# What will be the "Einstein field equations" for two or three bodies?

In general theory of relativity the Einstein field equations e.g. relate the geometry of space-time with the distribution of one body within it. $$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=\dfrac{8\pi G}{c^4}T_{\mu\nu}.$$

What will be the "Einstein field equations" for two or three bodies?

• There are no bodies as such in that equation. The tensor T represents various matter and energy related "densities" in some infinitesimal element of a total field. Describing this field and solving it numerically it is brutally difficult. See einsteintoolkit.org if you want to know more about this. Commented Jul 17, 2020 at 9:08

$$T_{\mu\nu}$$ represents the sources. So, if you have n bodies, you must take them into account into $$T_{\mu\nu}$$.
The Einstein-Infeld-Hoffman equations, derived in 1938, describe the motion of $$N$$ gravitating point masses in General Relativity. They are an expansion in powers of $$1/c$$, so they are useful when the particles are not highly relativistic. The leading term (with no power of $$1/c$$) is simply Newtonian gravity.