Covariant derivative in coordinate basis

AI Thread Summary
The discussion focuses on evaluating the covariant derivative of a vector field in a coordinate basis, specifically proving that the expression for the divergence of a vector field, ##\nabla_{\mu} A^{\mu}##, can be rewritten in terms of the partial derivative of the determinant of the metric tensor. The derivation shows that the first and third terms of the Christoffel symbols cancel, leading to a simplified expression involving the determinant of the metric. The user expresses confusion about the assumptions made, questioning why the result seems applicable only to coordinate bases and whether the expression is general. Additionally, there is a request for clarification on the most general form of the Christoffel symbols that does not depend on a specific frame. The discussion highlights the nuances of working with covariant derivatives in different bases.
LCSphysicist
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I need to evaluate ##\nabla_{\mu} A^{\mu}## at coordinate basis. Indeed, i should prove that ##\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})##.

So, $$\nabla_{\mu} A^{\mu} = \partial_{\mu} A^{\mu} + A^{\beta} \Gamma^{\mu}_{\beta \mu}$$

The first and third terms of Christoffel will cancel, so $$ = \partial_{\mu}A^{\mu} + A^{\beta} \frac{g^{\mu x}}{2}(\partial_{\beta}g_{x \mu})$$

Now, using the fact that ##\delta g = g g^{\mu v} \delta g_{\mu v}##, we can easily find that $$\frac{g^{\mu x}}{2}(\partial_{\beta}g_{x \mu}) = \frac{\partial_{\beta}(|g|^{1/2})}{|g|^{1/2}}$$

After substitute this at our main expression, we can recover ##\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})##.

The problem is, i have no idea what assumption i have made so that my result applies only to coordinate basis! That is, the problem ask for prove it at coordinate basis , so i guess it should be true only at these type of basis. But i haven't assumed nothing, just manipulate the terms and got the result. What am i missing? Is this expression really general like i have found?
 
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You have used the coordinate expression for the Christoffel symbols.
 
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Orodruin said:
You have used the coordinate expression for the Christoffel symbols.
I know there are another expressions for Christoffel symbol, like:
$$\Gamma^{i}_{jk} = - \partial e^{i}/\partial x^{j} e_{k}$$
But i didn't know that this expression i have used is not the most general. What is the most general expression for it so? (namely, the one that makes no reference to any frame)
 
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