Is space covariant or contravariant?-
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From: JEMebius on 17 Jul 2010 17:30 rudi wrote: 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 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. Does anyone have any ideas about that? For origin and explanation of the term "contravariant" and many other things, see Wikipedia: http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors Lesson the Zeroth(*) in theoretical physics: Space is Space. You were damn right! Lesson the First in theoretical physics is learning (maybe for you, certainly for me) and teaching (for me, later also for you) coordinate transformations in vector spaces and Euclidean spaces. Lesson the Second: coordinate transformations in differentiable manifolds (Ricci, Levi-Cività ; Einstein). Compare and contrast object transformations and basis transformations: (A) The basis remains in place and the space is linearly transformed into itself, in older literature sometimes denoted as "alibi transformation". Alibi (Latin): at another place and/or time. (B) The vectors remain in place and the basis is changed, denoted as "alias transformation". The vector components change to new values. Alias (Latin): a different name; in the case of basis transformations: different values of components. If the basis changes according to a matrix A and vectors are to remain in place then their components change according to the matrix A^-1 (or to its transpose, depending on conventions of notation); this is contra-variant behaviour, so to say. BTW, if your professors indeed insist that space itself is contravariant, then ask them the question "contravariant - oh dear, with respect to =what=?" I bet they cannot answer that question, or that they reply with some fashionable meta-cosmological gibberish. Happy studies: Johan E. Mebius ------------------------------------------------------------------------------------------ (*) "zeroth" - this means: the statement "space is space" is far above and far beyond the nitty-gritty of coordinate systems, vectors, tensors and all that, and so is exempt from ordinary counting. |