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General relativity, integration over a manifold exercise

  1. Feb 6, 2010 #1
    1. The problem statement, all variables and given/known data

    The problem I am facing is 2.9 in Sean Carroll's book on general relativity (Geometry and Spacetime) I should note that I am not studying this formally and so a full solution would not be unwelcome, though I understand that forum policy understandably prohibits it. The full problem is stated as

    In Minkowski space, suppose that [tex]*F = q \sin{\theta} d\theta\wedge d\phi[/tex].

    a.) Evaluate [tex]d*F=*J[/tex]
    b.) What is the two-form [tex]F[/tex] equal to?
    c.) What are the electric and magnetic fields equal to for this solution?
    d.) Evaluate [tex]\int_V d*F[/tex], where [tex]V[/tex] is a ball of radius [tex]R[/tex] in Euclidean three-space at a fixed moment of time.

    2. Relevant equations

    In the above the asterisk denotes the Hodge dual and the [tex]d[/tex] denotes the exterior derivative. The definitions of these operators should be well known.

    3. The attempt at a solution

    I think I have the first three parts solved. For each part a.) I arrived at the lengthy result of

    [tex]
    \frac{1}{2}[\partial_\mu(*F)_{\mu\rho} - \partial_\nu(*F)_{\mu\rho} + \partial_\nu(*F)_{\rho\mu} - \partial_\rho(*F)_{\nu\rmu} + \partial_\rho(*F)_{\mu\nu} - \partial_\mu(*F)_{\rho\nu} = \epsilon^\sigma_{\mu\nu\rho}J_\sigma
    [/tex]

    For part b.) I got (with the help of my TI-89) the two-form, in matrix form,

    [tex]
    F = -(**F) = \left[
    \begin{array}{cccc}
    0 & \frac{q}{r^4 \sin{\theta}} & 0 & 0\\
    \frac{-q}{r^4 \sin{\theta}} & 0 & 0 &0\\
    0 & 0 & 0 & 0\\
    0 & 0 & 0 & 0
    \end{array}
    \right]
    [/tex]

    and for c.) I arrived at, directly from part b.),

    [tex]E_r = \frac{-q}{r^4 \sin{\theta}}[/tex]
    [tex]E_\theta = 0[/tex]
    [tex]E_\phi = 0[/tex]

    [tex]B_\mu = 0[/tex]

    For [tex]\mu=1,2,3[/tex].

    Verification of these would be appreciated but I am especially confused about the last part d.). In particular the integrand is a three-form and I have no idea how to integrate this and how to transform the integral into a familiar volume integral that can be computed by standard multivariable calculus.
     
  2. jcsd
  3. Feb 14, 2010 #2
    This needs to be moved to the Special and General Relativity forum.

    Perhaps there is will get some attention.

    Thanks
    Matt
     
  4. Feb 14, 2010 #3
    Thanks, I will try my luck over there.
     
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