Proper time to reach the singularity of a Schwarschild black hole

[Reggid]

TL;DR

I have a question on how to calculate the proper time to reach the singularity of a Schwarzschild black hole using Schwarzschild coordinates.

When calculating the proper time along a timelike radial geodesic, with the initial condition that object the starts at rest at some Schwarzschild coordinate $r_0>r_S$, i.e.
$$\frac{\mathrm{d}r}{\mathrm{d}\tau}\Bigg|_{r=r_0}=0\;,$$
after using the equations of motion one finds
$$\mathrm{d}\tau=-\mathrm{d}r\,\sqrt{\frac{r_0}{r_S}}\sqrt{\frac{r}{r_0-r}}\;.$$
So the proper time to fall down to some Schwarzschild coordinate $r<r_0$ is
$$\tau (r)=-\sqrt{\frac{r_0}{r_S}}\int_{r_0}^{r}\mathrm{d}r^\prime\,\sqrt{\frac{r^\prime}{r_0-r^\prime}}=\sqrt{\frac{r_0}{r_S}}\biggl(\sqrt{r(r_0-r)}+r_0\cosh ^{-1}\Bigl(\sqrt{\frac{r}{r_0}}\Bigr)\biggr)\;.$$
From this result one can see that the proper time to reach the event horizon $\tau(r=r_S)$ is finite.

I know that until here everything is fine, but the above integral is also perfectly finite for $r=0$, such that the proper time to reach the singularity is
$$\tau(r=0)=\frac{\pi r_0^{3/2}}{2r_S^{1/2}}\;.$$
Now my question is: is this calculation also reliable beyond the horizon?
On the one hand everything is finite and everything is expressed in terms of invariant proper time (no reference to coordinate time $t$ that diverges at the horizon is needed).
But on the other hand I know that Schwarzschild coordinates only describe the patch of spacetime that lies outside the horizon, so maybe one would first have to go to coordinates that can be extended to the region beyond the horizon to be able to derive that result.
I have the feeling that the result could be correct, but still the way I obtained it is somewhat "sloppy" or not not really OK.

Can somebody help?

Thanks for any answers.

 

[Ibix]

It works because $r$ is definable all the way in (it's the areal radius of the spherical shells reflecting the symmetry of the spacetime), so $dr/d\tau$ is defined everywhere. I agree that using the Schwarzschild coordinate expression of the geodesic equation to get to that point is a bit sketchy because the coordinates don't work at the horizon (although they work above and below it).

I've not done this myself, but I think that formally you could translate everything into a coordinate system that does work at the horizon - e.g. Eddington-Finkelstein coordinates. Since they share a definition of the radial coordinate, $r$, with Schwarzschild coordinates, that ought to give you the same $dr/d\tau$.

 

[Reggid]

Thank you for your answer.

Yes you are right, using Eddington-Finkelstein coordinates it is actually very easy to see that they give you the same expression for $\mathrm{d}r/\mathrm{d}\tau$ (as they should).

 

[pervect]

So the proper time to fall down to some Schwarzschild coordinate $r<r_0$ is
$$\tau (r)=-\sqrt{\frac{r_0}{r_S}}\int_{r_0}^{r}\mathrm{d}r^\prime\,\sqrt{\frac{r^\prime}{r_0-r^\prime}}=\sqrt{\frac{r_0}{r_S}}\biggl(\sqrt{r(r_0-r)}+r_0\cosh ^{-1}\Bigl(\sqrt{\frac{r}{r_0}}\Bigr)\biggr)\;.$$
From this result one can see that the proper time to reach the event horizon $\tau(r=r_S)$ is finite.

I'm not sure this integral is good.

$\cosh^{-1}(x)$ is only real valued if x>1, but $r/r_0$ is <1

The equation to be integrated passes the sanity check that $\frac{dr}{d\tau} = \sqrt{E^2 - 1 + 2M/r} = \sqrt{E^2-1 + r_s/r}$ for some constant E, but I've got my doubts about the integration process.

 

[Reggid]

I'm not sure this integral is good.

cosh−1⁡(x) is only real valued if x>1, but r/r0 is <1

The equation to be integrated passes the sanity check that drdτ=E2−1+2M/r=E2−1+rs/r for some constant E, but I've got my doubts about the integration process.

That's because it should be a $\cos^{−1}$, not a $\cosh^{−1}$.

Sorry for the typo (and thanks for finding it)