Concepts of Hyperbolicity and Relativistic Continuum Mechanics

The notion of hyperbolicity of a partial differential equation (PDE), or a system of PDE’s, is central for the field theories of mathematical physics. It is closely related to the well-posedness of the Cauchy problem and to the causal structure underlying these theories. In standard theories describing relativistic fields in vacuo this causal structure is that given by the spacetime metric, a second-order symmetric tensor of Lorentzian signature. If matter is included, things become both more complicated and more subtle. In fact, the awareness of some of those complications predates Relativity by centuries. An example is afforded by the phenomenon, already studied by Huygens, of birefringence in crystal optics . There is currently an increase of attention in the field of Relativity, due in part to demands from Numerical Relativity, devoted to certain notions of hyperbolicity applied to the Einstein equations (for an excellent review see [18]). There the main challenge, not discussed in the present notes at all, comes from the fact that, already in vacuum, the Einstein equations by themselves, i.e. prior to the imposition of any gauge conditions, are not hyperbolic. The main burden, then, is to find a “hyperbolic reduction” turning the Einstein equations, or a subset thereof, into a hyperbolic system appropriate for the purpose at hand. However the complications in the causal structure one finds in continuum mechanics, which are our main focus here, are absent in the Einstein vacuum case at least for the reductions proposed so far. Of course, these complications do come into play ultimately once matter-couplings are included. These notes attempt an elementary introduction to some notions of hyperbolicity and the “characteristic geometry” associated with or underlying these notions. The section following this one is devoted to the general notion of a hyperbolic polynomial, which in our case of course arises as the characteristic polynomial of a PDE. It is interesting that this notion is on one hand restrictive enough to encode essentially all the required features of a theory in order to be “causal” on the other hand flexible enough to account for an amazing variety of phenomena relativistic or nonrelativistic ranging from gravitational radiation to water waves or phonons