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sergiokapone

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## 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?

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