Cristoffel Symbol of spherical coordinates

[space-time]

I just derived the 3-D Cristoffel symbol of the 2nd kind for spherical coordinates. I don't think I made any careless mistakes, but once again, I just want to verify that I am correct and I can't find any place on line that will give me the components of the symbol so I can check myself.

Here are the components that I derived: (I won't post the 0 components, nor will I post repeat components. By repeat components I mean: If I post \Gamma²₁₂ then I already know that \Gamma²₂₁ will be the same thing because you can switch around the bottom two indicies.)

\Gamma¹₂₂ = -r

\Gamma¹₃₃ = -rsin²(θ)

\Gamma²₁₂= 1/r

\Gamma²₃₃= -sin(θ)cos(θ)

\Gamma³₁₃= 1/r

\Gamma³₂₃ = cot(θ)

I used the metric tensor and derivatives of metric tensors formula to derive these components.

Can someone please look at these components that I derived and verify for me if I am right or not. I can provide the metric tensor that I used upon request if you wish to see further work.

 

[PeterDonis]

I can provide the metric tensor that I used upon request if you wish to see further work.

You should provide the metric tensor, yes. Otherwise we won't know how to check your work.

 

[space-time]

You should provide the metric tensor, yes. Otherwise we won't know how to check your work.

Here is my metric tensor g_ij :

g₁₁ = 1

g₂₂ = r²

g₃₃= r²sin²(θ)

All other elements were 0. It was a 3 by 3 matrix.

If you want my inverse version (the contravariant version that appears in the formula), then it is below:

g¹¹= 1

g²²= 1/r²

g³³ = 1/(r²sin²(θ))

Once again all other elements were 0 and it was a 3 by 3 matrix.

 

[PeterDonis]

All other elements were 0. It was a 3 by 3 matrix.

Ah, ok. You mentioned "0 components" in the OP, but if you're just working in 3-dimensional Euclidean space, which you are with this metric, there are no "0" components. Your results look OK to me.

 

[space-time]

Ah, ok. You mentioned "0 components" in the OP, but if you're just working in 3-dimensional Euclidean space, which you are with this metric, there are no "0" components. Your results look OK to me.

Thank you very much.

By the way, do you know of any places where I can just quickly check this stuff on line (for future reference)?

Edit: Oh and by 0 components, I didn't mean 0th dimension like time or anything like that. I just meant elements that were 0. Sorry for the confusion.

 

[pervect]

Thank you very much.

By the way, do you know of any places where I can just quickly check this stuff on line (for future reference)?

There are some programs that automate such calculations, some of them are free like Maxima.

 

[PeterDonis]

By the way, do you know of any places where I can just quickly check this stuff on line (for future reference)?

Googling will sometimes turn up an online reference, but I don't know of any site that specifically tabulates this sort of thing in a systematic fashion for lots of different coordinate charts.

To make these computations easier, I highly recommend learning how to use a symbolic math package. I use Maxima; other popular ones are Maple and MATLAB (which have the disadvantage of being a lot more expensive than Maxima, which is free ). You can find out more about Maxima here:

http://en.wikipedia.org/wiki/Maxima_(software )

Maxima also has a package available called GRTensor that is specifically for computing things like Christoffel symbols.