Understanding the Inverse of the Metric Tensor

[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.

 

[robphy]

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.

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.

 

[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?

 

[robphy]

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?

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