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Is space covariant or contravariant?

  1. Jul 16, 2010 #1
    I often meet the question whether the (physical) space is 'covariant' or 'contravariant'.

    I once replied to that question with: Space is space. The COMPONENTS of a tensor are covariant/contravariant if the basis is CHOSEN TO BE contravariant/covariant. As far as I know tensors the 'covariance' or 'contravariance' of space (or tensor) itself is not even the question.

    The thing is that the professors were not satisfied with the answer. They said the space is contravariant.

    So, I am confused. As far as I know tensors, or better saying tensor components, the 'covariance' or 'contravariance' depends on the choice of basis, i.e. contravariant or covariant basis. In other words, tensor cannot be covariant or contravariant, it can only be represented in covariant or contravariant BASIS and therefore having contravaiant or covariant components.

    Does anyone have any ideas about that?
  2. jcsd
  3. Jul 16, 2010 #2


    Staff: Mentor

    Coordinates are contravariant. I suspect that is all that the professors meant to ask.
  4. Jul 16, 2010 #3
    the coordinates can be covariant or contravariant if the basis is chosen to be contravariant or covariant. The are also cases when tensor is represented in mixed coordinates, i.e. as a linear combination of base vectors, some from tangent space and some from its dual space. So I don't understand why can an answer 'contravariant' be 'correct'.
    Last edited: Jul 16, 2010
  5. Jul 16, 2010 #4


    Staff: Mentor

    I think you should ask the professors. I am really bad at reading your professor's minds.

    However, if you look at your explanation you will note that justifying your answer required use of basis dual vectors from the dual space whereas the professor's question was about space. If he was intending to emphasize the distinction then you missed his point.
    Last edited: Jul 16, 2010
  6. Jul 16, 2010 #5
    thanks for the effort. It is obvious that reading someone’s mind is difficult if not impossible. But still, I found the answer on imechanica forum. Prof. Sia Nemat-Nasser clearly confirms my understanding of the topic. Here is the link to the post http://imechanica.org/node/4356. I recommend reading it. And once again thanks for your responses. I appreciate it.
    Last edited by a moderator: Apr 25, 2017
  7. Jul 16, 2010 #6
    But still I am afraid I am missing something. That is why I am asking for some help on physics forum instead of (solely) on mathematical one.
  8. Jul 16, 2010 #7


    Staff: Mentor

    You need to speak to your professor. He obviously wants you to learn something that you have not learned. Looking up websites that you think somehow agree with you may make you feel better but is not going to help you learn a concept you are struggling with.
  9. Jul 16, 2010 #8


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Keep in mind that, some time ago, the meaning of covariant and contravariant was reversed. I'm not sure if the usage is consistent these days.
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