  Our results provide a first step to make rigorous the formal analysis in
terms of $\frac{1}{c^2}$ proposed by Chandrasekhar \cite{Chandra65b},
\cite{Chandra65a}, motivated by the methods of Einstein, Infeld and Hoffmann,
see Thorne \cite{Thorne1}. We consider the non-relativistic limit for the local
smooth solutions to the free boundary value problem of the cylindrically
symmetric relativistic Euler equations, when the mass energy density includes
the vacuum states at the free boundary. For large enough (rescaled) speed of
light $c$ and suitably small time $T,$ we obtain uniform, with respect to $c,$
\lq\lq a priori\rq\rq estimates for the local smooth solutions. Moreover, the
smooth solutions of the cylindrically symmetric relativistic Euler equations
converge to the solutions of the classical compressible Euler equation, at the
rate of order $\frac{1}{c^2}$.
