No Ghost State in the Brane World

We discuss the role of the trace part of metric fluctuations $h_{MN}$ in the Randall-Sundrum scenario of gravity. Without the matter, this field ($h={\eta }^{MN}{h}_{MN}$) is a gauge-dependent term, and thus it can be gauged away. But, including the uniform source ${\tilde{T}}_{MN}$, this field satisfies the linearized equation ${\mathrm{\square }}_{4}h=16\pi G_5{T}_{\mu }^{\mu }$. This may correspond to the scalar ${\xi }^5$ in the bending of the brane due to the localized source. Considering the case of longitudinal perturbations ($h_{5\mu} =h_{55}=0$),onefindsthesourcerelation$\tilde{T}^{\mu}{\mu}=2\tilde{T}{55}$, which leads to the ghost states in the massive modes. In addition, if one requires $T_{44}=2(T_{22}+T_{33})$, it is found that in the limit of $m_h^2\to 0$ we have the massless spin-2 propagation without the ghost state. This exactly corresponds to the same situation as in the intermediate scales of Gregory-Rubakov-Sibiryakov (GRS) model.