components of Christoffel symbol


by latentcorpse
Tags: christoffel, components, symbol
latentcorpse
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#1
Oct28-09, 12:07 PM
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The metric of Euclidean [itex]\mathbb{R}^3[/itex] in spherical coordinates is [itex]ds^2=dr^2+r^2(d \theta^2 + \sin^2{\theta} d \phi^2)[/itex].
I am asked to calculate the Christoffel components [itex]\Gamma^{\sigma}{}_{\mu \nu}[/itex] in this coordinate system.

i'm not too sure how to go about this.

it talks about [itex]ds^2[/itex] being the metric but normally the metric is of the form [itex]g_{ab}[/itex] i.e. a 2-form but ds^2 isn't a 2-form. are these metrics different or do i make [itex]g_{\mu \nu}=ds^2 \omega_{\mu} \omega_{\nu}[/itex] where [itex]\omega_i[/itex] is a 1 form?

i think i'm missing some key point here....
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gabbagabbahey
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#2
Oct28-09, 12:16 PM
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The metric is often defined according to the equation [itex]ds^2=g_{ab}dx^adx^b[/itex]...In this case, you have [itex]x^a\in\{r,\theta,\phi\}[/itex]....so what are the components of [itex]g_{ab}[/itex]?
latentcorpse
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#3
Oct28-09, 12:44 PM
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[tex]g_{ab}=\left[ \begin {array}{ccc} 1&0&0 \\ 0&r&0
\\ 0&0& {\sin}{\theta}\end {array} \right] [/tex]

i didn't know how else to right it.

would that work? because say
[tex]g_{33}=g_{\phi \phi} = \sin^2{\theta} dx^\phi dx^\phi[/tex]

gabbagabbahey
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#4
Oct28-09, 12:58 PM
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components of Christoffel symbol


Don't you mean:

[tex]g_{ab}=\begin{pmatrix}1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & \sin^2\theta \end{pmatrix}[/tex]
latentcorpse
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#5
Oct28-09, 03:34 PM
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yeah sorry. okay so that would work out for the formula [itex]ds^2=g_{ab} dx^a dx^b[/itex]

now i guees i'm supposed to use 3.1.30 in Wald:

[tex]\Gamma^{\sigma}{}_{\mu \nu}=\frac{1}{2} \sum_{\rho} g^{\sigma \rho} \left( \frac{\partial g_{\nu \rho}}{\partial x^{\mu}} + \frac{\partial g_{\mu \rho}}{\partial x^{\nu}} - \frac{\partial g_{\mu \nu}}{\partial x^{\sigma}} \right)[/tex]

im confused about how this sum is going to work though.
[itex]\sigma,\nu,\mu \in \{ r, \theta, \phi \}[/itex] and so they aren't fixed variables...which is confusing
also what values does [itex]\sigma[/itex] take?
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#6
Oct28-09, 05:21 PM
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Quote Quote by latentcorpse View Post
[itex]\sigma,\nu,\mu \in \{ r, \theta, \phi \}[/itex] and so they aren't fixed variables...which is confusing
also what values does [itex]\sigma[/itex] take?
No, all of [itex]\mu,\nu,\sigma,\rho\in\{1,2,3\}[/itex], with [itex]x^{1}=r[/itex], [itex]x^{2}=\theta[/itex] and [itex]x^{3}=\phi[/itex]

Also, your last term in [itex]\Gamma^{\sigma}{}_{\mu\nu}[/itex] has a typo.

So, for example,

[tex]\Gamma^{1}{}_{23}=\frac{1}{2} \sum_{\rho} g^{1\rho} \left( \frac{\partial g_{3\rho}}{\partial \theta} + \frac{\partial g_{2\rho}}{\partial \phi} - \frac{\partial g_{23}}{\partial x^{\rho}} \right)=\frac{1}{2} g^{11} \left( \frac{\partial g_{31}}{\partial \theta} + \frac{\partial g_{21}}{\partial \phi} - \frac{\partial g_{23}}{\partial r} \right)=0
[/tex]
latentcorpse
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#7
Oct29-09, 07:56 AM
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ahh i think i get it. the sum reduces to just the [itex]\rho=1[/itex] term because [itex]g_{12}=g_{13}=0[/itex] which zeroes the whole expression in the cases of [itex]\rho=2[/itex] or [itex]\rho=3[/itex].

