Register to reply 
Derivation of the Christoffel symbol 
Share this thread: 
#1
Jan2909, 11:07 AM

P: 1

How can I derive the Christoffel symbol from the vanishing of the covariant derivative of the metric tensor? can somebody write the calculation, I read that I have to do some permutation and resumming but I don't get the result! Thank you!



#2
Jan2909, 06:40 PM

Emeritus
Sci Advisor
PF Gold
P: 9,279

Did you obtain a result like
[tex]g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\la mbda_{\mu\sigma}g_{\rho\lambda}[/tex]already? In that case, consider the quantity [tex]g_{\mu\sigma,\rho}+g_{\mu\rho,\sigma}g_{\rho\sigma,\mu}[/tex]and I think you'll be able to figure out the rest. Don't forget that a LeviCivita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower indices. 


Register to reply 
Related Discussions  
Transformation law for Christoffel symbol of first kind  Differential Geometry  4  
Definition of a Christoffel symbol  Calculus & Beyond Homework  0  
What is the mathematical relationship between Christoffel symbol and  Math & Science Software  2  
Derivation of Christoffel symbol  Special & General Relativity  10  
Christoffel symbol as tensor  Differential Geometry  20 