Covariant and Contravariant Rank-2 Tensors

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SUMMARY

The discussion centers on the necessity of a metric tensor for transforming covariant tensors to contravariant tensors. Schwartz Vandslire and HallsofIvy confirm that a metric tensor is essential for this transformation, specifically in the context of Riemann spaces. The conversation also touches on the implications of these transformations in Minkowski Space and raises questions about the transformation of 4x4 matrices and the nature of tensor multiplication.

PREREQUISITES
  • Understanding of covariant and contravariant tensors
  • Familiarity with metric tensors in Riemannian geometry
  • Basic knowledge of Minkowski Space and its properties
  • Concepts of tensor multiplication and operations
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  • Study the properties and applications of metric tensors in Riemannian geometry
  • Explore transformations between covariant and contravariant tensors in Minkowski Space
  • Learn about tensor multiplication techniques and their implications
  • Investigate the role of Riemann connections in general spaces
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Dear Fellows,

Do anyone 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.
 
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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.:cool:
 

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