Linearized Fields for Causal Variational Principles: Existence Theory and Causal Structure

The existence theory for solutions of the linearized field equations for causal variational principles is developed. We begin by studying the Cauchy problem locally in lens-shaped regions, defined as subsets of space-time which admit foliations by surface layers satisfying hyperbolicity conditions. We prove existence of weak solutions and show uniqueness up to vectors in the orthogonal complement of the jets used for testing. The connection between weak and strong solutions is analyzed. Global solutions are constructed by exhausting space-time by lens-shaped regions. We construct advanced and retarded Green's operators and study their properties.