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Conservation of Energy-Momentum Tensor

  1. Apr 13, 2009 #1
    1. The problem statement, all variables and given/known data

    1) Use conservation of Energy-Momentum Tensor to show that

    [tex]\partial_{0}^{2}T^{00}=\partial_{m}\partial_{n}T^{mn}[/tex]

    2. Relevant equations

    [tex]\partial_{\nu}T^{\mu\nu}=0[/tex]

    3. The attempt at a solution

    [tex]\partial_{\nu}T^{\mu\nu}=0[/tex]

    [tex]\partial_{\mu}\partial_{\nu}T^{\mu\nu}=\partial_{\mu}0=0[/tex]

    [tex]\partial_{0}\partial_{\nu}T^{0\nu}+\partial_{1}\partial_{\nu}T^{1\nu}+\partial_{2}\partial_{\nu}T^{2\nu}+\partial_{3}\partial_{\nu}T^{3\nu}=0[/tex]

    [tex]\partial_{0}\partial_{0}T^{00}+\partial_{0}\partial_{n}T^{0n}+\partial_{1}\partial_{\nu}T^{1\nu}+\partial_{2}\partial_{\nu}T^{2\nu}+\partial_{3}\partial_{\nu}T^{3\nu}=0[/tex]

    [tex]\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{1}\partial_{\nu}T^{1\nu}-\partial_{2}\partial_{\nu}T^{2\nu}-\partial_{3}\partial_{\nu}T^{3\nu}[/tex]

    [tex]\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{1}\partial_{0}T^{10}-\partial_{2}\partial_{0}T^{20}-\partial_{3}\partial_{0}T^{30}-\partial_{1}\partial_{n}T^{1n}-\partial_{2}\partial_{n}T^{2n}-\partial_{3}\partial_{n}T^{3n}[/tex]

    [tex]\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{1}\partial_{0}T^{01}-\partial_{2}\partial_{0}T^{02}-\partial_{3}\partial_{0}T^{03}-\partial_{m}\partial_{n}T^{mn}[/tex]

    [tex]\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{0}\partial_{1}T^{01}-\partial_{0}\partial_{2}T^{02}-\partial_{0}\partial_{3}T^{03}-\partial_{m}\partial_{n}T^{mn}[/tex]

    [tex]\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{0}\partial_{n}T^{0n}-\partial_{m}\partial_{n}T^{mn}[/tex]

    [tex]\partial_{0}^{2}T^{00}=-2\partial_{0}\partial_{n}T^{0n}-\partial_{m}\partial_{n}T^{mn}[/tex]

    This result have an extra term and a negative sign respect the disired result. What am I doing wrong?
     
  2. jcsd
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