Inverting the Metric Tensor

1. Oct 3, 2005

arunma

My cosmology textbook tells me that if I raise the indicies on the metric tensor (from subscript to superscript), then all I have to do is divide one by each element. But from what I know about inverting matricies, the process is quite a bit more involved. When I raise the indicies on the metric tensor, is this analogous to taking the inverse of a matrix? If not, then what is the mathematical meaning of this procedure?

Any hints would be appreciated. Thank you.

2. Oct 3, 2005

robphy

Yes raising the indices of the metric is analogous to taking the inverse of a matrix:
$$(g)_{ab}(g^{-1})^{bc}=g_{ab}g^{bc}=\delta_a{}^c=(I)_a{}^c$$

If your cosmology book says "divide one by each element", your metrics are probably diagonal in the basis used.

3. Oct 3, 2005

arunma

Oh, I think I understand now. Are you saying that in the special case of a diagonal matrix, the inverse can be found by dividing one by each element?

4. Oct 3, 2005

robphy

Yes, if the metric has an inverse (i.e., is non-degenerate). This is very easy to check!