Levi-Civita symbol

  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. clem

    clem 1,276
    Science Advisor

    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. What does LT stand for? Thank you!
  5. haushofer

    haushofer 1,073
    Science Advisor

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

    Fredrik 10,637
    Staff Emeritus
    Science Advisor
    Gold Member

    He was joking. A leprous tyrannosaurus would be a tyrannosaurus with leprosy.
  8. Ha ha~~ That disease is scary...
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