Metrics of rotating sphere in GR

[sergiokapone]

Homework Statement

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.

Homework 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}

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?

 

[_coyote_]

After finding the metrics, I need to find the dragging of inertial systems, which I am not sure how to start.