Differential systems, moving frames, structure-preserving submersions and geometrical problems in physics
The present work applies the theories of exterior differential systems, method of equivalence and moving frames to the study of geometrical problems arising in physics, especially the class of problems that can be described as “structure-preserving submer- sions”. A novel feature of our approach is the formulation of an algorithm which we have named “the method of involutive seeds”. By using this method, we can rapidly determine the number of free functions that we must specify in order to completely specify the problem, which we will call the “degree of arbitrariness” of the problem, and which for many physical systems is linked to the physical degree of freedom. This algorithm is especially helpful in dealing with systems with many constraints such as structure-preserving submersions. We also give other examples of calculations using this algorithm: in particular, we used it to investigate the degree of arbitrariness of the theory of general very special relativity, based on Riemannian geometry with a holonomy constraint, and thus argue that such a theory is not a suitable candidate for a physical theory. As for structure-preserving submersions, which we propose as a generalisation for Riemannian submersions to other geometrical structures, after investigating the prop- erties and degrees of arbitrariness of the general construction we use it to study the problem of flows, especially rigid flows in relativity. We generalise the classical Her- glotz–Noether theorem, which states that rotational rigid flow in Minkowski spacetime must be isometric, to all dimensions and to all conformally flat spacetimes in all di- mensions, and also to shear-free flows in conformally flat spacetimes; we generalize a partial result of the Ellis conjecture that a self-gravitating shear-free perfect fluid in geodesic motion must be either expansion-free or vorticity-free to all dimensions, and we will see clearly the origin of this result from the group structure of spacetime; we discuss an approach for the general Ellis conjecture, and show the relation between the Herglotz–Noether theorem and the Ellis conjecture; we show that for a free point particle lagrangian to have a Galilean boost symmetry, it is necessary and sufficient that we have a totally flat direction decoupled from the rest; finally, we give a rough, heuristic reasoning for why some of the Pauli reductions in which we attempt to get a larger gauge group than usually allowed from dimensional reduction are consistent.