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Diff operators are Covariant

  1. Jul 16, 2004 #1


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    I am doing the excercises on Chapter 2 of Ziebach's new book A First Course in String Theory. Part (b) of Problem 2.3 asks us to show that the objects [tex]\partial/{\partial x^{\mu}} [/tex] transform under a boost along the [tex]x^1[/tex] axis in the same way as the [tex]a_{\mu}[/tex] do.

    In other words, to show the differential operators are covariant in special relativity. I haven't done this demonstration before and all the tricks I have come up with don't seem to get there. Can anyone help?

    His "lower index objects" are produced from upper index objects by multiplying with the inverse of the contravariant metric tensor : [tex] a_{\mu} = \eta_{\mu\nu} a^{\nu} [/tex]. This has the effect of changing the sign of the 0 (time) component of the vector. Signature is -+++.
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  3. Jul 16, 2004 #2


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    For any transformation of coordinates, let [tex]x^{mu}[/tex] be the old coordinates and [tex]x^{mu}'[/tex] the new coordinates. Use the chain rule:
    [tex]\frac{\partial }{\partial x^{mu}'}= \frac{\partial x^{mu}}{\partial x^{mu}'}\frac{\partial }{\partial x^{mu}}[/tex].
    Last edited: Jul 16, 2004
  4. Jul 17, 2004 #3


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    Hah! Thanks! I missed that because in this case
    [tex]\frac{\partial{x^{\mu}}}{\partial{x'^{\mu}}} [/tex]
    is constant, being a particular Lorentz trannsform.

    Thanks again Halls.
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