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

[fu11meta1]

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

Schwarzschild 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!

 

[mark726]

Hi there,

First of all, it's great that you're teaching yourself GR and seeking out challenging problems to work on! It's always helpful to have a mentor or professor to guide you, but it's also admirable to take the initiative to learn on your own.

To solve this problem, you'll need to use the Schwarzschild metric and the equation for proper time (dτ^2 = -ds^2). You're correct in that you need to solve for dr/dt, but you also need to take into account the fact that the particle is inside the event horizon, which means r < r_s.

To start, you can substitute the Schwarzschild metric into the equation for proper time and use the fact that ds^2 = 0 for a particle in free fall. This will give you an expression for dτ^2 in terms of dr and dt.

Next, you can use the fact that the particle is in free fall to set the acceleration (d^2r/dt^2) equal to zero. This will give you a differential equation for dr/dt in terms of r.

Finally, you can solve this differential equation to get an expression for dr/dt in terms of r. This will give you the minimum rate at which the radial coordinate must decrease for a particle inside the event horizon.

For the second part of the problem, you'll need to integrate this expression from r = 2GM/c^2 to r = 0 to find the maximum proper time. You can then convert this proper time into seconds using the mass of Sagittarius A* and the speed of light.

I hope this helps guide you in the right direction. Good luck with your studies!