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Levi-Civita symbol

  1. Jan 15, 2012 #1
    I am reading Landau and Lifgarbagez's Classical Theory of Fields, 4th edition. In the beginning of page 18, the completely antisymmetric unit tensor is said to be a pseudotensor, because none of it components changes sign when we change the sign of one or three of the coordinates.

    Then, in the 2nd paragraph, the product [tex]e^{iklm}e^{prst}[/tex] is a tensor of rank 8 and it is a true tensor! Why?

    We know that [tex]e^{iklm}[/tex] does not change sign when one of the coordinates changes its sign. Either does [tex]e^{prst}[/tex]. Then the product does change its sign either. How could it be possible that the product is a true tensor?

    I totally cannot understand. I need your help, your hints. Thank you!
  2. jcsd
  3. Jan 16, 2012 #2


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    A pseudotensor has the determinant of the LT included in its transformation.
    This gives a minus sign compare to the transformation of a true tensor.
    If a pseudotensor is combined with another pseudotensor, the determinant is squared and always gives +1.
  4. Jan 16, 2012 #3
    What does LT stand for? Thank you!
  5. Jan 16, 2012 #4


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    Lorentz Transformation, or Leprous Tyrannosaurus. I think here the first one is ment.
  6. Jan 16, 2012 #5
    Should be the first one. Is the second one an English name?
  7. Jan 16, 2012 #6


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    He was joking. A leprous tyrannosaurus would be a tyrannosaurus with leprosy.
  8. Jan 16, 2012 #7
    Ha ha~~ That disease is scary...
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