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Conserved Gravitational Charges and Question on Equal Divergences

  1. Mar 12, 2014 #1
    Hello,

    I am working on a project but I stuck at a point. I found that divergence of two objects are equal
    [itex]\nabla_{a}J^{ab}=\nabla_{a}K^{ab}[/itex]. They are also equal to zero, so I try to construct conserved quantities out of them by using Stoke's theorem.
    [itex]\int_{M}\nabla_{a}J^{ab}d^{d-1}x=\int_{\partial{M}}J^{ab}d\Sigma_{a}=0[/itex]
    Assuming that the J vanishes sufficiently fast at spatial infinity we can find
    [itex]\int_{\Sigma_{1}}J^{ab}\sqrt{-g}d^{d-2}x=\int_{\Sigma_{2}} J^{ab}\sqrt{-g}d^{d-2}x[/itex]
    meaning that [itex]J^{ab}[/itex] is a conserved quantity. Same holds for [itex]K^{ab}[/itex] too. However I wonder that what is the relation between the two charges. Since the divergence of both tensor are equal to each other(at this point I should note that, I can transform them to each other only by using Bianchi identities without adding 0's), what can I say about their original forms. Any help would be appreciated. Thanks.
     
  2. jcsd
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