Cauchy Problem for Gott Spacetime

Abstract

Gott recently has constructed a spacetime modeled by two infinitely long, parallel cosmic strings which pass and gravitationally interact with each other. For large enough velocity, the spacetime will contain closed timelike curves. An explicit construction of the solution for a scalar field is presented in detail and a proof for the existence of such a solution is given for initial data satisfying conditions on an asymptotically null partial Cauchy surface. Solutions to smooth operators on the covering space are invariant under the isometry are shown to be pull back of solutions of the associated operator on the base space. Projection maps and translation operators for the covering space are developed for the spacetime, and explicit expressions for the projection operator and the isometry group of the covering space are given. It is shown that the Gott spacetime defined is a quotient space of Minkowski space by the discrete isometry subgroup of self-equivalences of the projection map.