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Definition of Kronecker delta

  1. Jul 15, 2008 #1
    Can anyone give me a coordinate-independent definition of [itex]\delta^a_b[/itex] on curved manifolds?

    Should it be defined as [itex]\delta^a_b = g^{ac}g_{bc}[/itex] where abstract index notation has been used?
     
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  3. Jul 16, 2008 #2

    CompuChip

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    You could define it that way. Then again, you could take that as the definition for [itex]g^{ab}[/itex] (the inverse metric) :smile:
    One problem, is that we are used to thinking of the Kronecker delta as "the thing which is 1 iff the indices are equal, and 0 otherwise" which of course, introduces coordinates right away. I am wondering if "the unit tensor" (e.g. dxd unit matrix) is a coordinate independent statement... :smile:
     
  4. Jul 16, 2008 #3

    Mentz114

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    I find that,

    [tex]g_b^a = g^{an}g_{nb} = \delta_b^a[/tex]

    but I can't prove it.
     
  5. Jul 16, 2008 #4

    CompuChip

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    I though that was the definition of the inverse metric. Basically you have written down that
    [tex]g^{-1} g = I[/tex]
     
  6. Jul 16, 2008 #5

    Mentz114

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    Yep. Going round in cicles. Break the circle at any point and select a definition.

    M
     
  7. Jul 16, 2008 #6
    If your manifold has a metric, you can give a perfectly good coordinate-independent definition of the Kronecker tensor (and, indeed, its generalizations) in terms of the so-called "musical isomorphism" between the tangent space and cotangent space.

    This is pretty basic stuff, but beyond a yearning for strict coordinate-independence, I can't see any actual advantage in using such a definition.
     
  8. Jul 17, 2008 #7

    DrGreg

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    You are nearly there. The coordinate-free version of a matrix is a linear operator, a function that maps vectors to vectors (or covectors to covectors).

    So the Kronecker delta is just the identity operator [itex] \delta (\textbf{X}) = \textbf{X} [/itex] acting on any tangent space (or cotangent space).

    This follows from the coordinate expression [itex] \delta^a_b X^b = X^a [/itex] which is true in every coordinate system.
     
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