This week I am at a workshop on mathematical relativity at the Mathematical Research Institute in Oberwolfach which I am organizing together with Piotr Chrusciel and Jim Isenberg. I was a co-organizer of similar conferences here in 2000, 2003 and 2006. The institute organizes workshops fifty weeks in the year on all areas of mathematics and participation is generally by invitation only. The isolated setting of the institute in the Black Forest tends to create an intense research environment. Work is also stimulated by the fact that the institute has the best mathematics library in Germany which is no doubt also one of the best in the world.
There have been a lot of excellent talks here. One of these which was of particular interest to me personally was by Todd Oliynyk. His subject was connected with the Newtonian limit of general relativity. I mentioned this topic in a previous post as having been something whose importance was emphasized by Jürgen Ehlers. Unfortunately Jürgen did not live long enough to see some of his questions answered by the work which Oliynyk has been doing recently.
So what is the Newtonian limit? General relativity is, among other things, a theory of gravity which is fully relativistic. In standard textbooks on the subject we can read that Newtonian physics arises as a limit of general relativity when typical velocities in the system are small compared with the speed of light. Unfortunately it is quite unclear what this means mathematically. For instance, in general relativity gravity is described by the metric, a tensor with ten components, while in Newtonian gravity it is described by a scalar function. How can the former converge to the latter? The conceptual basis of the Newtonian limit was elucidated in work by many people over many years and these ideas were synthesized by Ehlers. On this basis I was able to prove a theorem about convergence to the Newtonian limit in 1994. This concerned asymptotically flat spacetimes (in physical terms isolated systems) and the matter was described by kinetic theory (Vlasov equation). I chose this type of matter since a more commonly used description, the perfect fluid, suffers from technical difficulties. This is because the equations degenerate when the fluid density becomes small and in an isolated system the density has to become small somewhere. The Vlasov equation is immune to these difficulties.
The key mathematical problems involved in the analysis of the Newtonian limit should be independent of the details of the matter model chosen. We just need some matter model which does not place obstacles in our way. In a way Oliynyk took this more literally than I did myself. I had proved an existence theorem for certain types of fluid bodies in general relativity by extending ideas introduced by Tetu Makino in the Newtonian case. These were far away from the generality which would be desirable from a physical point of view but they are good enough to play the role of matter sources when studying the Newtonian limit. This has been exploited by Oliynyk who used ‘Makino fluids’ as matter source in his results. The formulation of the Newtonian limit involves a family of solutions of he Einstein-matter equations depending on a parameter , roughly corresponding to where is the speed of light. The Newtonian limit is then the limit . What I proved in 1994 was the existence of families which are continuous in at . It is also interesting to know how smooth the family is at . The derivatives, when they exist, define higher order approximations to general relativity called the post-Newtonian approximations (PN for derivatives). I only got 0PN. Oliynyk has in the meantime reached 2PN. Results have been obtained for the asymptotically flat case which is the one most frequently considered in physics. It is well known that after 2PN the simple expansion breaks down. How can this be understood? My explanation (a little vague) is as follows. We are trying to approximate something which is bounded in its dependence on the spatial variables. Unfortunately above the 2PN level the approximation is not uniform and the coefficients in the expansion want to be unbounded. If you try to force them to be bounded by assumption the expansion breaks down. These coefficients are supposed to solve Poisson equations but the right hand sides have poor decay. The physicists typically try to represent the solutions of the Poisson equations in terms of the fundamental solution and get divergent integrals.
In his talk here Oliynyk reported that he has been able to treat post-Newtonian expansions of arbitrary order in the cosmological (spatially compact) case. It was a pleasant surprise for me that this works at all. When solving the Poisson equations in the cosmological case the right hand sides must have integral zero. It is remarkable that this works out at all. Once it is known that the procedure works at all, even for low orders, the intuition presented above makes it plausible that the obstructions familiar from the asymptotically flat case will not come up.
Returning to the asymptotically flat case, methods based on the post-Newtonian approximations are used to do the theoretical modelling for gravitational wave detectors whose cost is of the order of a billion dollars. It is an interesting comment on the role of mathematics in applications that nobody seems to worry too much about the almost entire lack of a rigorous mathematical foundation for these methods. In any case, the work I have been reporting on here represents the first steps on the road to changing this.