actually no black-hole horizon.

Junction conditions {#subsec:junction_conditions}
-------------------

Any spacetime obtained by gluing together two different geometries, via a hypersurface $\Sigma$, is well defined if—and only if—it satisfies the Israel junction conditions [@Isr66; @Isr67]: both geometries must induce (a) the same 3-metric, and (b) the same extrinsic curvature on $\Sigma$.

The junction hypersurface $\Sigma$ is the world sheet of a comoving 2-sphere, as imposed by the symmetry of the problem. Hence, it is defined by $\chi=\chi{_{\text{h}}}=\text{cst}$ in FL coordinates, and by $r=r{_{\text{h}}}(t)$ in Kottler coordinates. Both points of view are depicted in Fig. \[fig:junction\_hypersurface\].

![The junction hypersurface as seen from the FL point of view with equation $\chi=\chi{_{\text{h}}}$ (left); and from the Kottler point of view with equation $r=r{_{\text{h}}}(t)$ (right).[]{data-label="fig:junction_hypersurface"}](matchingsurf.pdf){width="\columnwidth"}

In the FL region, the normal vector to the hypersurface is given by $n^{({\rm FL})}_\mu=\delta_\mu^\chi/a$. The 3-metric and the extrinsic curvature induced by the FL geometry are respectively $$\begin{aligned}
{{\rm d}}s^2_\Sigma &= - {{\rm d}}T^2 + a^2(T)f_K^2(\chi{_{\text{h}}})\,{{\rm d}}\Omega^2,\\
K^{({\rm FL})}_{ab}{{\rm d}}x^a {{\rm d}}x^b &= a(T)f_K(\chi{_{\text{h}}})f'_K(\chi{_{\text{h}}})\,{{\rm d}}\Omega^2.\end{aligned}$$ where $(x^a) = (T,\theta,{\varphi})$ are natural intrinsic coordinates for $\Sigma$. We stress carefully that, in the following and as long as there is no ambiguity, a dot can denote a time derivative with respect 