Metrics of rotating sphere in GR

In summary, the conversation discusses finding the metric inside and outside a thin spherical shell with a total mass uniformly distributed, assuming a small departure from flat space-time. The equations used are based on the general view of metrics in linearized general relativity. The vector-potential for a rotating sphere is determined to only have a ##\hat\phi## component. The ##A_{\phi}## value is then found using a similar approach to electromagnetic theory. The potential for the shell is also determined. The next step is to find the dragging of inertial systems, but it is unclear where to start.
  • #1
sergiokapone
302
17

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?
 
Last edited:
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  • #2
After finding the metrics, I need to find the dragging of inertial systems, which I am not sure how to start.
 

Related to Metrics of rotating sphere in GR

What is a rotating sphere in General Relativity (GR)?

A rotating sphere in GR refers to the concept of a massive, spinning object within the framework of General Relativity. In GR, the shape of spacetime is affected by the presence of mass and energy, and a rotating sphere is one example of how this can manifest.

What are the metrics used to describe a rotating sphere in GR?

The most commonly used metric for a rotating sphere in GR is the Kerr metric, which describes the spacetime around a rotating, massive object. It takes into account both the mass and angular momentum of the object.

How do the metrics of a rotating sphere differ from a non-rotating sphere in GR?

The metrics for a rotating sphere are more complex than those for a non-rotating sphere in GR. This is because the rotation of the object adds an extra degree of freedom, which must be accounted for in the equations.

What are the applications of studying the metrics of a rotating sphere in GR?

Studying the metrics of a rotating sphere in GR has many practical applications, such as in astrophysics where it can help us understand the behavior of rotating objects like black holes. It also has theoretical implications for our understanding of the nature of spacetime and gravity.

Can the metrics of a rotating sphere in GR be tested or observed?

Yes, the predictions made by the Kerr metric for a rotating sphere have been confirmed through various observations and experiments. For example, the frame-dragging effect, where the rotation of a massive object affects the spacetime around it, has been observed through the Gravity Probe B mission.

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