Difference/convert between covariant/contravariant tensors

[roberto85]

Homework Statement

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.

Homework Equations

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 I am 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 - let's 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

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

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 I am 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:
$[g^{ij}][g_{ij}] = δ^{i}_{j}$
where $δ^{i}_{j}$ = 1 if i=j or 0 if i ≠ j

2. Combining a vector with the metric - or its dual, which is the matrix inverse - let's us go from one form to another.
$A_{/mu} = Σg_{/mu/alpha} A^{/alpha}$
$A^{/mu} = Σg^{/mu/alpha} A_{/alpha}$

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

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

 

[roberto85]

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

Apologies Ray, have edited now, appreciate the pointers.