Creating Metric Describing Large Disk

[Onyx]

TL;DR Summary

How can I create a metric describing the space outside a large disk, like an elliptical galaxy?

How can I create a metric describing the space outside a large disk, like an elliptical galaxy? In cylindrical coordinates, $\phi$ would be the angle restricted the the plane, as $\rho$ would be the radius restricted to the plane. I think that if $z$ is suppressed to create an embedding function with just $\rho$ and $\phi$, it would look very much like the Schwarzschild case, since it is a circle in the plane. But if I suppressed $\phi$, I think the embedding function of that plane would have arguments of both $\rho$ and $z$, and it would look more oblong. So I feel like the metric must have these features, but I'm not sure specifically in what arrangment.

 

[renormalize]

TL;DR Summary: How can I create a metric describing the space outside a large disk, like an elliptical galaxy?

Take a look at this 1996 Helvetica Physica Acta paper and its references:

Relativistically rotating dust
by G. Neugebauer, A. Kleinwachter and R. Meinel
Abstract: Dust configurations play an important role in astrophysics and are the simplest models for rotating bodies. The physical properties of the general–relativistic global solution for the rigidly rotating disk of dust, which has been found recently as the solution of a boundary value problem, are discussed.

Available here: https://arxiv.org/pdf/gr-qc/0301107.pdf

 

[Onyx]

This seems right if considering a significantly rotating disk, but unless I'm mistaken most galaxies don't rotate very fast in proportion to their disk radius. I found another metric describing a stationary and static ellipse, as described in this paper. However, I don't understand why the $g_{tt}$ term is still the same as in the Schwarzschild case; since this metric does not have spherical symmetry, I would have thought it would be different.

 

[renormalize]

I found another metric describing a stationary and static ellipse, as described in this paper.

I wouldn't rely on this paper. It appears in International Journal of the Physical Sciences from "Academic Journals", which is on Beall's list of predatory publishers. Can you find this same metric in a proper journal or textbook?