 Israel junction conditions

Across a thin timelike matter layer, the induced metric is continuous while the jump in its extrinsic curvature is determined by the surface stress-energy tensor. Choose one common spacelike normal and $K_{ab}=\frac{1}{2}{\partial }_y{h}_{ab}$ in Gaussian normal coordinates. The singular tangential Einstein tensor is $-([K_{ab}]-h_{ab}[K])\delta (y)$; equating it to ${\kappa }_D^2{S}_{ab}\delta (y)$ gives the displayed conditions. Reversing the normal or the sign convention for extrinsic curvature changes the corresponding jump signs consistently. These conditions determine the cosmological expansion law of a thin gravitating brane.