Christoffel symbols for gravitational waves

[e^ipi=-1]

Homework Statement

Determine the Christoffel symbol $$\Gamma^{t}_{xx}$$ for the metric $$ds^2 = -c^2dt^2 + (1+h\sin(\omega t))dx^2 + (1-h\sin(\omega t))dy^2 + dz^2$$

The answer should be: $$\frac{h\omega}{2} \cos(\omega t)$$

Homework Equations

For the evaluation we have to use $$\frac{d^2x^\alpha}{d\tau^2}+\Gamma^\alpha_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}$$

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:
$$-2\frac{d^2(ct)}{d\tau^2}-c^{-1}\omega h\cos(\omega t)((\frac{dx}{d\tau})^2 - (\frac{dy}{d\tau})^2) = 0$$
Which leaves us with the equation of motion
$$\frac{d^2 t}{d\tau^2}+\frac{1}{2c^2}\omega h\cos(\omega t)((\frac{dx}{d\tau})^2 - (\frac{dy}{d\tau})^2) =0$$
So the answer is:
$$\Gamma^{t}_{xx}=\frac{h\omega}{2c^2} \cos(\omega t)$$
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.

 

[sgd37]

are you sure you're not working in natural units because I get the same answer as you. i don't think it cancels. you could redefine t' = ct but that still gives you factors of c

 

[WannabeNewton]

Don't know if you already figured it out but the problem probably wanted it in natural units. Also, when you only have one specific Christoffel symbol to calculate it is much easier to just use the equation for the Christoffel symbols in terms of permutations of first derivatives of the metric; I am sure you know which one this is (too lazy to do all the latex business).