Derivation of the value of christoffel symbol

[Boltzmann2012]

Hi,
I am new to general relativity and as I would like to find out how we could derive the value of christoffel symbol in terms of the metric tensor.
I have also heard that it was given as a definition for the christoffel symbol and would like a clarification on that.

Regards
Bltzmn2012

 

[jtbell]

Does this help?

http://en.wikipedia.org/wiki/Christoffel_symbols

 

[Boltzmann2012]

By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols as a function of the metric tensor:

I didnt quite get that(although I was given the formula underneath).could you please explain what is permuting the indices. I am really new to this stuff.

Regards
Bltzmnn2012

 

[elfmotat]

$$\Gamma^\rho_{~\mu \nu}=\frac{1}{2} g^{\rho \lambda} (\partial_\mu g_{\nu \lambda}+\partial_\nu g_{\mu \lambda}-\partial_\lambda g_{\mu \nu})$$

It's relatively straightforward to calculate the components of the Christoffel symbols from the components of the metric. It can get pretty tedious though.

 

[pervect]

$$\Gamma^\rho_{~\mu \nu}=\frac{1}{2} g^{\rho \lambda} (\partial_\mu g_{\nu \lambda}+\partial_\nu g_{\mu \lambda}-\partial_\lambda g_{\mu \nu})$$

It's relatively straightforward to calculate the components of the Christoffel symbols from the components of the metric. It can get pretty tedious though.

Yes - a few comments on notation for the OP
$$\partial_\mu = \frac{\partial}{\partial_\mu}$$

Summation over lambda is implied by the Einstein convention, (which is that you sum over repeated indices), i.e: [correction to index]

$$\Gamma^\rho_{~\mu \nu}=\sum_{\lambda=0}^{\lambda=3} \frac{1}{2} g^{\rho \lambda} (\partial_\mu g_{\nu \lambda}+\partial_\nu g_{\mu \lambda}-\partial_\lambda g_{\mu \nu})$$

 

[Nabeshin]

$$\Gamma^\rho_{~\mu \nu}=\sum_{\lambda=0}^{\lambda=4} \frac{1}{2} g^{\rho \lambda} (\partial_\mu g_{\nu \lambda}+\partial_\nu g_{\mu \lambda}-\partial_\lambda g_{\mu \nu})$$

The sum should extend from 0-3, not 0-4 :)

 

[pervect]

The sum should extend from 0-3, not 0-4 :)

Ooops - yes, good point, space-time has 4 dimensions, not 5. I corrected the original, for whatever it's worth.