- #1

- 640

- 153

- Homework Statement
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- Relevant Equations
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The metric is $$ds^2 = \frac{dr^2 + r^2 d\theta ^2}{r^2-a^2} - \frac{r^2 dr^2}{(r^2-a^2)^2}$$

I need to prove the geodesic is: $$a^2 (\frac{dr}{d \theta})^2 + a^2 r^2 = K r^4$$

My method was to variate the action ##\int\frac{(\frac{dr}{d\theta})^2 + r^2 }{r^2-a^2} - \frac{r^2 (\frac{dr}{d\theta})^2}{(r^2-a^2)^2} d \theta##

But the equation i am getting is $$a^2 (\frac{dr}{d \theta})^2 + r^2 = K (r^2-a^2)^2$$

Which can not be reduced to the answer.

I am a little confused, i could simpy calculate the Christoffel symbol, but i think this variation method easier, yet i am not sure how to use it.

So basically, my guess on why i have got the wrong answer is that the "lagrangean" i am variating is wrong. So what lagrangean indeed give us the right answer? Only ##g_{ab} \frac{dx^a}{ds} \frac{dx^b}{ds}= 1 ## and ##\sqrt{g_{ab} \frac{dx^a}{d\lambda} \frac{dx^b}{d\lambda}}##? Or the lagrangean i used above is right, and i have done some algebric error?

I need to prove the geodesic is: $$a^2 (\frac{dr}{d \theta})^2 + a^2 r^2 = K r^4$$

My method was to variate the action ##\int\frac{(\frac{dr}{d\theta})^2 + r^2 }{r^2-a^2} - \frac{r^2 (\frac{dr}{d\theta})^2}{(r^2-a^2)^2} d \theta##

But the equation i am getting is $$a^2 (\frac{dr}{d \theta})^2 + r^2 = K (r^2-a^2)^2$$

Which can not be reduced to the answer.

I am a little confused, i could simpy calculate the Christoffel symbol, but i think this variation method easier, yet i am not sure how to use it.

So basically, my guess on why i have got the wrong answer is that the "lagrangean" i am variating is wrong. So what lagrangean indeed give us the right answer? Only ##g_{ab} \frac{dx^a}{ds} \frac{dx^b}{ds}= 1 ## and ##\sqrt{g_{ab} \frac{dx^a}{d\lambda} \frac{dx^b}{d\lambda}}##? Or the lagrangean i used above is right, and i have done some algebric error?