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Covariant Derivatives (1st, 2nd) of a Scalar Field

  1. Nov 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it

    2. Relevant equations
    Suppose we have
    $$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C \phi $$
    it should be proven that
    $$\nabla_A \: X^{AB} =0$$
    with ##\phi## is a scalar field

    3. The attempt at a solution

    Naturally, we would expand the equations.
    \begin{align}\nabla_A \: X^{AB} &= \nabla_A \left[ \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C \phi \right] \\
    &= \nabla^B \phi \left[ \nabla_A \nabla^A \phi \right] + \nabla^A \phi \left[ \nabla_A \nabla^B \phi \right] - \frac{1}{2} g^{AB} \nabla_C \phi \left[\nabla_A \nabla^C \phi\right] - \frac{1}{2} g^{AB} \nabla^C \phi \left[\nabla_A \nabla_C \phi\right]
    \end{align}

    We know that the covariant derivative of a scalar is its partial derivative ## \nabla_A \phi = \partial_A \phi##

    \begin{align}\nabla_A \: X^{AB} &= \partial^B \phi \left[ \nabla_A \partial^A \phi \right] + \partial^A \phi \left[ \nabla_A \partial^B \phi \right] - \frac{1}{2} g^{AB} \partial_C \phi \left[\nabla_A \partial^C \phi\right] - \frac{1}{2} g^{AB} \partial^C \phi \left[\nabla_A \partial_C \phi\right]
    \end{align}

    Now the 2nd Cov. Der. would depends on the christoffel symbol where
    \begin{align}
    \nabla_A \partial_B \phi &= \partial_A \partial_B \phi - \partial_C \phi \Gamma_{AB}^C \\
    \nabla_A \partial^B \phi &= \partial_A \partial^B \phi + \partial^C \phi \Gamma_{AC}^B
    \end{align}

    so that
    \begin{align}\nabla_A \: X^{AB} &= \partial^B \phi \left[ \partial_A \partial^A \phi + \partial^C \phi \Gamma_{AC}^A \right] + \partial^A \phi \left[ \partial_A \partial^B \phi + \partial^C \phi \Gamma_{AC}^B \right] \\
    \nonumber
    & - \frac{1}{2} g^{AB} \partial_C \phi \left[\partial_A \partial^C \phi + \partial^D \phi \Gamma_{AD}^C\right] - \frac{1}{2} g^{AB} \partial^C \phi \left[\partial_A \partial_C \phi - \partial_D \phi \Gamma_{AC}^D\right]
    \end{align}

    Now I'm stuck as to where should I go? How could t be proven that last (and quite long) equation be equal to zero?
    Any help would be much appreciated
    Thanks in advance
     
  2. jcsd
  3. Nov 27, 2016 #2

    andrewkirk

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    That doesn't sound correct to me. IIRC the covariant derivative of the covariant derivative of scalar field ##\phi## is a ##\pmatrix{0\\2}## tensor whose ##(\alpha,\beta)## element is
    $$\phi_{;\alpha\beta}=\partial_\beta\partial_\alpha\phi-\Gamma^\mu_{\alpha\beta}(\partial_\mu\phi)$$
    which is not in general zero.
     
  4. Nov 27, 2016 #3
    All in all, said tensor ##X^{AB}## in my post was actually an Energy momentum tensor, which should have a covariant derivative of zero ##\nabla_A X^{AB} = \nabla_A T^{AB} = 0 ##

    or is there a special case where that would be zero?
     
  5. Nov 29, 2016 #4

    nrqed

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    You have forgotten to apply the covariant derivative to the metric [itex] g^{AB} [/itex].
     
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