Stability of the GRS Model

We discuss the compatibility between the weaker energy condition and the stability of Gregory, Rubakov and Sibiryakov (GRS) model. Because the GRS spacetime violates the weak energy condition, it may cause the instability. In the GRS model, the four dimensional gravity can be described by the massive KK modes with the resonance. Hence, instead of considering the weaker energy condition, we require for the stability of this model: no tachyon and no ghost condition for graviton modes ($h_{\mu \nu }$). No tachyonic condition ($m^2_h \geq 0$)issatisfiedbecausetheloweststate$m_h=0$ is supersymmetric vacuum state. Further, no ghost state condition is achieved if one requires some relations for the matter source: $2T_{55}=T_{\mu }^{\mu }=3(T_{22}+T_{33})$. It turns out that, although the GRS spacetime does not satisfy the weaker energy condition, it is stable against small perturbation.