Israel first junction condition I am studying Eric Poisson's book A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics and it states that the metric, $g_{\alpha \beta}$, can be expressed as a distributions-valued tensor:
$$g_{\mu\nu} = \Theta(\ell) g^{+}_{\mu\nu}+\Theta(-\ell) g^{-}_{\mu\nu}\tag{1} \, .$$
According to the book, differentiating this equation, one should get the following:
$$g_{\mu\nu , \gamma} = \Theta(\ell) g^{+}_{\mu\nu , \gamma}+\Theta(-\ell) g^{-}_{\mu\nu, \gamma} + \epsilon \delta(\ell)\left[ g_{\mu \nu} \right] n_{\gamma}\tag{2} \, .$$
However, I can't reach the last term. The book also says that from $(1)$ to $(2)$, the following identity is used:
$$n_{\alpha}  = \epsilon \partial_{\alpha}\ell \, .$$ Nonetheless, I can't calculate the last term of Eq.$(2)$. Can you help me, please?

*

*$\ell$ is the proper distance (or time) along the geodesics.
$$
n^{\alpha} n_{\alpha} = \epsilon 
= \begin{cases}
1, \text{if the hypersurface is timelike}\\
-1, \text{if the hypersurface is spacelike}\\
\end{cases}
$$
 A: I found the "trick".
So, after differentiating we have the following expression:
$$g_{\mu\nu , \gamma} = \Theta(\ell) g^{+}_{\mu\nu , \gamma}+\Theta(-\ell) g^{-}_{\mu\nu, \gamma} + \left(\partial_{\gamma}\Theta(\ell)\right) g^{+}_{\mu\nu} + \left(\partial_{\gamma}\Theta(-\ell)\right) g^{-}_{\mu\nu }\, .$$
We know the next identity $\frac{d\Theta(\ell)}{d\ell} = \delta(\ell)$, so, the idea is to manipulate $\partial_{\gamma}\Theta(\ell)$ to that identity.
$$\partial_{\gamma}\Theta(\ell) = \frac{\partial\Theta(\ell)}{\partial x^{\gamma}} =\frac{\partial \ell}{\partial x^{\gamma}}\frac{\partial \Theta(\ell)}{\partial \ell} =\frac{\partial \ell}{\partial x^{\gamma}}\frac{d \Theta(\ell)}{d \ell} = \frac{\partial \ell}{\partial x^{\gamma}}\delta(\ell) =  \epsilon n_{\gamma} \delta(\ell) \, ,$$ where we have the following trick. Consider the next identity:
$$n_{\gamma}  = \epsilon \partial_{\gamma}\ell \iff \epsilon n_{\gamma} = \epsilon^{2} \partial_{\gamma}\ell \, ,$$ since $\epsilon = 1 \vee -1 \rightarrow \epsilon^{2} = 1$, thus we may write:
$$ \epsilon n_{\gamma} = \epsilon^{2} \partial_{\gamma}\ell \iff \frac{\partial \ell}{\partial x^{\gamma}}= \epsilon n_{\gamma}$$
Finally, we obtain:
$$ g_{\mu\nu , \gamma} = \Theta(\ell) g^{+}_{\mu\nu , \gamma}+\Theta(-\ell) g^{-}_{\mu\nu, \gamma} + \delta(\ell)n_{\gamma} \left(g_{\mu \nu}^{+} - g_{\mu \nu}^{-} \right)
\\
\iff g_{\mu\nu , \gamma} = \Theta(\ell) g^{+}_{\mu\nu , \gamma}+\Theta(-\ell) g^{-}_{\mu\nu, \gamma} + \epsilon \delta(\ell) \left[g_{\mu \nu} \right]n_{\gamma}
\, ,$$
as we wanted.
