  The initial data of the gravitational field produced by a loop thick string
is considered. We show that a thick loop is not a geodesic on the initial
hypersurface, while a loop conical singularity is. This suggests that there is
the ``{\it critical thickness}'' of a string, at which the linear perturbation
theory with a flat space background fails to describe the gravity of a loop
cosmic string. Using the above initial data, we also show that the linear
perturbation around flat space is plausible if the string thickness is larger
than $\sim5\times10^{-3}a$, where $a$ is the curvature radius of the loop.
