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Normalization of a delta function in curved spacetime

  1. Aug 23, 2008 #1
    Which of the following are true in curved spacetime?

    [itex]\int d^4 x \delta^4(x - x_0) = 1[/itex] (1)

    [itex]\int d^4 x \sqrt{-g} \delta^4(x - x_0) = 1[/itex] (2)

    I think the first one is incorrect in curved spacetime, or in general when the metric is non-constant. I would argue this by saying that the delta function does not transform, whereas the fourth-order differential transforms in the opposite way to [itex]\sqrt{-g}[/itex], so the whole thing transforms as a scalar as it must.

    I've also heard that [itex]\delta^4[/itex] is not a scalar, which suggests that (1) is the correct statement. However, this seems strange to me as I would think that (1) will fail to hold in curvilinear coordinates e.g.
     
  2. jcsd
  3. May 14, 2011 #2
    actually the right difinition is
    [tex]\int_M F(x^{\mu})[\frac{\delta^{(4)}(x^{\sigma}-y^{\sigma})}{\sqrt{-g}}]\sqrt{-g}d^4x=F(y^{\sigma})[/tex]
     
  4. May 15, 2011 #3

    Bill_K

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    The delta function is a scalar density. You don't need curved space or four dimensions to see this, it follows from the identity δ(f(x)) = (1/|f '(x0)|) δ(x-x0) where f(x0) = 0. For example δ(3x) = (1/3) δ(x). Or take plane polar coordinates: δ(x) ≡ δ(x) δ(y) = (1/r) δ(r) δ(Θ).
     
    Last edited: May 15, 2011
  5. May 15, 2011 #4

    haushofer

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    I'd say that

    [itex]\int d^4 x \delta^4(x - x_0) = 1[/itex]

    is the usual definition. The RHS is trivially a scalar. The measure on the LHS is a density. So the delta distribution is also a density, as was mentioned by others here.

    You can "tensorize" the delta distribution by defining

    [tex]
    \delta^4(x - x_0) \rightarrow \frac{\delta^4(x - x_0) }{\sqrt{g}}
    [/tex]
     
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