of a collapsing one, is of course better described by a Schwarschild metric than by a FRW one. We note that the metric of a collapsing homogenous sphere of dust is given by the Oppenheimer-Snyder solution [@OS; @BS], which is a Schwarzschild metric outside the sphere, matched with a closed FRW metric $$\label{eq:OSmetric}
ds^2 = -dt^2 + a(t)^2(d\chi^2 + \sin^2 \chi d\Omega^2)$$ inside it. The scale factor $a(t)$ in the above metric can be expressed parametrically through the conformal time $\eta$

$$\begin{aligned}
a(t(\eta)) &= \frac12\sqrt{\frac{R^3_0}{2GM_0}}(1+\cos \eta), \label{eq:OSa}\\
t(\eta) &=  \frac12\sqrt{\frac{R_0^3}{2GM_0}}(\eta + \sin \eta), \label{eq:OSt}\end{aligned}$$

with $M_0>0$ the ADM mass of the collapsing sphere and $R_0 \geq 2GM_0$ its initial areal radius. The conformal time at which the sphere is completely inside its Schwarzschild radius is given by $\eta_0 = \cos^{-1}(4GM_0/R_0 -1)$.

The continuous match between the exterior Schwarzschild metric and the interior FRW one and the results of [@DMP] recalled above seem therefore to justify the ansazt of replacing $\lambda _P$ in  by $ \lambda _P a(0)/a(t)$. Indeed such an expression for the effective Planck length converges to the usual value $\lambda_P$ in the limit $M \to 0$ in which the FRW metric becomes Minkowski, as one can easily verify 