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Sean Carroll's Spacetime and Geometry Chapter 5. Questions 3

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Homework Statement


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.


Homework Equations


Schwarzchild metric
ds^2 = 0 = -(1 - 2GM/r) dt^2 + (1 -2GM/r)^-1 dr^2

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!
 

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Answers and Replies

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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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