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The S-stars discovered in the Galactic center (GC) are expected to provideunique 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 therelated relativistic effects, a comprehensive understanding of thegravitational perturbations of the stars and stellar remnants around the MBH isquite essential. Here, we study the perturbations on the observables of atypical target star, i.e., the apparent orbital motion and the redshift, due toboth the spin-induced relativistic effects and the Newtonian attractions of asingle or a cluster of disturbing object(s). We find that, in most cases, theNewtonian perturbations on the observables are mainly attributed to theperturbed orbital period of the target star, rather than the Newtonian orbitalprecessions. The Newtonian perturbations have their unique features when theypeak around the pericenter passage in each revolution, which is quite differentfrom those of the spin-induced effects. Looking at the currently detected starS2/S0-2, we find that its spin-induced effects on both the image position andredshift are very likely obscured by the gravitational perturbations from thestar S0-102 alone. We also investigate and discuss the Newtonian perturbationson a hypothetical S-star located inside the orbits of the currently detectedones. By considering a number of possible stellar distributions near thecentral MBH, we find that the spin-induced effects on the apparent position andthe redshift dominate over the stellar perturbations for target stars withorbital semimajor axis smaller than $100-400$AU if the MBH is maximallyspinning. Our results suggest that, in principle, the stellar perturbations canbe removed as they have distinctive morphologies comparing to those of therelativistic Kerr-type signatures.

On the Newtonian and Spin-induced Perturbations felt by the Stars Orbiting around the Massive Black Hole in the Galactic Center