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Metrics of rotating sphere in GR

  1. Dec 31, 2015 #1
    1. The problem statement, all variables and given/known data
    A thin spherical shell of radius ## R ## rotates with angular velocity ## \Omega ##. Its total mass ## M ## is uniformly distributed. Find metric inside and outside the shell, assuming its small departure from the flat space-time. Find the angular velocity ## \omega ## dragging of the inertial systems within the enclosure.

    2. Relevant equations

    General view of metrics in linearized GR is:
    \begin{equation}
    ds^2 = (1-2\phi) dt^2 - (1+2\phi) (dx^2 + dy^2+dz^2)+2A_i dt dx^i
    \end{equation}

    with
    \begin{align}
    \phi = \int\limits_V \frac{\rho}{r}dV\\
    A_i = 4\int\limits_V \frac{\rho v_i}{r}dV
    \end{align}


    3. The attempt at a solution

    First, it need to go to spherical coordinate system
    \begin{equation}
    dx^2 + dy^2+dz^2 = dr^2+r^2d\theta^2 + r^2\sin^2\theta d\phi^2
    \end{equation}

    As we know from electromagnetism, for the rotating sphere, vector-potential has only ##\hat\phi## component, then
    \begin{equation}
    A_i dx^i = A_{\phi} r\sin\theta d\phi
    \end{equation}
    Am I right here?

    The ##A_{\phi}## I can find similar to emectrodynamics (Griffiths, David J. (2007) Introduction to Electrodynamics, 3rd Edition, example 5.11) as
    \begin{align}
    A_{\phi} = \frac{2Mr \Omega}{3r^2}\sin\theta, r<R \\
    A_{\phi} = \frac{2M R^2 \Omega}{3r^2}\sin\theta, r>R
    \end{align}

    For a ##\phi##-potential
    \begin{align}
    \phi = 0, r<R \\
    \phi = \frac{M}{r}, r>R
    \end{align}

    Am I right for starting?
     
    Last edited: Dec 31, 2015
  2. jcsd
  3. Jan 5, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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