Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes

Abstract

Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.2010 Mathematics Subject Classification. 58J45,35L45,35L51,35L55,81T20. Key words and phrases. Globally hyperbolic Lorentzian manifolds, Green-hyperbolic operators, wave operators, normally hyperbolic operators, Dirac-type operators, Green's operators, support system, symmetric hyperbolic system, Cauchy problem, energy estimate, finite propagation speed, locally covariant quantum field theory. 1 1.1. Cauchy hypersurfaces. A subset Σ ⊂ M is called a Cauchy hypersurface if every inextensible timelike curve in M meets Σ exactly once. Any Cauchy hypersurface is a topological submanifold of codimension 1. All Cauchy hypersurfaces of M are homeomorphic. If M possesses a Cauchy hypersurface then M is called globally hyperbolic. This class of Lorentzian manifolds contains many important examples: Minkowski space, Friedmann models, the Schwarzschild model and deSitter spacetime are globally hyperbolic. Bernal and Sánchez proved an important structural result [6, Thm. 1.1]: Any globally hyperbolic Lorentzian manifold has a Cauchy temporal function. This is a smooth function t : M → R with past-directed timelike gradient ∇t such that the levels t −1 (s) are (smooth spacelike) Cauchy hypersurfaces if nonempty.1.2. Future and past. From now on let M always be globally hyperbolic. For any x ∈ M we denote by J + (x) the set all points that can be reached by future-directed causal curves emanating from x. For any subset A ⊂ M we put J + (A) := x∈A J + (x). If A is closed so is J + (A). We call a subset A ⊂ M strictly past compact if it is closed and there is aWe denote by I + (x) the set of all points in M that can be reached by future-directed timelike curves emanating from x.Interchanging the roles of future and past, we similarly define J − (x), J − (A), I − (x), strictly future compact and past compact subsets of M. If A ⊂ M is past compact and future compact then we call A temporally compact. For any compact subsetscompact. Similarly, strictly future compact sets are future compact. If we want to emphasize the ambient manifold M, then we write J + M (x) instead of J + (x) and similarly for J − M (x), J ± M (A), and I ± M (A). GREEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 3 Example 1.1. Let M be Minkowski space and let C ⊂ M be an open cone with tip 0 containing the closed cone J − (0) \ {0}. Then A = M \ C is past compact but not strictly past compact. Indeed, for each x ∈ M, the set J −is not strictly past compact because the in...

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