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

AI Thread Summary
The discussion revolves around a problem from Wald's book on General Relativity concerning particles in the Schwarzschild solution's region II (r < 2M). It emphasizes that any particle must decrease its radial coordinate at a specific rate, leading to the conclusion that observers in this region will inevitably fall into the singularity at r = 0 within a maximum proper time of τ = πM. The conversation also touches on the necessity of using either Schwarzschild or Kruskal coordinates for calculations in this region. Participants express uncertainty about deriving the explicit expression for dr/dτ and whether to consider the timelike nature of the particle's world-line. The focus remains on understanding the implications of these conditions on particle motion near a black hole singularity.
camipol89
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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.
 
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For part 1, the world-line of a free-falling particle must be timelike. Write down the appropriate condition for this to be true.
 
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?
 
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