The Helically - Reduced Wave Equation as a Symmetric - Positive System

Motivated by the partial differential equations of mixed type that arise in the reduction of the Einstein equations by a helical Killing vector field, we consider a boundary value problem for the helically-reduced wave equation with an arbitrary source in 2+1 dimensional Minkowski spacetime. The reduced equation is a second-order partial differential equation which is elliptic inside a disk and hy-perbolic outside the disk. We show that the reduced equation can be cast into symmetric-positive form. Using results from the theory of symmetric-positive differential equations, we show that this form of the helically-reduced wave equation admits unique, strong solutions for a class of boundary conditions which include Sommerfeld conditions at the outer boundary. Physical systems are typically governed by partial differential equations (PDEs) of a fixed type: elliptic, hyperbolic, or parabolic. The mathematical properties of such equations have been extensively investigated (see, e.g., Refs. [1,2]). Considerably less is known about PDEs of mixed type, by which we mean equations whose type is different in different subdomains of the domain of interest, e.g., elliptic in one region and hyperbolic in another [3]. Compared to elliptic, hyperbolic or parabolic equations, mixed type equations are rather unusual, both in the boundary conditions that can be imposed to get existence and uniqueness of solutions as well as in the regularity of solutions that are obtained. Moreover , the lower-order terms in equations of mixed type take on a more significant role than in equations of fixed type. This latter feature means that it is difficult to obtain general results about PDEs of mixed type; to a large extent, one must investigate each set of equations, each set of boundary conditions, etc. separately. In relativistic field theory on a fixed spacetime, mixed type equations occur after performing a symmetry reduction of hyperbolic PDEs with respect to an isometry group which has an infinitesimal generator that changes type from timelike to spacelike. In generally covariant theories such symmetry reductions may yield PDEs of mixed type in appropriate gauges. An important example of the latter type, currently of considerable interest in gravitational physics, arises in the quasi-stationary approximation to the 2-body problem in general relativity [4,5,6]. There one is interested in solving the Einstein equations for spacetimes admitting a helical Killing vector field. The helical Killing vector field, which represents a rotating reference frame, will be timelike near the bodies, and spacelike far from the bodies. The …