The Einstein-Infeld-Hoffmann (EIH) Method

The Einstein-Infeld-Hoffmann (EIH) method is a way of deriving the equations of particle motion for general relativity and similar theories, by finding approximate two-particle solutions of the field equations. 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 electro-vac 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 equation.

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 electro-vac 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. And apart from dubious arguments about cosmic censorship of naked singularities, there is no reason why these charged point particles can't have light masses like an electron.

Here are some papers on the EIH method.

So why is all of this important? It is little known among physicists that electro-vac Einstein-Maxwell theory can predict charged point particles AND their interactions just by finding solutions (via the EIH method) of the field equations. It seems important, yet it is not in any general relativity or electromagnetics textbook that I am aware of, and it was not taught in any class that I took while getting my PhD. In fact, the references above are almost the only ones I am aware of that discuss it in any detail. It is important most of all because it illuminates the search for a more complete theory. It suggests that the purely classical version of a complete unified field theory should also lack a source term in the Lagrangian, and that fermions such as electrons should originate as point-particle solutions of this Lagrangian. Einstein-Maxwell theory can be generalized to Einstein-Maxwell-Yang-Mills theory to get the additional bosons of the real universe (i.e. Z, W+, W- bosons and gluons, in addition to photons/electromagnetic waves). The point-particle solutions of electro-vac Einstein-Maxwell-Yang-Mills theory are found by generalizing the Reisner-Nordstrom solution so the charge is any of the N^2 square-roots of the identity matrix (for example the identity matrix plus the Pauli matrices for N=2), and the EIH method can be applied to such solutions (see the papers link above). It would be most beautiful if the point-particle solutions of Einstein-Maxwell-Yang-Mills theory were associated with the additional fermions of the real universe (i.e. neutrinos and quarks, in addition to electrons). Of course any theory must somehow be quantized to represent the real universe. The results above suggest that in addition to 2nd quantization, there should also be a 1st quantization step where fermion wave-functions are substituted for point-particle solutions, and where a Dirac type of source term is included in the Lagrangian. At present, it is not known how to 2nd quantize Einstein-Maxwell-Yang-Mills theory with sources, and this is certainly an obstacle to this idea, but one can only hope that a way around it will be found eventually. The usual practice for 30-40 years has been to assume that point-particle solutions of Einstein-Maxwell-Yang-Mills theory have no connection with fermions, and that there is no such thing as 1st quantization. I think this is wrong.