Calculating Time to Singularity in a Schwarzschild Black Hole

[latentcorpse]

An observer falls radially into a Schwarzschild black hole of mass $M$. She starts from rest (i.e. $\frac{dr}{d \tau} = 0$) at $r = 10M$. How much time elapses on her clock before she hits the singularity?

Is my first step to take the metric equation
$ds^2 = - ( 1 - \frac{2M}{r} ) dt^2 + \frac{dr^2}{(1 - \frac{2M}{r})} + r^2 ( d \theta^2 + \sin^2{\theta} d \phi^2 )$
and find the Euler Lagrange equations of motion?

Or is this barking up the wrong tree?

 

[VanOosten]

just did this in class today and after substituting E=(1-2M/r)dt/dT and d(phi) and d(theta)=0 into the metric and after some algebra and assuming E=1 so dr/dT=0 at infinity we got:
(dr/dT)^2=2M/r and therefore
T=integral(sqrt(2M/r) dr) from R0 to R1

please let me know if this is what your looking for or if there is a better easier way

 

[latentcorpse]

just did this in class today and after substituting E=(1-2M/r)dt/dT and d(phi) and d(theta)=0 into the metric and after some algebra and assuming E=1 so dr/dT=0 at infinity we got:
(dr/dT)^2=2M/r and therefore
T=integral(sqrt(2M/r) dr) from R0 to R1

please let me know if this is what your looking for or if there is a better easier way

ok. I'm a bit confused about the E substitution, I get a metric eqn of

$ds^2 = - (1 - \frac{2M}{r}) dt^2 + \frac{dr^2}{1- \frac{2M}{r}}$

Now if I didvide through by $d \tau^2$ I get:

$\frac{ds^2}{d \tau^2} = - ( 1- \frac{2M}{r} ) ( \frac{dt}{d \tau})^2 + \frac{1}{1- \frac{2M}{r}} ( \frac{dr}{d \tau} )^2$

How do I substitute E into that? Do I need to multiply through by $1- \frac{2M}{r}$ first?

Doing this, I get

$(1-\frac{2M}{r})(\frac{ds}{d \tau})^2 = - E^2 + (\frac{dr}{d \tau})^2$
Why do we know to find E by looking at dr/dT at infinity?

Also, I assum you changed $ds= c d \tau = d \tau$
Thank you.

 

[VanOosten]

substitute (1-2M/r)(dt/dT)^2=(E^2)/(1-2M/r) to get
(dr/dT)^2=E^2-(1-2M/r)

and we look at r at infinity because at this limit dr/dT=0 which tells us E must be 1
also yes i assumed ds = dT
hope that helps

 

[latentcorpse]

substitute (1-2M/r)(dt/dT)^2=(E^2)/(1-2M/r) to get
(dr/dT)^2=E^2-(1-2M/r)

and we look at r at infinity because at this limit dr/dT=0 which tells us E must be 1
also yes i assumed ds = dT
hope that helps

Yes but surely we don't want dr/dT=0 at r=infinity. The question tells us that dr/dT=0 at r=10M.
If you follow this through you get E^2=4/5.
This then leads to the following (much more complicated) integral that I'm sure must be wrong!
$\tau = \int_{10M}^0 ( \frac{2M}{r} - \frac{1}{5} ) ^{\frac{1}{2}} dr$

 

[latentcorpse]

bump.