  The S-stars discovered in the Galactic center (GC) are expected to provide
unique dynamical tests of the Kerr metric of the massive black hole (MBH)
orbited by them. In order to obtain unbiased measurements of its spin and the
related relativistic effects, a comprehensive understanding of the
gravitational perturbations of the stars and stellar remnants around the MBH is
quite essential. Here, we study the perturbations on the observables of a
typical target star, i.e., the apparent orbital motion and the redshift, due to
both the spin-induced relativistic effects and the Newtonian attractions of a
single or a cluster of disturbing object(s). We find that, in most cases, the
Newtonian perturbations on the observables are mainly attributed to the
perturbed orbital period of the target star, rather than the Newtonian orbital
precessions. The Newtonian perturbations have their unique features when they
peak around the pericenter passage in each revolution, which is quite different
from those of the spin-induced effects. Looking at the currently detected star
S2/S0-2, we find that its spin-induced effects on both the image position and
redshift are very likely obscured by the gravitational perturbations from the
star S0-102 alone. We also investigate and discuss the Newtonian perturbations
on a hypothetical S-star located inside the orbits of the currently detected
ones. By considering a number of possible stellar distributions near the
central MBH, we find that the spin-induced effects on the apparent position and
the redshift dominate over the stellar perturbations for target stars with
orbital semimajor axis smaller than $100-400$AU if the MBH is maximally
spinning. Our results suggest that, in principle, the stellar perturbations can
be removed as they have distinctive morphologies comparing to those of the
relativistic Kerr-type signatures.
