Linking doughnuts, soda straws and energy.
Knowing how magnetic field lines in a neutron star are tangled or linked gives scientists a good estimate of the minimum energy such a system can have. That surprising result is one consequence of a recently proved mathematical theorem linking the topology, or geometric form, of lines representing the flow of an incompressible fluid (whether an ideal liquid or a magnetic field) and the energy embodied in that fluid flow.
"What's of interest is the relationship between the topology of the lines and the energy in the system," says mathematician Michael H. Freedman of the University of California, San Diego.
One way to picture the situation is to think of a solid torus, or doughnut sh ape, made up of stretched rubber bands, each running once around the doughnut's hole. The idealized rubber bands want to shrink as much as possible, but because the material is also incompressible, they can't make the doughnut shrink to nothing. However, the rubber bands can release their energy by forcing the doughnut's shape into that of a long, thin soda straw. Like the doughnut, the soda straw has a central hole, but because of the straw's much narrower configuration, each rubber band has a significantly shorter distance to stretch.
When one doughnut winds through the hole of a second doughnut, the situation changes dramatically. The soda-straw solution is no longer possible. If one doughnut were to spring into its soda-straw configuration, the change would stretch the other doughnut in a way that increases the energy of its rubber bands. In other words, the linking of the two doughnuts prevents the dissipation of energy. Similar argunents hold for more complicated links and knots.
In the case of a neutron star, the material within the star moves around so as to let magnetic field lines straighten out and separate as much as possible. But any linking and knotting of these lines gets in the way, limiting how much the system can "relax" and setting a lower bound on its energy. Similarly, a solid torus of spinning fluid may give up energy by elongating like a soda straw, but such a shift is prevented when several tori link.
Freedman, working with graduate student Zheng-Xu He, builds on a theorem proved by Soviet mathematical physicist V.I. Arnold, who showed that energy bounds exist for the special case when a quantity called the "linking number" can be computed for a given tangle of lines. Using ideas from knot theory, Freedman and He extended Arnold's theorem to cover a much broader range of knots and links.
"Geometric linking, independent of linking number, can be used to estimate a lower bound on the energy of an incompressible flow," Freedman says. "This suggests wider applicability of the topological lower-bound principle than was visible from Arnold's work."
The new results provide an intriguing hint of how to keep a hot, ionized gas, or plasma, from leaking out of a magnetic "bottle" -- a problem faced in fusion-energy research. In this case, creating local "tangles" in a confining magnetic field many keep energy from dissipating below a certain level.
"Any real-world system is much more complicated than what we addressed," Freedman says. "But that's often the nature of mathematical physics. You don't try to build a model as complicated as the world. You try to build something you can analyze, and then you take it from there."