Metric matrix for binary star system?

[exmarine]

What does the metric matrix look like for a binary star system? Does each follow its usual geodesic about the other? It seems like the solution would have to be different somehow than that for a tiny planet circling a big sun.

 

[pervect]

There isn't any exact 2-body metric in GR. We do have an approximate metric for the n-body solar system, though, based on the theory of harmonic coordinates. I think it is of order 1.5. A terse and not terribly understandable presentation of the resulting metric can be found in the IAU (International Astronomical Union) resolution B1.3 for the year 2000, see for instance http://syrte.obspm.fr/IAU_resolutions/Resol-UAI.htm. There are a variety of sources that try to explain the resolution in more detail. See for instance "THE IAU 2000 RESOLUTIONS FOR ASTROMETRY, CELESTIAL MECHANICS, AND METROLOGY IN THE RELATIVISTIC FRAMEWORK: EXPLANATORY SUPPLEMENT", http://iopscience.iop.org/1538-3881/126/6/2687/fulltext/202343.text.html. The "explanations" are still not light reading.

An earlier (I forget the year) IAU version had a scalar gravitational vector potential u, which can be thought of as the Newtonian potential, defined by a Newtonian-like integral, and a set of corresponding metric coefficients. It is rather similar to the PPN formula you'll find in many papers and textbooks. The year 2000 version has a 4-vector potential, broken down by the resolution into a scalar part w, and a 3-vector part $w^i$. The metric coefficeints are are written as functions of u (in the simple earlier version), and the 4-potential $(w, w^i)$ in the current 2000 version. There are already papers that discuss higher-order, higher accuracy approximations for some upcoming experiments