1. The problem statement, all variables and given/known data I'm not in grad school but I've been trying to teach myself some GR and I asked a professor what problems he thought would be good to study. He mentioned this one. I'd ask him for help, but he's out of town this week. I've also attached a picture to this problem. (It seems many professors use this problem) Consider a particle (not necessarily on a geodesic) that has fallen inside the event horizon of a black hole, r < r_s Show that the radial coordinate must decrease at a minimum rate given by dr/dt = (2GM/r - 1 )^1/2 Calculate the maximum lifetime for a particle along a trajectory from r = 2GM/c^2 to r = 0. Express this in seconds for the supermassive black hole Sagittarius A∗ in the center of our galaxy, whose mass is about 8 · 1036 kg. Show that this maximum proper time is achieved by a radial free fall. 2. Relevant equations Schwarzchild metric ds^2 = 0 = -(1 - 2GM/r) dt^2 + (1 -2GM/r)^-1 dr^2 3. The attempt at a solution I simply solve for dr/dt but I ended up with dr/dt = (1 - 2GM/r) I believe I'm missing some information. I've read the chapter a few times, but maybe I've missed something. As for the second part. I believe you just integrate from those boundaries. 2GM to zero. Any help setting me on the right track would be great! Thanks!