Derivation of the Christoffel symbol

In summary, to derive the Christoffel symbol from the vanishing of the covariant derivative of the metric tensor, one must perform some permutation and resumming. If one has already obtained the result g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\lambda_{\mu\sigma}g_{\rho\lambda}, then they should consider the quantity g_{\mu\sigma,\rho}+g_{\mu\rho,\sigma}-g_{\rho\sigma,\mu} and take into account the fact that a Levi-Civita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower
  • #1
elenuccia
1
0
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!
 
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  • #2
Did you obtain a result like

[tex]g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\lambda_{\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 Levi-Civita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower indices.
 
  • #3


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.

 

Related to Derivation of the Christoffel symbol

What is the Christoffel symbol?

The Christoffel symbol is a mathematical concept used in differential geometry to represent the connection between points on a curved surface. It is a set of numbers that describes how a vector field changes as it moves along a curved surface.

Why is the Christoffel symbol important?

The Christoffel symbol is important because it helps us understand the geometry of curved spaces, which is essential in many fields of science, including physics and astrophysics. It also plays a crucial role in the mathematical formulation of Einstein's theory of general relativity.

How is the Christoffel symbol derived?

The Christoffel symbol is derived from the metric tensor, which describes the distance between points on a curved surface. It is calculated using the partial derivatives of the metric tensor and involves a series of mathematical calculations.

What are the practical applications of the Christoffel symbol?

The Christoffel symbol is used in various fields of science, such as physics, engineering, and computer graphics. It is used to calculate the curvature of spacetime in general relativity, simulate fluid dynamics in engineering, and model the deformation of objects in computer graphics.

Are there any limitations to the Christoffel symbol?

One limitation of the Christoffel symbol is that it only applies to curved spaces and cannot be used in flat spaces. Additionally, it only describes the local behavior of a curved surface and does not take into account global properties. It also becomes more complex and difficult to calculate in higher dimensions.

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