Inner product for vector field in curved background

[Einj]

Hello everyone, I would like to know if anyone knows what is the inner product for vector fields $A_\mu$ in curved space-time. Is it just:

$$
(A_\mu,A_\mu)=\int d^4x A_\mu A^\mu =\int d^4x g^{\mu\nu}A_\mu A_\nu
$$
? Do I need extra factors of the metric?

Thanks!

 

[Orodruin]

You do not take inner products of vector fields, you take inner products of vectors. In that sense, the way of making sense of the inner product of two vector fields would be a scalar field constructed by pointwise taking the inner product of the vector fields.

If for some reason you wish to integrate a scalar field over a manifold, you need to multiply by $\sqrt g$ to get a coordinate independent volume element. Edit: $\sqrt{-g}$ if the metric determinant is negative.

 

[Einj]

I was thinking about some analogous of the Klein-Gordon inner product for scalar fields where one defines:

$$ (\phi_1,\phi_2)=i \int d^3x \left(\phi_1^*\pi_2-\pi^*_1\phi_2\right)$$

with $\pi_i=\partial \mathcal{L}/\partial \dot\phi_i$ is the conjugate momentum. Is there anything similar?