- #1
Vrbic
- 407
- 18
Homework Statement
Originally the statement:
Find a length of two points on sphere. It was easy.
##\int \sqrt{g_{\phi\phi}}d\phi##
I hope you agree :-) But I have idea, how to find a length of path which is NOT a part of arc (circle). For example sinusoid. Is possible to find length of sinusoid on the sphere and how?
Homework Equations
##ds^2=g_{rr}dr^2+g_{\theta\theta}d\theta^2+g_{\phi\phi}d\phi^2##
The Attempt at a Solution
My attempt hit the snag very early :-)
A took ##\theta=\pi/2+\sin{\phi}##
##d\theta=\cos{\phi}d\phi##
##ds^2=0+r^2d\theta^2+r^2\sin^2{\theta}d\phi^2##
##ds=r\sqrt{\cos^2{\phi}+\sin^2{(\pi/2+\sin{\phi})}}d\phi##
And now I don't know. I'm not sure if my procedure is so naive, and it exists better, or such problem doesn't have an analytical solution.
Please advice.