to $T$ or $t$, so that $\dot a = {{\rm d}}a/{{\rm d}}T$ and $\dot r{_{\text{h}}}={{\rm d}}r{_{\text{h}}}/{{\rm d}}t$, while a prime can denote a derivative with respect to $\chi$ or $r$, so that $f_K' = {{\rm d}}f_K/{{\rm d}}\chi$ and $A'={{\rm d}}A/{{\rm d}}r$.

The 3-metric induced on $\Sigma$ by the Kottler geometry is $${{\rm d}}s^2 = - \kappa^2(t)\,{{\rm d}}t^2 + r{_{\text{h}}}^2(t)\,{{\rm d}}\Omega^2,$$ where $$\kappa(t) \equiv \sqrt{\frac{A^2[r{_{\text{h}}}(t)]-\dot r{_{\text{h}}}^2(t)}{A[r{_{\text{h}}}(t)]}} .
\label{eq:def_kappa}$$ Therefore, the first junction condition implies $$\begin{aligned}
r{_{\text{h}}}(t) &= a(T)f_K(\chi{_{\text{h}}}),
\label{eq:relation_between_radii}\\
\frac{{{\rm d}}T}{{{\rm d}}t} &= \kappa(t),
\label{eq:relation_between_times}\end{aligned}$$ which govern the dynamics of the hole boundary, and relate the time coordinates of the FL and Kottler regions.

The extrinsic curvature of $\Sigma$ induced by the Kottler geometry, but expressed in $(x^a)$ coordinates, reads $$K^{({\rm K})}_{ab}{{\rm d}}x^a {{\rm d}}x^b = - \frac{\ddot r{_{\text{h}}} + \kappa^2 A'(r{_{\text{h}}})/2 }{\kappa^3}\,{{\rm d}}T^2 
                                    + \frac{r{_{\text{h}}} A(r{_{\text{h}}})}{\kappa} \, {{\rm d}}\Omega^2 .$$ Hence, the second junction condition is satisfied only if $$\label{e.kappa}
\kappa = \frac{A(r{_{\text{h}}})}{f'_K(\chi{_{\text{h}}})}, 
\qquad \text{whence} \qquad
\frac{{{\rm d}}T}{{{\rm d}}t} = \frac{A[a(T)f_K(\chi{_{\text{h}}})]}{f'_K(\chi{_{\text{h}}})} .$$ It is straightforward to show that Eq. , together with the Friedmann equation , imply that the Kottler and FL regions have the same cosmological constant, and $$\label{e.massK}
M = \frac{4\pi}{3} \rho \, a^3 f_K^3(\chi{_{\text{h}}}).$$

Summary
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Given 