# Metrics of rotating sphere in GR

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1. Dec 31, 2015

### sergiokapone

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:

ds^2 = (1-2\phi) dt^2 - (1+2\phi) (dx^2 + dy^2+dz^2)+2A_i dt dx^i

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

dx^2 + dy^2+dz^2 = dr^2+r^2d\theta^2 + r^2\sin^2\theta d\phi^2

As we know from electromagnetism, for the rotating sphere, vector-potential has only $\hat\phi$ component, then

A_i dx^i = A_{\phi} r\sin\theta d\phi

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. Jan 5, 2016