Particle falling into a black hole singularity within a finite proper time

[camipol89]

Hello,
Can anyone help me with that?
It's a problem taken from Wald book on General Relativity,in the section of Schwarzschild solution
Thanks


Show that any particle (not necessarily in geodesic motion) in region II (r <
2M ) of the extended Schwarzschild spacetime, Figure 6.9, must decrease
its radial coordinate at a rate given by |dr/dτ | ≥ [2M/r − 1]1/2 . Hence,
show that the maximum lifetime of any observer in region II is τ = πM
[∼ 10−5 (M/M⊙ ) s], i.e., any observer in region II will be pulled into the
singularity at r = 0 within this proper time. Show that this maximum time
is approached by freely falling (i.e., geodesic) motion from r = 2M with
E → 0.

 

[pervect]

For part 1, the world-line of a free-falling particle must be timelike. Write down the appropriate condition for this to be true.

 

[camipol89]

I thought about it but I'm not sure wether should I use schwarzschild coordinates nor kruskal coordinate,since I'm supposed to be in a regon with r<2M.
The proper time for the obsverver is,I think, d/dτ= [(2M/r) − 1]^-1/2.
The thing is is,when I try to write down the explicit expression for dr/dτ I don't know what to do...Should I use the fact that the velocity I'm calculating is a timelike vector and thus has norm = -1?