- #1

JD_PM

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- TL;DR Summary
- This thread is aimed at gaining insight on the Christoffel symbol in GR context.

What I would like to discuss is the following:

1) Is there a Physics reasoning on why for curvilinear coordinate systems not all Christoffel symbol components vanish? Honestly, this fact goes against my logic.

2) Could you give me an extra exercise that gives me more insight on Christoffel symbols? This technique has really worked for me in PF. See below for evidence.

I was reading chapter 3 of Carroll's book up to page 100, where he mentions that the coefficients of Christoffel symbol in flat spacetime vanish in cartesian coordinates but not in curvilinear coordinate systems.

I was thinking on why this happens. My logic tells me that these coefficients should also vanish while using other coordinate system as the spacetime remains flat anyway.

I wanted to check that the coefficients indeed do not vanish using curvilinear coordinate systems and I picked as an example spherical coordinates.

The metric in spherical coordinates is

$$ds^2 = dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta d \phi^2$$

We also know that the metric tensor in spherical coordinates is

$$g_{\mu \nu} = \begin{pmatrix}1&0&0\\0&r^2&0\\0&0&r^2 \sin^2\theta\\ \end{pmatrix}$$

The definition of the Christoffel symbol is

$$\Gamma_{\mu \nu}^{\sigma} = \frac 1 2 g^{\sigma \rho}(\partial_{\mu}g_{\nu \rho} + \partial_{\nu}g_{\rho \mu} - \partial_{\rho}g_{\mu \nu})$$

OK so based on the Christoffel symbol definition we see that it is certainly possible to get non-zero terms iff one of the indices appears twice! Actually I see 6 independent terms (by independent I mean that there are more nontrivial components that can be obtained out of these 6 due to the lower symmetry of the Christoffel symbol: ##\Gamma_{\mu \nu}^{\sigma} = \Gamma_{\nu \mu}^{\sigma}##).

For the sake of completeness I computed them:

$$\Gamma_{\theta \theta}^{r} = -r$$

$$\Gamma_{\phi \phi}^{r} = -r \sin^2 \theta$$

$$\Gamma_{\phi \phi}^{\theta} = -\sin \theta \cos \theta$$

$$\Gamma_{\theta r}^{\theta} = 1/r$$

$$\Gamma_{\phi \theta}^{\phi} = \cot \theta$$

$$\Gamma_{\phi r}^{\phi} = 1/r$$What I would like to discuss is the following:

1) Is there a Physics reasoning on why for curvilinear coordinate systems not all Christoffel symbol components vanish? Honestly, this fact goes against my logic.

2) Could you give me an extra exercise that gives me more insight on Christoffel symbols? This technique has really worked for me in PF. Evidence:

1) Orodruin's #39 here https://www.physicsforums.com/threa...rem-to-get-a-constant-of-motion.982721/page-2

2) Orodruin's #19 here https://www.physicsforums.com/threa...leaves-the-given-lagrangian-invariant.984601/

3) PeterDonis #2 here https://www.physicsforums.com/threa...-timelike-vectors-and-inertial-frames.984948/

Thank you very much.

JD.

I was thinking on why this happens. My logic tells me that these coefficients should also vanish while using other coordinate system as the spacetime remains flat anyway.

I wanted to check that the coefficients indeed do not vanish using curvilinear coordinate systems and I picked as an example spherical coordinates.

The metric in spherical coordinates is

$$ds^2 = dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta d \phi^2$$

We also know that the metric tensor in spherical coordinates is

$$g_{\mu \nu} = \begin{pmatrix}1&0&0\\0&r^2&0\\0&0&r^2 \sin^2\theta\\ \end{pmatrix}$$

The definition of the Christoffel symbol is

$$\Gamma_{\mu \nu}^{\sigma} = \frac 1 2 g^{\sigma \rho}(\partial_{\mu}g_{\nu \rho} + \partial_{\nu}g_{\rho \mu} - \partial_{\rho}g_{\mu \nu})$$

OK so based on the Christoffel symbol definition we see that it is certainly possible to get non-zero terms iff one of the indices appears twice! Actually I see 6 independent terms (by independent I mean that there are more nontrivial components that can be obtained out of these 6 due to the lower symmetry of the Christoffel symbol: ##\Gamma_{\mu \nu}^{\sigma} = \Gamma_{\nu \mu}^{\sigma}##).

For the sake of completeness I computed them:

$$\Gamma_{\theta \theta}^{r} = -r$$

$$\Gamma_{\phi \phi}^{r} = -r \sin^2 \theta$$

$$\Gamma_{\phi \phi}^{\theta} = -\sin \theta \cos \theta$$

$$\Gamma_{\theta r}^{\theta} = 1/r$$

$$\Gamma_{\phi \theta}^{\phi} = \cot \theta$$

$$\Gamma_{\phi r}^{\phi} = 1/r$$What I would like to discuss is the following:

1) Is there a Physics reasoning on why for curvilinear coordinate systems not all Christoffel symbol components vanish? Honestly, this fact goes against my logic.

2) Could you give me an extra exercise that gives me more insight on Christoffel symbols? This technique has really worked for me in PF. Evidence:

1) Orodruin's #39 here https://www.physicsforums.com/threa...rem-to-get-a-constant-of-motion.982721/page-2

2) Orodruin's #19 here https://www.physicsforums.com/threa...leaves-the-given-lagrangian-invariant.984601/

3) PeterDonis #2 here https://www.physicsforums.com/threa...-timelike-vectors-and-inertial-frames.984948/

Thank you very much.

JD.