- #1
whatisreality
- 290
- 1
Homework Statement
Given that ##ds^2 = r^2 d\theta ^2 + dr^2## find the geodesic equations.
Homework Equations
The Attempt at a Solution
I think the ##g_{\mu\nu} =
\left( \begin{array}{ccc}
1& 0 \\
0 & r^2 \end{array} \right)##
Then I tried to use the equation
##\tau = \int_{t_1}^{t_2} \sqrt{ g_{\mu\nu}(x(t)) \frac{ dx^{\mu}}{dt}\frac{dx^{\nu}}{dt} } dt##
Which if I expand the sum gives
##\tau = \int_{t_1}^{t_2} \sqrt{ 1 + \left(\frac{ dr}{dt}\right)^2 +r^2 \left(\frac{d\theta}{dt}\right)^2 }## ##dt##
Unfortunately I don't know anything about Lagrangians, which seems to be the normal way to proceed... so I get a bit stuck here. There is an example without Langrangians in the lecture notes but I don't understand what he did to get from here to his ##d\tau##. We've had two lectures on GR so far and I think I've already missed something massively important, that explains how to do this!
Thank you for any help, I really appreciate it.