Radial Infall
Why do (shooting) stars fall down from the sky? From the previous post , we know that τ=∫dr±√˜E2−(1−2Mr)(1+˜L2r2). We study the simplified case with ˙ϕ such that ˜L=0, and ˜E=1. Then, the proper time is τ(r)=±∫dr√1−(1−2Mr)=±∫√r2Mdr=±23√2Mr3/2+const Let r=r0 when τ=0. \bbox[5px,border:2px solid #666]{ \tau(r) = \frac{\pm 2}{3\sqrt{2M}}\left(r^{3/2} - r_0^{3/2}\right) } The \pm sign is due to taking square root on the previous equation. If the testing particle is falling in, r decreases with time, then we should take a minus sign. Up until now we have been using the proper time of the testing particle. Now we would like to see a...