The Cost of Geodesics

Michael Nielsen has a nice post about the geodesic equation when there is a cosmological constant. This reminded me of an interesting property of geodesics in general relativity.
What Michael showed was how you can obtain the equation of geodesic motion in quick and easy steps, once you assume a form for the stress-energy tensor. A simple question to ask is to ask whether you can obtain the geodesic equation without such an assumption. And this is exactly what Einstein, Infield, and Hoffman showed, in 1938 (Annals of Mathematics, 39, p. 62). What these authors showed was that if you introduce singularities in the field equations, these singularities indeed follow geodesics. In these models, the geodesic equation indeed comes for “free” without any assumption about the stress-energy tensor.
It’s interesting to consider the analogous situation for Maxwell’s equations and for the equations of motion in a non-viscous fluid. Maxwell’s equations in vacuum, when we add singularities to the fields, do not lead properly to the equations of motion for charged particles. On the other hand, for the equation of motions for vortices, i.e. singularities, in a non-viscous fluid are determined by the equations of motion for a non-viscous fluid alone. It seems that the essential reason why this works is that the former equations are linear, while the later equations are non-linear.
Oh, and another interesting fact about the Einstein, Infeld, Hoffman derivation is that it doesn’t give a sign for the mass of the singularity: it gives no reason why gravitation is always attractive!