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

## 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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