so [itex]\Gamma^{\sigma}{}_{\mu \nu}[/itex] will have [itex]3^3=27[/itex] copmonents, correct? i can't write my final answer as a matrix can i?
i'd just have to write them out explicitly as:
[itex]\Gamma^1_{11}= ...[/itex]
[tiex]\Gamma^1_{12}= ...[/itex]
etc.

that doesn't look very concise though?
gabbagabbahey
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#8
Oct29-09, 08:13 AM
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It will look more concise once you realize just how many of those 27 components are zero (Also, you should keep in mind that [itex]g^{ab}[/itex] is the inverse of [itex]g_{ab}[/itex] when doing your calculations)\

As a matter of convention, [itex]\Gamma^1{}_{23}[/itex] is often written as [itex]\Gamma^r_{\theta\phi}[/itex] and so on; which may be what was confusing you earlier.
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#9
Oct29-09, 08:18 AM
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Also, my earlier matrix contains a typo, it should be:

[tex]g_{ab}=\begin{pmatrix}1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\theta \end{pmatrix}[/tex]
latentcorpse
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#10
Oct29-09, 08:27 AM
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why is [itex]g^{ab}[/itex] the inverse of [itex]g_{ab}[/itex] and how will that be useful?

also when u say [itex]\sigma \in \{ 1,2,3 \}[/itex] and [itex]x^1=r, x^2= \theta, x^3 = \phi[/itex]
this means that if [itex]\sigma=1[/itex] then [itex]\sigma=r[/itex] and that's why we can write [itex]\Gamma^{1}{}_{23}=\Gamma^r{}_{\theta \phi}[/itex]
doesn't that imply that [itex]1=r[/itex] rather than [itex]x^1=r[/itex]?
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#11
Oct29-09, 08:49 AM
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Quote Quote by latentcorpse View Post
why is [itex]g^{ab}[/itex] the inverse of [itex]g_{ab}[/itex] and how will that be useful?
Because of its definition; [itex]g^{ab}g_{ac}=\delta^{b}{}_{c}[/itex]....which tells you that multiplying the matrix [itex]g^{ab}[/itex] with the matrix [itex]g_{ac}[/itex] produces the identity matrix....i.e. [itex]g^{ab}[/itex] is the inverse of [itex]g_{ab}[/itex].

It's useful, because you will need to know the components of [itex]g^{ab}[/itex] to compute the Christoffel symbols; and you can get those components just by taking the inverse of [itex]g_{ab}[/itex]

also when u say [itex]\sigma \in \{ 1,2,3 \}[/itex] and [itex]x^1=r, x^2= \theta, x^3 = \phi[/itex]
this means that if [itex]\sigma=1[/itex] then [itex]\sigma=r[/itex] and that's why we can write [itex]\Gamma^{1}{}_{23}=\Gamma^r{}_{\theta \phi}[/itex]
doesn't that imply that [itex]1=r[/itex] rather than [itex]x^1=r[/itex]?
That's why that notation is often confusing; nevertheless, it is still the convention....
latentcorpse
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#12
Oct29-09, 09:04 AM
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so because [itex]1=r[/itex] and [itex]x^1=r[/itex], don't you mean to write that [itex]\sigma \in \{ x^1,x^2,x^3 \}[/itex]?

i used maple to quickly get [itex]g^{ab}=\left[ \begin {array}{ccc} 1&0&0\\ \noalign{\medskip}0&{r}^{-2}&0
\\ \noalign{\medskip}0&0&{\frac {1}{{r}^{2}{\sin}^{2}\theta}}
\end {array} \right] [/itex]
latentcorpse
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#13
Oct29-09, 09:53 AM
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scratch that above post.

i couldn't think of any quick way to do it so i just did all 27 calculations and found the non zero terms are :

