# Dimension of Fields in Physics: What is it?

• jfy4
In summary, the dimension of a manifold is determined by the contraction of the metric tensor with its inverse. This applies not only to the gravitational field, but also to other fields in physics such as A_{\mu}. In string theory, there is the concept of emergent space where energy becomes a dimension, but this is not fully understood.
jfy4
Hi,

I was not entirely sure where to post this, but I think this will work.

With the gravitational field we have that
$$g^{\alpha\beta}g_{\alpha\beta}=4$$
which is the dimension of the manifold I believe. I have normally heard of $g_{\alpha\beta}$ being interpreted as the gravitational field quantity (or the tetrad). For the other fields in physics (like $A_{\mu}$), how does one compute the dimension, or does such a quantity not exist for anything other than the gravitational field?

Thanks in advance,

The dimension is a property of the underlying manifold (ie. how many coordinates do you need to specify a point). All fields inherit their dimensionality from the manifold. In string theory, apparently you can get emergent space - where energy becomes a dimension - but I have no understanding of this!

jfy4 said:
Hi,

I was not entirely sure where to post this, but I think this will work.

With the gravitational field we have that
$$g^{\alpha\beta}g_{\alpha\beta}=4$$
which is the dimension of the manifold I believe. I have normally heard of $g_{\alpha\beta}$ being interpreted as the gravitational field quantity (or the tetrad). For the other fields in physics (like $A_{\mu}$), how does one compute the dimension, or does such a quantity not exist for anything other than the gravitational field?

Thanks in advance,

The contraction of the metric with its inverse indeed gives the dimension of the spacetime you are looking at. You can see this by taking a look at how the metric tensor is contracted twice with some vectorial quantity $A_{\nu}$ in a d-dimensional space:

$g_{\mu\rho}g^{\mu\nu}A_{\nu}=A_{\rho}$.

Therefore we can make the identification with a Kronecker delta of dimension d:

$g_{\mu\rho}g^{\mu\nu}={\delta_{\rho}}^{\nu}$.

The trace of the Kronecker delta gives just the dimension of the space, therefore

$g_{\mu\nu}g^{\mu\nu}=d$.

Last edited:

## 1. What is the concept of dimension of fields in physics?

The dimension of a field in physics refers to the number of independent parameters needed to describe the field at any given point in space. This is a fundamental aspect of understanding the behavior and properties of various physical fields, such as electric, magnetic, and gravitational fields.

## 2. How is the dimension of a field determined?

The dimension of a field is determined by its units of measurement. For example, the dimension of an electric field is measured in volts per meter, while the dimension of a magnetic field is measured in teslas (or webers per square meter). These units are derived from the fundamental units of length, mass, and time.

## 3. What is the significance of dimension in fields?

The dimension of a field is significant because it provides a way to understand and compare different types of fields. Fields with the same dimension can be combined and compared, while fields with different dimensions cannot. Additionally, the dimension of a field can provide insight into its behavior and how it interacts with other fields.

## 4. Can the dimension of a field change?

Yes, the dimension of a field can change depending on the conditions in which it is measured. For example, the dimension of an electric field may change in the presence of a magnetic field. However, the fundamental units of measurement and the concept of dimension remain constant.

## 5. How does the dimension of a field relate to its physical properties?

The dimension of a field is closely related to its physical properties. For example, the dimension of an electric field is directly related to its strength and direction, while the dimension of a gravitational field is related to the mass and distance of the objects involved. Understanding the dimension of a field is crucial in understanding its behavior and how it affects the surrounding environment.

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