Derivation of the Christoffel symbol

[elenuccia]

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!

 

[Fredrik]

Did you obtain a result like

$$g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\lambda_{\mu\sigma}g_{\rho\lambda}$$​

already? In that case, consider the quantity

$$g_{\mu\sigma,\rho}+g_{\mu\rho,\sigma}-g_{\rho\sigma,\mu}$$​

and I think you'll be able to figure out the rest. Don't forget that a Levi-Civita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower indices.

 

[Entropia]

The Christoffel symbol is a mathematical object used in differential geometry to represent the connection between tangent spaces on a manifold. It is defined as:

Γ^a_bc = (1/2) g^ad [∂bg_dc + ∂cg_db - ∂dg_bc]

where g^ad is the inverse of the metric tensor g_ab and ∂ denotes the partial derivative with respect to the coordinate x^d.

To derive this expression from the vanishing of the covariant derivative of the metric tensor, we first need to understand the concept of the covariant derivative. The covariant derivative is a way of taking derivatives of tensors on a curved manifold, where the usual partial derivative is not well-defined. It is defined as:

∇_aT^b...d = ∂_aT^b...d + Γ^b_aeT^e...d + ... + Γ^d_aeT^b...e

where T^b...d is a tensor and Γ^b_ae are the Christoffel symbols. This expression essentially takes into account the curvature of the manifold when taking derivatives.

Now, if we consider the metric tensor g_ab, we can take its covariant derivative and set it equal to zero, since the metric is a constant on the manifold. This gives us:

∇_ag_bc = 0

Expanding this expression using the definition of the covariant derivative and the symmetry of the metric tensor, we get:

∂_ag_bc + Γ^d_ag_bc + Γ^b_ag_dc = 0

Now, using the definition of the Christoffel symbols and rearranging, we get:

Γ^d_ag_bc = (1/2)(∂bg_dc + ∂cg_db - ∂dg_bc)

This is the same expression as the one for the Christoffel symbol that we started with, confirming that it can be derived from the vanishing of the covariant derivative of the metric tensor.

To calculate the Christoffel symbols, we need to perform some permutation and summing as mentioned in the question. This involves substituting the values of the metric tensor and its inverse into the expression for the Christoffel symbol and simplifying the resulting expression. This can be a tedious process, but it is a straightforward application of the definition of the Christoffel symbol. Alternatively, you can use a computer algebra system to perform the calculation for you.