The Einstein-Infeld-Hoffmann (EIH) method is a way of deriving the equations of motion for general relativity and similar theories. For Einstein-Maxwell theory (ordinary general relativity with electromagnetism), the EIH method allows the equations of motion for both charged and neutral pariticles to be derived directly from the electro-vac Einstein-Maxwell field equations. For neutral particles the method has been verified to Post-Newtonian order, and it was how the Post-Newtonian equations of motion were first derived in 1938. For charged particles the method has been verified to Post-Coulombian order, meaning that it gives the same result as the Darwin Lagrangian. The EIH method is valuable because it does not require any additional assumptions, such as the postulate that neutral particles follow geodesics, or the ad hoc inclusion of source terms in the Lagrangian density and field equations.

In Einstein-Maxwell theory, the exact solutions for stationary and rotating bodies are the Reisner-Nordstrom and Kerr-Newman solutions (which become the Schwarzschild and Kerr solutions when uncharged). Einstein-Maxwell theory is non-linear, so a two-particle solution does not result by summing together two one-particle solutions at different locations. Two-particle solutions exist, but only if the two particles are accelerating relative to each other, and it just happens that this acceleration obeys the Lorentz-force equation. This contrasts with pre-Einstein electrodynamics where the Lorentz force equation is postulated independent of Maxwell's equations. In pre-Einstein electrodynamics all the field equations are linear, so a two-particle solution is found by summing two stationary Q/r^2 solutions at separate locations, but this solution is essentially ignored because it contradicts the Lorentz force equaton.

The EIH method is made to work with non-linear theories like Einstein-Maxwell theory. With the EIH method, one does not just find equations of motion, but rather one finds approximate solutions g_ik and F_ik of the field equations which correspond to a system of two or more particles. These approximate solutions will in general contain 1/r^p singularities, and these are considered to represent particles. It is required that the 1/r^p fields in these approximate solutions should approach exact solutions like the Reisner-Nordstrom solution asymptotically, and this requirement constrains the motions of the singularities. With this requirement, it is reasonable to assume that the motions of the 1/r^p singularities should approximate the motions of singularities in exact two-particle solutions. Any event horizon or other unusual feature of exact solutions at small radii is irrelevant because the singularities are assumed to be separated by much larger distances, and because the method relies greatly on surface integrals done at large distances from the singularities.

For Einstein-Maxwell theory the EIH method predicts the Lorentz force equation to Coulombian order, and the equations resulting from the Darwin Lagrangian to post-Coulombian order. The fact that these results agree with measurement demonstrates that the electro-vac Einstein-Maxwell field equations by themselves describe a universe with some resemblance to reality. Such a universe contains charged point particles (in the form of Reisner-Nordstrom or Kerr-Newman solutions) which interact with one another according to the correct classical equations of motion, and it also contains electromagnetic waves and part of neutral matter general relativity. Of course this imaginary universe lacks the very important quantum mechanical aspect of the real universe, and it also excludes the strong and weak forces and most of the Standard Model particles.

Here are some papers on the EIH method.

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