= Israel junction conditions
{c}
{title2=$[h_{ab}]=0,\quad[K_{ab}]-h_{ab}[K]=-\kappa_D^2S_{ab}$}
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 $K_{ab}=\tfrac12\partial_yh_{ab}$ in Gaussian normal coordinates. The singular tangential <Einstein tensor> is $-([K_{ab}]-h_{ab}[K])\delta(y)$; equating it to $\kappa_D^2S_{ab}\delta(y)$ 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>.
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