# Inner product for vector field in curved background

• Einj
In summary, the conversation discusses the concept of inner product for vector fields in curved space-time. It is clarified that inner products are taken for vectors, not vector fields. However, if one wishes to integrate a scalar field over a manifold, they need to multiply by the metric determinant to get a coordinate-independent volume element. An analogy is also mentioned for the Klein-Gordon inner product for scalar fields.
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!

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.

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?

## 1. What is an inner product for vector fields in a curved background?

An inner product for vector fields in a curved background is a mathematical operation that assigns a number (scalar) to every pair of vector fields in a given space. It is used to measure the angle and length of vector fields and is an essential tool in differential geometry.

## 2. How is the inner product defined in a curved background?

In a flat (Euclidean) space, the inner product of two vector fields is simply the dot product of their components. In a curved background, the inner product is defined using the metric tensor, which encodes the information about the curvature of the space. The inner product is then given by the contraction of the tensor with the components of the two vector fields.

## 3. Why is the inner product important in studying vector fields in a curved background?

The inner product allows us to define important geometric concepts such as length, angle, and orthogonality for vector fields in a curved background. It also plays a crucial role in formulating equations and laws in differential geometry, such as the geodesic equation and the Einstein field equations.

## 4. How is the inner product used in practical applications?

The inner product is used in various fields, including physics, engineering, and computer graphics. In physics, it is used to calculate the energy of a system, momentum, and other physical quantities. In engineering, it is used in the study of fluid flow and stress analysis. In computer graphics, it is used to render 3D objects and simulate light and shadow effects.

## 5. Can the inner product be generalized to higher dimensions?

Yes, the inner product can be defined for vector fields in spaces of any dimension. In fact, the concept of inner product can be extended to more abstract mathematical structures, such as vector spaces and function spaces. This allows for a deeper understanding and application of inner products in various fields of mathematics and science.

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