In the thin shell formalism, the components of the surface stress-energy tensor are found using the next equation:
\begin{equation} S^{a}_{\, \, \, b} = const\left([K^{a}_{\, \, \, b}] - \delta^{a}_{\, \,\, b}[K] \right) ,\tag{12.125} \end{equation} where $K = h^{ab}K_{ab}$ and $[K] = K_{+} - K_{-} $ defines extrinsic curvature's jump.
Moreover, using the following equations, one can easily encounter the extrinsic curvature components:
\begin{equation} K_{\tau\tau} = -n_{\mu}a^{\mu} \\ K_{\phi\phi} = - \Gamma^{\mu}_{\phi\phi}n_{\mu} \end{equation}
However, I am not understanding how to compute the jump after having the expression for the extrinsic curvature. How do you compute this explicitly for any two spacetimes metrics?
Reference:
Padmanabhan's book - "Gravitation: Foundations and Frontiers". The surface stress-energy tensor is defined on page 554.
