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I have read a few different ways of looking at this problem, and it's one of those things where I am happywith the answer, just not how to get there using proper mathematics. My lecturer described this with some complex integrals involving E (but I'm not sure what that is!) but I have found a simpler treatment in a textbook, which revolves around this idea:

Looking at the Schwarzschild metric (in natural units with c = 1), we can get to

dt/dr = (1 - 2GM/r)

so t is the integral of (1 - 2GM/r) from infinity (distant starting point) to 2GM (the event horizon or Schwarzschild radius).

I know the answer is infinity as to a distant observer a falling object never reaches the event horizon, and I understand the physics of why that happens. I am not sure how to properly evaluate this integral to prove it though. My calculus is pretty poor, so I always try and break these things down as much as possible. I got

t = ∫(1 - 3GM/r) dr

t = ∫1 - ∫(2GM/r) dr

t = ∫1 - 2GM∫(1/r) dr

t = [r] - 2GM[ln r]

So when r = 2GM I get

t1 = 2GM - 2GM ln 2GM or just t = x - x ln x

and when r = infinity

t2 = ∞ - 2GM ln ∞

and infall time t = t2 - t1

That all looks like total rubbish to me, and certainly doesn't look like a clear answer that dt = ∞ which is what I'd like to get to. Is there a better way to approach this or have I made a stupid mistake/drastic oversimplification somewhere?

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# Evaluating time for falling body to reach event horizon

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