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Contracting tensors

  1. Feb 24, 2013 #1
    Am i right in thinking:

    [itex] g^{\mu\nu}g_{\mu\nu}=4 \mbox{ and } g^{\mu\nu}T_{\mu\nu}=T [/itex] ?
     
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  3. Feb 24, 2013 #2

    WannabeNewton

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    Re: contracting tenors

    The first one is certainly true in 4 dimensions. The second is correct yes assuming by [itex]T[/itex] you mean the trace.
     
  4. Feb 25, 2013 #3
    Re: contracting tenors

    sorry I mean T is the scalar stress energy tensor (maybe the same thing). Yeah sorry in 4-D.
     
  5. Feb 25, 2013 #4

    Fredrik

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    Re: contracting tenors

    The left-hand side is a scalar. The right-hand side is not.
     
  6. Feb 25, 2013 #5
    Re: contracting tenors

    sorry shouldn't it be [itex] T^{\mu \nu}_{\mu \nu} = T [/itex]?
     
  7. Feb 25, 2013 #6

    Fredrik

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    Re: contracting tenors

    Components of the stress-energy tensor have two indices, not four. You could however define T by ##T=T^\mu{}_\mu##. The right-hand side is defined by ##T^\mu{}_\mu =T^{\mu\nu}g_{\mu\nu}##.
    $$g^{\mu\nu}T_{\mu\nu} =g^{\mu\nu} g_{\mu\rho} T^{\rho\sigma} g_{\sigma\nu} =\delta^\nu_\rho T^{\rho\sigma} g_{\sigma\nu} = T^{\nu\sigma} g_{\sigma\nu} = T^\nu{}_\nu=T.$$
     
  8. Feb 25, 2013 #7
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