# Covariant and Contravariant Rank-2 Tensors

## Main Question or Discussion Point

Dear Fellows,

Do any one have an idea of whether there must be a system tensor in order to be able to transform from the covariant form of a certain tensor to its contravariant one?

This is a bit important to get rigid basics about tensors.

Schwartz Vandslire

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Either to it correctly as required, or to pass it as required.

## Answers and Replies

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HallsofIvy
Homework Helper
If by "system" tensor you mean "metric" tensor, then yes, in order to have covariant and contravariant vectors and tensors, you must have a metric tensor such that ai= gijaj.

A more general Riemann space may have a "Riemann connection" rather than a metric tensor but my understanding is that we do no talk about covariant and contravariant vectors and tensors in such a space.

Oh! Thanks!

But, HallsofIvy, let's talk, in special, in Minkoweski Space? I know that the latter is a physical concept, but, it refers to a special case which is 4 indices. What about transforming 4x4-Matrices (or Tensors, to be more precise), not only Vectors?

But I have a question please. Does the previous relation also apply to transforming position vectors (They are the basics of the coordinate system)?

What is the meaning of transforming a contravariant tensor to a covariant one?

Is there more than one type of multiplication WRITTEN IN TENSOR EQUATIONS?

Schwartz VANDSLIRE.