I'm looking for applying the Israel junction conditions to the collapse of a spherical region filled with pressure-free dust. The metric inside is a k=1 Friedmann model $$ds^2_{-}=−d\tau^2 +a^2(\tau)􏰋d\chi^2 +sin^2\chi d\Omega^2􏰌$$ The metric outside is the Schwarzschild metric $$ds^2_+ = −fdt^2 + f^{−1}dr^2 + r^2d\Omega^2 \qquad f = 1 − 2M/r $$ The surface separating the two regions is $\Sigma$. My reference book (Gravitation by Padmanabhan (pag.554)) tells that

The outward normal to Σ, treated as a differential form, is given by $n_− = ad\chi$ on the inside and $n_+ = −\dot{R}dt + \dot{T}dr$ on the outside.

Can someone explain me how can I compute them?


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