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.

 

[paranormal]

for your help.

Dear fellow scientist,

I understand your confusion regarding the use of the metric connection and the geodesic equations in General Relativity. Let me try to provide some clarification and help you with your calculations.

Firstly, the indices in the metric connection represent the coordinates of the space-time manifold. In your case, the indices b and c represent the cylindrical polar coordinates (r, θ, z) while the index d represents the dummy index that runs from 1 to 3. This means that when you calculate the geodesic equation for R^{3} in cylindrical polars, you will have to use the metric connection with respect to these coordinates.

Secondly, the geodesic equation is used to calculate the path of a free-falling particle in curved space-time. In order to do so, we need to use the metric connection to calculate the Christoffel symbols, which are then used in the geodesic equation. The Christoffel symbols are essentially the components of the metric connection and they help us understand how the coordinates of a particle change as it moves along a geodesic.

In your case, the geodesic equation for R^{3} in cylindrical polars would be:

\frac{d^{2}x^{a}}{d\lambda^{2}}+\Gamma_{bc}^{a}\frac{dx^{b}}{d\lambda}\frac{dx^{c}}{d\lambda}=0

where x^{a} represents the coordinates of the particle and λ represents an affine parameter along the geodesic. The first term on the left-hand side represents the acceleration of the particle, while the second term involves the Christoffel symbols, as calculated using the metric connection.

I hope this helps in understanding how to use the metric connection and the geodesic equations in General Relativity. Keep exploring and learning, and don't hesitate to reach out for further clarification. All the best in your studies!

Sincerely,