- #1
e^ipi=-1
- 5
- 0
Homework Statement
Determine the Christoffel symbol [tex]\Gamma^{t}_{xx}[/tex] for the metric [tex]ds^2 = -c^2dt^2 + (1+h\sin(\omega t))dx^2 + (1-h\sin(\omega t))dy^2 + dz^2[/tex]
The answer should be: [tex]\frac{h\omega}{2} \cos(\omega t)[/tex]
Homework Equations
For the evaluation we have to use [tex]\frac{d^2x^\alpha}{d\tau^2}+\Gamma^\alpha_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}[/tex]
The Attempt at a Solution
I keep getting c's where they shouldn't be. I calculated the Euler Lagrange Equation for the time to be:
[tex]-2\frac{d^2(ct)}{d\tau^2}-c^{-1}\omega h\cos(\omega t)((\frac{dx}{d\tau})^2 - (\frac{dy}{d\tau})^2) = 0[/tex]
Which leaves us with the equation of motion
[tex]\frac{d^2 t}{d\tau^2}+\frac{1}{2c^2}\omega h\cos(\omega t)((\frac{dx}{d\tau})^2 - (\frac{dy}{d\tau})^2) =0[/tex]
So the answer is:
[tex]\Gamma^{t}_{xx}=\frac{h\omega}{2c^2} \cos(\omega t)[/tex]
Where have I gone wrong? Also, I don't understand whether you are supposed to take t or ct as the zero'th coordinate and whether it gives a different answer.