  We construct spherically symmetric solutions to the Einstein-Euler equations,
which give models of gaseous stars in the framework of the general theory of
relativity. We assume a realistic barotropic equation of state. Equilibria of
the spherically symmetric Einstein-Euler equations are given by the
Tolman-Oppenheimer-Volkoff equations, and time periodic solutions around the
equilibrium of the linearized equations can be considered. Our aim is to find
true solutions near these time-periodic approximations. Solutions satisfying so
called physical boundary condition at the free boundary with the vacuum will be
constructed using the Nash-Moser theorem. This work also can be considered as a
touchstone in order to estimate the universality of the method which was
originally developed for the non-relativistic Euler-Poisson equations.
