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Defiunition of kroneker delta as a tensor

  1. May 3, 2013 #1
    hi,

    the delta symbol as a tensor (in the minkovski space, in case one has to be specific), what is it exactly?



    is it
    [itex]\delta^a_b = \frac{\partial{x^a}}{\partial{x^b}}[/itex]

    is it
    [itex]\delta^a_b = g^{ac} g_{cb}[/itex]


    or is there some other definition?


    thanks
     
  2. jcsd
  3. May 3, 2013 #2

    jambaugh

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    Science Advisor
    Gold Member

    The Kroneker delta is the the indexed components of the Identity operator which looks the same in all bases.

    [tex] \mathbf{1}\mathbf{x} = \mathbf{x}: x^\mu = \delta^\mu_\nu x^\nu[/tex]
     
  4. May 3, 2013 #3
    It is both. The kroneker delta represents the mixed (covariant/contravariant) components of the metric tensor, and applies to both flat as well as curved spacetime.
     
  5. May 3, 2013 #4
    I believe the technical definition is:

    [itex] \delta^a_b = 0 (a != b), \delta^a_a = 1 [/itex]

    In other words, it's just the identity matrix. The two "definitions" you gave don't define δ, rather you can show those two expressions equal δ as I defined it.
     
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