\Lambda_{\rm eff} h_{ab} = \frac{1}{M_{\rm P}^2 } \tau_{ab} +
\frac{1}{1 + 2 \sigma m^2 /3 M^6 } \left( M^6 Q_{ab} - W_{ab} \right) \, ,$$ where $$\Lambda_{\rm eff} = \frac{\Lambda_{\rm RS}}{1 + 2 \sigma m^2 / 3 M^6} \, ,
\quad \Lambda_{\rm RS} = \frac{\Lambda_{\rm b}}{2} + \frac{\sigma^2}{3 M^6} \,
, \quad M_{\rm P}^2 = m^2 + \frac{3 M^6}{2 \sigma} \, ,$$ $$Q_{ab} = \frac13 E E_{ab} - E_{ac} E^{c}{}_b + \frac12 \left(E_{cd} E^{cd} -
\frac13 E^2 \right) h_{ab}$$ is the quadratic expression with respect to the ‘bare’ Einstein equation $E_{ab} \equiv m^2 G_{ab} - \tau_{ab}$ on the brane, $E = h^{ab} E_{ab}$, and $W_{ab} \equiv h^c{}_a h^e{}_b W_{cdef} n^d n^f$ is a projection of the bulk Weyl tensor $W_{abcd}$ to the brane. The limit of $m \to 0$ leads us to the original result of [@SMS].

Contracting Eq. (\[effective\]) once again, one obtains the following scalar equation on the brane: $$\label{2nd}
M^{6}(R-2\Lambda_{\rm b})+(m^2G_{ab}+\sigma h_{ab}-\tau_{ab})(m^2G^{ab}+\sigma
h^{ab}-\tau^{ab})-\frac{1}3(m^2R-4\sigma+\tau)^2=0\, ,$$ where $\tau=\tau_{ab}h^{ab}$. It is argued in [@Shtanov] that, if one is interested only in the evolution on the brane and if no additional boundary or regularity conditions are specified in the bulk (as often is the case), then one is left only with 