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Energy-Momentum tensor components for complex Klein-Gorden field

  1. Oct 28, 2014 #1
    Hey guys,

    So I have the stress energy tensor written as follows in my notes for the complex Klein-Gordon field:

    [itex]T^{\mu\nu}=(\partial^{\mu}\phi)^{\dagger}(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial^{\nu}\phi^{\dagger})-\mathcal{L}g^{\mu\nu}[/itex]

    Then I have the next statement that [itex]T^{0i}[/itex] is given by

    [itex]T^{0i}=-[(\partial_{t}\phi)^{\dagger}\nabla^{i}\phi+(\partial_{t}\phi)\nabla^{i}\phi^{\dagger}][/itex]

    And I was wondering how this comes about. I can sort of see what's happened here. Obviously you replace mu and nu with 0 and i respectively to split the spatial and time derivatives. However I have a couple of questions:

    1) is it true that [itex]\partial^{t}\phi=-\partial_{t}\phi[/itex]?
    2) why does the Lagrangian density at the end, [itex]\mathcal{L}g^{\mu\nu}[/itex] drop out?

    Thanks a lot guys :)
     
  2. jcsd
  3. Oct 28, 2014 #2
    Okay maybe I know why the Lagrangian density drops out...is it cos [itex]g^{\mu\nu}[/itex] is diagonal and so [itex]g^{0i}[/itex], for i = 1, 2, 3, are off-diagonal elements which are 0?
     
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