- #1

- 642

- 160

- Homework Statement
- .

- Relevant Equations
- .

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?