Proper distance in Schwarzschild metric

[WendysRules]

Homework Statement

Let the line element be defined as $ds^2 = -(1-\frac{2m}{r})dt^2+\frac{dr^2}{1-\frac{2m}{r}}+r^2 d\theta^2 + r^2 \sin^2{\theta} d\phi^2$

a) Find a formula for proper distance between nearby spherical shells, assuming only the radius changes, and $r > 2m$
b) Now look at as r approaches 2m from the positive side. How far away is the horizon? Will you ever reach the horizon?
c) Now use your formula to find the distance between two concentric spherical shells around a black hole with mass = 5km. The first shell has circumference = $2\pi r$ the second shells has circumference $2\pi(r+\Delta r)$ where $\Delta r = 100 cm$

Estimate the radial distance for:
$r = 50km$
$r = 15 km$
$r = 10.5 km$

Homework Equations

The Attempt at a Solution

a) If only radius changes, can I just assume that $dt = d\theta = d\phi = 0$? If I can, then $ds = \frac{dr}{\sqrt{1-\frac{2m}{r}}}$ would be the formula for nearby spherical shells.

b) As r approaches 2m, our horizon seems to get bigger and bigger from the formula? I don't think we could ever reach the horizon because we get closer and closer to "infinity".

c) Uhh, this one I'm the most unconfident on. My formula gives me the distance between the shells, but, I'm not sure exactly which numbers to use. If $r = 50km$ and $m = 5km$ then $ds = \frac{dr}{\sqrt{1-\frac{20km}{50km}}} = \frac{dr}{.775}$ which would say that my curvature looks bigger because $\frac{dr}{.775 km} > dr$ but I didn't use anything with my shells. So, I don't think that's how I should do these.

My other thought was to say that $dr = C_2 - C_1 = 2\pi(50.1 km) - 2\pi(50km) = .2 \pi km$ so then using the work above, we would say that $ds = \frac{.2 \pi km}{.775 km} = .811 km$ but then again to say that dr is the difference of my circumference is a little weird and I'm sure is not right. Any help is appreciated!

 

[Tio Barnabe]

Question (c) seems unusual for me, because as long as I know circumferences are defined only for 2D objects. I think what the question asks is to consider viewing the system black hole + spherical shells through a "top view" in space. In that case, we could talk about a circumference for the shells. Then, the distance would be given by $$\int_{\Delta r} \frac{dR}{\sqrt{1- \frac{2m}{R}}}$$ and this would be the same for the three given values of $r$.