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Question: Interpretation of the Hodge Dual of Antisymmetric Tensors

  1. Feb 10, 2014 #1
    I am a grad student who has had a couple semesters of GR. I am currently perusing a book about Two Spinors in Spacetime by Penrose and Rindler, as background for an essay on Spinor Methods in GR.
    My question relates to the concept of taking the Hodge Dual of a antisymmetric tensor. I understand that taking the Hodge Dual is somehow relating the k forms to the of 4-k forms, and that I can extend this notion to antisymmetric tensors, or even a subset of the tensors indices which are antisymmetric. But how should I view this mapping, what am I physically doing when i take the Hodge Dual?
    Also, the book notes that when viewing tensor operations as spinor operations (as the tensor algebra is a subalgebra of spinor algebra), the Hodge Dual of a antisymmetric tensor corresponds to exactly the same spinor transformation as the trace reversal of a symmetric tensor (up to a factor of i). That is, if we write a tensor Hab, as H[ab] + H(ab), and preform the required spin operation, we obtain i(*H[ab])+TraceReversed(H(ab)). Are trace reversal and hodge dualizing related in some more obvious physical/geometric way?
    Thanks a bunch guys, any info would be super helpful.
  2. jcsd
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