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[itex]ds^2=-\frac{dr^2}{(\frac{2M}{r}-1)} + ( \frac{2M}{r}-1) dx^2 + r^2 d \Omega^2[/itex]

Hence show that

*every*timelike curve in region 2 intersects the singularity at r=0 within a proper time no grater than [itex]\pi M[/itex]

So I did the first bit by observing that using the standard Schwarzschild metric, with r>2M, [itex]g_{tt}u^tu^t<0,g_{rr}u^ru^r>0[/itex] which tells us that t behaves as a timelike coordinate here and r as a spacelike coordinate as expected. However, for r<2M, [itex]g_{tt}u^tu^t>0,g_{rr}u^ru^r<0[/itex] and so t behaves in a spacelike manner and r in a timelike manner. Hence we can treat r as a time coordinate.

I have some doubts about this though. Firstly, I have "used" the Schwarzschild metric in the region r<2M where it doesn't apply. Surely, I should want to use ingoing Eddington Finkelstein coordinates or something - these have the advantage of being applicable for r<2M but if I use them then I get [itex]g_{rr}u^ru^r=0[/itex] since there is no rr component!

What have I done wrong here? If anything?

And does anyone have any advice on how to do the next two bits?

Thanks very much!