I started working on this four years ago and have had this solution worked out for a couple of years now and am surprised I have allowed that much time to pass. I took some time to delve into it further to see what extra insights might be had and to attempt to reduce it further or to find a way to integrate it more directly as well as experimenting with different coordinate choices to see where they might lead but I always ended back at the original solution with the single best coordinate choice to be made that allows us to ingrate and solve for all of the variables. As such, the original solution is still the simplest I have managed to come up with so I have decided to present it finally as is. I have taken a couple of days from work to do this and there is a lot to cover and it may be somewhat hurried so please bear with me. I will attempt to present it as concisely and clearly as I can.

Given the metric

we will find solutions for the variables A, B, and D for the internal metric of a static symmetrical body. These solutions will work for all nonrelativistic bodies. That is, all bodies other than black holes, so bodies that do not contain an event horizon, whereas the surface of the body lies above where the event horizon would otherwise be, normal stars and planets for instance. These equations could hypothetically be applied to a black hole as well, since the solutions are taken directly from EFE's, but they are found with the idea that an observer can remain static at some r within a body and make measurements, all the way to the center, which is not feasible within an event horizon. In this thread, peterdonis graciously gave me EFE's for the energy density and radial and tangent pressures, respectively, reworked in post #13 as

I will simplify this further using

to gain

The only assumption we shall make is that the radial and tangent pressures are isotropic so that p = s. Setting the last two equations equal, they can then be rewritten as

We shall now make our coordinate choice. We want to choose one that places the center of our coordinate chart at the center of the body where r = 0 in both cases and is as close to the Newtonian solution as possible and such that b and d can be integrated easily to determine B and D. I find that

works quite well, where k is a constant. In fact it is the only coordinate choice i have found so far that will allow us to integrate properly. There are a couple of other slight variations I have found that work also but they tend to be much more complicated. It appears that we are very limited on our coodinate choice in that case although I am still experimenting with other possibilities. So now we have

We shall now solve for the variables with what we have gained so far.

which gives us

\int