Principle of Local Lorentz
Invariance (\emph{PLLI}) that many authors claim to follow from the Equivalence
Principle. Using rigourous mathematics we introduce in the General Theory of
Relativity two classes of reference frames (\emph{PIRFs} and
\emph{LLRF}$\gamma$\emph{s}) which natural generalizations of the concept of
the inertial reference frames of the Special Relativity Theroy. We show that it
is the class of the \emph{LLRF}$\gamma$\emph{s} that is associated with the
\emph{PLLI.} Next we give a defintion of physically equivalent referefrence
frames. Then, we prove that there are models of General Relativity Theory (in
particular on a Friedmann universe) where the \emph{PLLI}is false. However our
find is not in contradiction with the many experimental claims vindicating the
\emph{PLLI}, because theses experiments do not have enough accuracy to detect
the effect we found. We prove moreover that \emph{PIRFs}are not physically
equivalent.
