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Trouble with Stokes' theorem 4-dim

  1. Aug 12, 2012 #1
    Well I am studying the book Relativisits' Toolkit by Eric Poisson and I am stuck in the part Stokes Theorem.

    Working under 4 dim. spacetime. greek indices runs from 0 to 3 latin indices runs from 1 to 3
    It defines surface element as

    dƩ[itex]_{\mu}[/itex]=ε[itex]_{\mu\alpha\beta\gamma}[/itex]e[itex]^{\alpha}_{1}[/itex]e[itex]^{\beta}_{2}[/itex]e[itex]^{\gamma}_{3}[/itex]d[itex]^{3}[/itex]y

    [itex]\int_{V}[/itex]A[itex]^{\alpha}_{;\alpha}[/itex][itex]\sqrt{-g}[/itex]d[itex]^{4}[/itex]x=[itex]\oint_{\partial V}[/itex]A[itex]^{\alpha}[/itex]dƩ[itex]_{\alpha}[/itex]

    It says "think a nest of closed hypersurfaces foliating V with boundary ∂V forming outer layer of the nest.let x[itex]^{0}[/itex] constant on each one of these hypersurfaces with x[itex]^{0}[/itex]=1 designating ∂V and x[itex]^{0}[/itex]=0 , the zero volume hypersurface at the center of V."


    [itex]\int_{V}[/itex]A[itex]^{\alpha}_{;\alpha}[/itex][itex]\sqrt{-g}[/itex]d[itex]^{4}[/itex]x= [itex]\int_{V}[/itex]([itex]\sqrt{-g}[/itex]A[itex]^{\alpha}[/itex]),[itex]_{ \alpha}[/itex]d[itex]^{4}[/itex]x
    [itex]^{*}[/itex]=[itex]\int[/itex]dx[itex]^{0}[/itex][itex]\oint[/itex]([itex]\sqrt{-g}[/itex]A[itex]^{0}[/itex]),[itex]_{0}[/itex]+[itex]\int[/itex]dx[itex]^{0}[/itex][itex]\oint[/itex]([itex]\sqrt{-g}[/itex]A[itex]^{a}[/itex]),[itex]_{a}[/itex]d[itex]^{3}[/itex]x

    [itex]^{*}[/itex]=[itex]\int[/itex]dx[itex]^{0}[/itex][itex]\frac{d}{dx^{0}}[/itex][itex]\oint[/itex]([itex]\sqrt{-g}[/itex]A[itex]^{0}[/itex])d[itex]^{3}[/itex]x

    [itex]^{*}[/itex]=[itex]\oint[/itex]([itex]\sqrt{-g}[/itex]A[itex]^{0}[/itex])[itex]_{0}[/itex]d[itex]^{3}[/itex]x l[itex]^{x^{0}=1 }_{x^{0}=0 }[/itex]
    [itex]^{*}[/itex]=[itex]\oint[/itex]([itex]\sqrt{-g}[/itex]A[itex]^{0}[/itex])[itex]_{0}[/itex]d[itex]^{3}[/itex]y
    "[itex]^{*}[/itex]=" means in the desired coordinate system.

    [itex]\oint_{∂V}[/itex]A[itex]^{\alpha}[/itex]dƩ[itex]_{\alpha}[/itex] [itex]^{*}[/itex]= [itex]\oint_{∂V}[/itex]A^{0}[itex]\sqrt{-g}[/itex]d[itex]^{3}[/itex]y

    What I did not get is probably quite easy but I'm stuck, I cannot figure it out.
    [itex]\int[/itex]dx[itex]^{0}[/itex][itex]\oint[/itex]([itex]\sqrt{-g}[/itex]A[itex]^{a}[/itex]),[itex]_{a}[/itex]d[itex]^{3}[/itex]x part is equal to zero in the frame x^{0}=constant ,so called radial part ,and this part is angular part. I cannot get why it is equal to zero.Any help greatly appreciated.
     
    Last edited: Aug 12, 2012
  2. jcsd
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