# Dust in Spherically Symmetric Gravitational Potential

How can I describe dust in a spherically symmetric gravitational field? I thought of using Schwarzschild, but then I found out that it is a solution of the Einstein Field Equations in Vacuum.

Essentially, the simplest dust universe is a FLRW model (spatially flat is the simplest to work with mathematically), containing a perfect fluid, which would be dust in your case. You would need to match this using the Israel junction conditions to the Schwarzschild metric.

In summary, on the "interior" is your FLRW dust model, on the "exterior" you have the Schwarzschild vacuum.

I suppose in a more general sense, it also depends on what exactly you mean by spherically symmetric. The FLRW models are spherically symmetric in that they are isotropic, i.e., they have 3 rotational Killing vectors. (They are also spatially homogeneous, so they have 3 additional translational Killing vectors). In this case, a simple FLRW dust model would satisfy your requirement as well. But, you must be careful what it is you mean by spherically symmetric.

The other interesting solutions are those of a perfect fluid in a spherically symmetric geometry. For dust you just take p=0.

See the paper at http://www.aei.mpg.de/~rezzolla/lnotes/mondragone/collapse.pdf It shows the use of some of the perfect fluid solutions to model collapsing round objects. Oppenheimer-Snyder had a solution that they used to do the first GR model of collapse to a black hole (BH), taking pressureless dust and constant density. Others have been used, and you can see them in the paper, with non-zero pressure models. In most cases you need to solve the equations numerically, but it has some analytical solutions also. Unless you have a model where the pressure increases enough as it compacts that it doesn't get to be too small (see the paper for how much), it'll contract to inside a radius that there is nothing that can then be done to not have it become a BH.

Of course, the FLRW solution for dust during a few billion years in an expanding universe is not a complete solution, as the dust universe becomes dominated over time by dark energy. If it was pure dust for a flat universe it would have to be the critical density.....but it is not, it's only about 25%. Change that to 100% and you'll get a perfectly flat space universe expanding forever. Get rid of the dark energy, and say any radiation (not physical when densities are higher), and you can adjust the dust density, essentially not very high density matter, and make it be hyperbolic and expands faster, or spherical and it'll re-collapse.

You may be able to create perfect fluid state models (p as function of $\rho$) that achieve stability as a spherical object, with the outside solution then Schwarzschild.

You can use the Tolman metric which is the most general spherically symmetric (but neither homogeneous like FLRW nor static like Schwarzschild) metric. I wrote my Bachelor thesis on a collapsing spherically symmetric dust inhomogeneity in a Friedmann universe and found some papers: check out

and

http://www.numdam.org/item?id=AIHPA_1978__29_2_207_0

(the latter three are in French, unfortunately).