  We present a detailed analysis for the classical stability of a four
dimensional Anti-de Sitter spacetime (AdS$_4$) by decomposing the first-order
perturbations of a spherical symmetric gravitational field into the so called
tensor harmonics which transform as irreducible representative of the rotation
group (Regge-Wheeler decomposition). It is shown that there is no nontrivial
stationary perturbation for the angular momentum $l < 2$. The stability
analysis forces the frequency of the gravitational modes to be constrained in a
way that the frequency of scalar modes are constrained.
