Contravariant vs. Covariant Tensors?

Question

Could someone explain, in as much detail as they are willing to give, How the contravariant tensors can be transformed into Covariant tensors? I understand that Contravariant tensors are like Tangents and Covariant are as normal vectors to the surface. Perhaps someone has a good reference for the relations because I am having a difficult time seeing it...

Brian

Answer 1

The key to the trransformation is having a metric around, i.e. a non-singular symmetric 2-tensor. A contravariant vector is a tangent vector and a covariant vector is technically a linear functional on the collection of tangent vectors (a covector). If you want to visualize a covariant vector, imagine a collection of parallel planes where spacing corresponds to the 'length of the covector' (rather than just visualizing a normal vector). Then, a contravariant vector will correspond to the collection of planes normal to that vector. To talk about 'normality' is what requires the metric (which is a type of inner product).

Answer 2

In addition to the previous post: there is a nice formalism which is entirely typographical--the "raised" and "lowered" index formalism of general relativity--which does much of the bookkeeping for you.

Also: if you think of the usual vector, there are components and basis vectors. Each component multiplies its own basis vector. If you put it through a linear transformation using a matrix, you are using the components in the existing basis to generate the components in the new basis. However, the basis vectors themselves transform the "other" way, eg via rightward on row rather than leftward on column multiplication. This does things like keeping inner products invariant. Well, covariant and contravariant are terms related to this duality: one way like the components (co) and one like the basis (contra).