[itex]\Gamma^1{}_{22}=-r[/itex]
[itex]\Gamma^1{}_{33}=-r \sin^2{\theta}[/itex]
[itex]\Gamma^2{}_{12}=r^3[/itex]
[itex]\Gamma^2{}_{21}=r^3[/itex]
[itex]\Gamma^2{}_{33}=-\frac{1}{2}r^4 \sin{2 \theta}[/itex]
[itex]\Gamma^3{}_{13}=r^3 \sin^4{\theta}[/itex]
[itex]\Gamma^3{}_{23}=r^4 \sin^3{\theta} \cos{\theta}[/itex]
[itex]\Gamma^3{}_{31}=r^3 \sin^4{\theta}[/itex]
[itex]\Gamma^3{}_{32}=r^4 \sin^3{\theta} \cos{\theta}[/itex]

i'm not sure if there's a pattern i was supposed to spot so i could save myself some time in working out the copmonents or what?

anyway, when it asks for the components of the Christoffel symbol, do i just leave it as a list of the non zero ones like i have done above or am i missing how to write the whole thing neatly as a matrix or something?
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#14
Oct29-09, 09:56 AM
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Quote Quote by latentcorpse View Post
so because [itex]1=r[/itex] and [itex]x^1=r[/itex], don't you mean to write that [itex]\sigma \in \{ x^1,x^2,x^3 \}[/itex]?
No, [itex]\sigma \in \{ 1,2,3 \}[/itex] with [itex]x^1=r[/itex], [itex]x^2=\theta[/itex] and [itex]x^3=\phi[/itex]....so the Christoffel symbols should be labeled [itex]\Gamma^1{}_{23}[/itex] etc... But, by convention they are often labeled [itex]\Gamma^r_{\theta\phi}[/itex] etc...it's sloppy notation to do this, but nevertheless, convention is convention.
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#15
Oct29-09, 10:01 AM
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Quote Quote by latentcorpse View Post
[itex]\Gamma^2{}_{12}=r^3[/itex]
[itex]\Gamma^2{}_{21}=r^3[/itex]
[itex]\Gamma^2{}_{33}=-\frac{1}{2}r^4 \sin{2 \theta}[/itex]
You seem to be missing a factor of [itex]1/r^4[/itex]


[itex]\Gamma^3{}_{13}=r^3 \sin^4{\theta}[/itex]
[itex]\Gamma^3{}_{23}=r^4 \sin^3{\theta} \cos{\theta}[/itex]
[itex]\Gamma^3{}_{31}=r^3 \sin^4{\theta}[/itex]
[itex]\Gamma^3{}_{32}=r^4 \sin^3{\theta} \cos{\theta}[/itex]
And for these you are missing a factor of [itex]1/(r^4\sin^4\theta)[/itex]

i'm not sure if there's a pattern i was supposed to spot so i could save myself some time in working out the copmonents or what?
You could save a little time by remembering that the Christoffel symbols are symmetric in the bottom pair of indices...so you only have to calculate 18 of them.

anyway, when it asks for the components of the Christoffel symbol, do i just leave it as a list of the non zero ones like i have done above or am i missing how to write the whole thing neatly as a matrix or something?
You can't really write them as a single matrix, but they are often written as a set of 3 matrices in the form:

[tex]\Gamma^r=\begin{pmatrix}0 & 0 & 0 \\ 0 & -r & 0 \\ 0 & 0 & -r\sin^2\theta\end{pmatrix}[/tex]

[tex]\Gamma^\theta=\begin{pmatrix}0 & \frac{1}{r} & 0 \\ \frac{1}{r} & 0 & 0 \\ 0 & 0 & -\sin\theta\cos\theta\end{pmatrix}[/tex]

[tex]\Gamma^\phi=\begin{pmatrix}0 & 0 & \frac{1}{r} \\ 0 & 0 & \cot\theta \\ \frac{1}{r} & \cot\theta & 0\end{pmatrix}[/tex]

Again, this is somewhat sloppy notation, but is still fairly common in the literature.
latentcorpse
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#16
Oct29-09, 10:04 AM
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are they ok then?
gabbagabbahey
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#17
Oct29-09, 10:18 AM
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See my edited post above^^^
latentcorpse
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#18
Oct29-09, 10:34 AM
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i dont see how im missing those factors. take for example

[itex]\Gamma^2{}_{12}=\frac{1}{2} g^{22} \left( \frac{\partial g_{22}}{\partial r} + \frac{\partial g_{12}}{\partial \theta} - \frac{\partial g_{12}}{\partial \theta} \right) = \frac{1}{2} r^2 \frac{\partial}{\partial r} \left(r^2 \right) = r^3[/itex]

i can't see where i'm missing this factor?


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