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How do I differentiate this Scalar Field?

  1. Apr 20, 2015 #1
    1. The problem statement, all variables and given/known data

    (a) Find the christoffel symbols (Done).
    (b) Show that ##\phi## is a solution and find the relation between A and B.


    2013_B5_Q2.png
    2. Relevant equations


    3. The attempt at a solution
    Part(b)
    [tex]\nabla_\mu \nabla^\mu \phi = 0[/tex]
    I suppose for a scalar field, this is simply the normal derivative:
    [tex]\frac{\partial^2 \phi}{\partial \eta^2} + \frac{\partial^2 \phi}{\partial r^2} = 0 [/tex]

    Starting with the ##\eta## component:
    [tex]\frac{\partial \phi}{\partial \eta} = exp() \left[ B + (A+B\eta)(-ic|k|) \right] [/tex]
    [tex] \frac{\partial^2 \phi}{\partial \eta^2} = exp() \left[ -2ic|k|B - |k|^2c^2(A+B\eta) \right] [/tex]

    Now for the ##r## component:
    [tex]\frac{\partial \phi}{\partial r} = exp() \left[ A+B\eta \right](\vec k \cdot \hat r)[/tex]
    [tex]\frac{\partial^2 \phi}{\partial r^2} = exp() \left[ -(A+B\eta)(\vec k \cdot \hat r)^2 \right] [/tex]

    Equating both real parts doesn't work; It gives ##(\vec k \cdot \hat r)^2 = |k|^2 c^2##..
     
  2. jcsd
  3. Apr 21, 2015 #2

    strangerep

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    What is ##\nabla_\mu## here? I would have thought they'd be covariant derivatives, not partial derivatives. (?)

    Even though a covariant derivative of a scalar field is just the partial derivative, the 2nd covariant derivative is acting on a vector....
     
  4. Apr 21, 2015 #3

    Orodruin

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    strangerep is correct, the second covariant derivative is not acting on a scalar, but on a covariant vector. I suggest rewriting it as
    $$
    \nabla^\mu \nabla_\mu \phi = g^{\mu\nu} \nabla_\mu \nabla_\nu \phi
    $$
    and apply the rules for covariant derivatives from there.
     
  5. Apr 21, 2015 #4
    I know that for a vector field, ##\nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma##. Does this apply to scalar fields as well? Or in the case of a scalar field, does it simply reduce to a normal derivative?

    I have also found that
    [tex]\nabla_\alpha \nabla_\beta V^\mu = \partial_\alpha \partial_\beta V^\mu + \left(\partial_\alpha \Gamma^\mu_{\beta \nu} \right)V^\nu + \Gamma^\mu_{\beta \nu} \left( \partial_\alpha V^\nu \right) + \Gamma^\mu_{\alpha \nu} \left[ \partial_\beta V^\nu + \Gamma^\nu_{\beta \epsilon} V^\epsilon \right] + \Gamma^\nu_{\alpha \beta} \left[ \partial_\nu V^mu + \Gamma^\mu_{\nu \epsilon} V^\epsilon \right][/tex]
     
  6. Apr 21, 2015 #5

    Orodruin

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    No, for scalar fields you have ##\nabla_\mu \phi = \partial_\mu \phi##. However, ##\nabla_\mu \phi## is a covector field and as such its covariant derivative ##\nabla_\nu\nabla_\mu \phi## will involve a Christoffel symbol.
     
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