# The dimension of fields

1. Jun 23, 2011

### 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?

2. Jun 23, 2011

### atyy

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!

3. Jun 23, 2011

### Polyrhythmic

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: Jun 23, 2011