- #1
Reggid
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- TL;DR Summary
- 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.
[tex]\frac{\mathrm{d}r}{\mathrm{d}\tau}\Bigg|_{r=r_0}=0\;,[/tex]
after using the equations of motion one finds
[tex]\mathrm{d}\tau=-\mathrm{d}r\,\sqrt{\frac{r_0}{r_S}}\sqrt{\frac{r}{r_0-r}}\;.[/tex]
So the proper time to fall down to some Schwarzschild coordinate ##r<r_0## is
[tex]\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)\;.[/tex]
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
[tex]\tau(r=0)=\frac{\pi r_0^{3/2}}{2r_S^{1/2}}\;.[/tex]
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.
[tex]\frac{\mathrm{d}r}{\mathrm{d}\tau}\Bigg|_{r=r_0}=0\;,[/tex]
after using the equations of motion one finds
[tex]\mathrm{d}\tau=-\mathrm{d}r\,\sqrt{\frac{r_0}{r_S}}\sqrt{\frac{r}{r_0-r}}\;.[/tex]
So the proper time to fall down to some Schwarzschild coordinate ##r<r_0## is
[tex]\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)\;.[/tex]
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
[tex]\tau(r=0)=\frac{\pi r_0^{3/2}}{2r_S^{1/2}}\;.[/tex]
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.