Post-Newtonian two-body and n-body problems with electric charge in general relativity

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Abstract
Starting with the Bażański two‐body post‐Newtonian Lagrangian with electric charge in general relativity, we construct a coordinate transformation (not involving center‐of‐mass coordinates) with two arbitrary parameters and obtain a Hamiltonian which is in agreement with one derived from quantum field theory. The field theory Hamiltonian corresponds to using an arbitrary parameter xp in the photon propagator as well as an arbitrary parameter xg in the graviton propagator. These results are also generalized to the case of n bodies. The condition for static balance ei=±G1/2mi is found to hold both for the exact Reissner–Nordstro/m ’’one‐body’’ problem and for the post‐Newtonian n‐body problem. An alternate condition for static balance ei=± (Gm1m2)1/2 is found to hold for the post‐Newtonian two‐body problem. The precession of the perihelion for the post‐Newtonian two‐body problem is given along with four special cases, one of which is the two‐body generalization of the ’’one‐body’’ special relativity result of Sommerfeld. Post‐Newtonian two‐body equations of motion (in center‐of‐mass coordinates) with the condition of static balance are also examined.