I How to fill the stress energy tensor for multi body systems

Adding to @PeterDonis point, It is true that, mathematically, you can universally go the other way: pick an arbitrary metric, compute the Einstein tensor from it, and then check to see if the implied SET is physically plausible (e.g. energy conditions), and what it describes. However, a random rank 2 tensor field as a candidate SET has probability zero of being a possible Einstein tensor, because of the differential identities and integrability conditions that hold for the Einstein tensor.

 

What you'd actually do most likely is to use a post-newtonian approximation (PPN) https://en.wikipedia.org/wiki/Post-Newtonian_expansion, rather than the full theory. I don't really recall the details, but it's discussed in a lot of texts, such as MTW's "Gravitation". And the wiki article of course, though I don't use PPN enough to know how accurate the Wiki article on it is.

There's been a lot of refinements since the discussion in those older texts anyways - not in the theory itself so much, but making the theory usable in practice with Earth-based measurements of Earth-based atomic time, right ascension, and declination. There's a long and very technical exposition in the IAU 2000 resolutions and their various and fairly numerous ammendemnts, in which the IAU define a pair of coordinate systems (the BCRS and the GCRS) by specifying a metric for each , and additoinally a way to convert (approximately but with a high degree of accuracy) from one set of coordinates to the other.

As far as the stress-energy tensor goes, the stress part of the tensor doesn't directly contribute appreciably to the gravitational field of the Earth, but a point mass and/or spherical model for the distribution of matter in the Earth is a poor approximation, one needs to do a multipole expansion. Eliminating the pressure terms means that only the density (and momentum) terms are really important, but the effects of the Earth not being spherical make this not as simple as it seems.

 

That looks like a pure Newtonian calculation.

 

No, I don't. I know that JPL Ephermedies does take into account the figure of the Earth, and also has a model to include the effects of Earth and lunar tides to calculate the orbital effects for the ephermis, but I don't know for a fact if the JPL Ephermedies (of which there are a bunch of versions) uses a simple ellipsoid model for the Earth's figure, or something more sophisticated. I seem to recall skimming a theory paper by the JPL group at one time, but I couldn't find it again.

From what I read, lunar laser rangefinding has been used to get a better model of the tide-induced part of the pertubation, but it's felt that it's only safely applied for times relatively close to our own era.

 

I did find a link on the JPL ephermedies. https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf covers de430 and de431. Unfortunately I'm not sure how well it answers your question in detail. A summary of my read on this is that the internal structure of the Earth, moon, planets, and sun does have an effect on their gravitational fields and orbital motions. (For instance, the Earth has an iron core, and the moon is believed to have one as well).

What has been directly measured, and what has been fit to make the simulations match the observations isn't really clear to me.

It appears that the Earth's moon has the most complex structure as far as uneven distribution of mass goes. I believe I've heard this referred to as "lunar mass concentratoins", sometimes abbreviated, and it was historically important to the Apollo missions. This is important because of the tight coupling between the Earth and the moon, and because our observations are (mostly?) Earth-based, so we need to know accurately how the Earth moves and how it's axis of rotation changes (precession of the equinoxes).

Some of the details:

An image I found that might help explain this (I had to look up the terminology).



So the mathematical tool used to handle the distribution of mass in the planets (and moon) is spherical harmonics.

I think I first ran into the spherical harmonics in the context of gravity in Goldstein's "Classical Mechanics" in the section on potential theory. Goldstein used the Earth-moon system as an example of potential theory. This was all in the context of Newtonian mechanics though.

So the PPN theory is a theory of point masses, and on top of this additional , basically Newtonian, corrections due to spherical harmonics of the gravitatioanl fields due to the uneven distribution of matter (including, but not limited to, the equatorial bulges of spinning objects) is added in as needed.

I think DIxon has a more formal treatment for extended bodies in GR, but while I know it exists, I'm not really familiar with the details. There may be better papers on the topic of extended bodies in GR than Dixon's as well. http://rspa.royalsocietypublishing.org/content/314/1519/499