Understanding Metric Connection and Geodesic Equations in General Relativity"

[Mr-R]

Dear all,

In my journey through learning General relativity. I have stumbled upon this problem. I have to calculate the geodesic equation for R$^{3}$ in cylindrical polars. I am not sure how to use the metric connection. The indices confuse me. I would appreciate it if someone could shade some light on it. Every time I try to calculate it I get zero, sometimes due to the first metric tensor and sometime the terms in the parentheses are zeros.

$\Gamma_{bc}^{a}=\frac{1}{2}g^{ad}(\partial_{b}g_{dc}+\partial_{c}g_{db}-\partial_{d}g_{bc})$

As I understand it, the index d is the dummy index and runs from 1 to 3 in this case, right? What about b and c? how do I use them?

Thanks in advance

 

[bloby]

As I understand it, the index d is the dummy index and runs from 1 to 3 in this case, right? What about b and c?

Yes and a, b and c can be 1, 2 or 3 giving 27 gamma's.

I am not sure how to use the metric connection. The indices confuse me.

You have to calculate all the gamma's, but a lot of them are equals or vanish.

Every time I try to calculate it I get zero, sometimes due to the first metric tensor and sometime the terms in the parentheses are zeros.

$\Gamma_{bc}^{a}=\frac{1}{2}g^{ad}(\partial_{b}g_{dc}+\partial_{c}g_{db}-\partial_{d}g_{bc})$

What did you find for the metric in these coordinates?

 

[Mr-R]

I actually got it now

Thanks bloby

 

[bloby]

Ok, you're welcome.