Gravitational Collapse Calculations Problem Solved

[ShayanJ]

I'm reading T. Padmanabhans General Relativity. In section 7.6, he describes the gravitational collapse of a sphere of pressureless dust(So $T_{\mu \nu}=\rho u_{\mu} u_{\nu}$). I should say his argument is the same as Landau's, but reading Landau's didn't help too.
At first, he assumes a line element of the form:
$ds^2=-d\tau^2+e^{\lambda(\tau,R)}dR^2+[r(\tau,R)]^2(d\theta ^2+\sin^2 \theta d\varphi^2)$
The rest of the calculations are shown in the following pictures.

My problem is with eq. 7.188,7.189 and 7.190. I can't understand why should we consider three cases for the function f. Because if this is only an arbitrary function we use in constructing the solution, then choosing one region would do the job and it makes no difference which region we choose. But then is it actually considering three regions for an arbitrary function? Or a physical interpretation is being attached to the function f and the three regions are somehow different parts of the collapse?
I'll appreciate any exlpanation.
Thanks

 

[PeterDonis]

if this is only an arbitrary function we use in constructing the solution, then choosing one region would do the job

No, it wouldn't, because you might not be covering all possible initial conditions. The three different possibilities for $f$ correspond to three different kinds of initial conditions: if I'm remembering correctly, $f > 0$, $f = 0$, and $f < 0$ correspond to collapses where the inward velocity of the dust a a given radius is greater than, equal to, or less than the "escape velocity" at that radius, which is $\sqrt{2M / r}$ ( $M$ is the total mass of the dust as measured from far away, which will be constant during the collapse). All three possibilities must be considered since they are all physically possible initial conditions.