Homework Help: Difference/convert between covariant/contravariant tensors

1. Dec 1, 2017

roberto85

1. The problem statement, all variables and given/known data
1. Explain the difference between a covariant tensor and a contravariant tensor, using the metric tensor as an example.
2. Explain how the components of a general covariant tensor may be converted into those of the equivalent contravariant tensor, and vice versa.

2. Relevant equations

3. The attempt at a solution
1. The difference is due to the alignment of the axes and the metric tells us what that alignment is. A contravariant tensor can be explained by drawing some cartesian axes and a vector from the origin to some point. Then components are obtained by counting how far from each axis we need to go to get from origin to the point. With a covariant tensor we isntead project the vector perpendicularly onto each axis in turn and measure how long each projection is.
But im not sure how to use the metric tensor as an example. I can only think of saying that the contravariant metric tensor is the inverse of the covariant metric tensor. Then to add:
$$[g^{ij}][g_{ij}] = \delta^{i}_{j}$$
where $$\delta^{i}_{j}$$ = 1 if i=j or 0 if i ≠ j

2. Combining a vector with the metric - or its dual, which is the matrix inverse - lets us go from one form to another.
$$A_{\mu} = \Sigma g_{\mu\alpha} A^{\alpha}$$
$$A^{\mu} = \Sigma g^{\mu\alpha} A_{\alpha}$$

Unsure if i have given enough detail in these answers, any help would be appreciated

Last edited: Dec 1, 2017
2. Dec 1, 2017

Ray Vickson

Please stop using a bold font: it looks like you are yelling at us. Also: read my response to your other post to see how to use LsTeX properly

3. Dec 1, 2017

roberto85

Apologies Ray, have edited now, appreciate the pointers.