aa r X i v : . [ m a t h - ph ] J u l GREEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLICSPACETIMES
CHRISTIAN B ¨ARA
BSTRACT . Green-hyperbolic operators are linear differential operators acting on sec-tions of a vector bundle over a Lorentzian manifold which possess advanced and retardedGreen’s operators. The most prominent examples are wave operators and Dirac-type oper-ators. This paper is devoted to a systematic study of this class of differential operators. Forinstance, we show that this class is closed under taking restrictions to suitable subregionsof the manifold, under composition, under taking “square roots”, and under the direct sumconstruction. Symmetric hyperbolic systems are studied in detail. I NTRODUCTION
Green-hyperbolic operators are certain linear differential operators acting on sections of avector bundle over a Lorentzian manifold. They are, by definition, those operators whichpossess advanced and retarded Green’s operators. The most prominent examples are nor-mally hyperbolic operators (wave equations) and Dirac-type operators. The reason forintroducing them in [2] lies in the fact that they can be quantized; one can canonicallyconstruct a bosonic locally covariant quantum field theory for them.The aim of the present paper is to study Green-hyperbolic operators systematically from ageometric and an analytic perspective. The underlying Lorentzian manifold must be wellbehaved for the analysis of hyperbolic operators. In technical terms, it must be globallyhyperbolic. In the first section we collect material about such Lorentzian manifolds. Weintroduce various compactness properties for closed subsets and show their interrelation.These considerations will later be applied to the supports of sections.In the second section we study various spaces of smooth sections of our vector bundle.The crucial concept is that of a support system. This is a family of closed subsets of ourmanifold with certain properties making it suitable for defining a good space of sectionsby demanding that their supports be contained in the support system. We observe a dualityprinciple; a distributional section has support in a support system if and only if it extendsto a continuous linear functional on test sections with support in the dual support system.Green’s operators and Green-hyperbolic differential operators are introduced in the thirdsection. We give various examples and show that the class of Green-hyperbolic operators isclosed under taking restrictions to suitable subregions of the manifold, under composition,under taking “square roots”, and under the direct sum construction. This makes it a largeand very flexible class of differential operators to consider. We show that the Green’soperators are unique and that they extend to several spaces of sections. We argue thatGreen-hyperbolic operators are not necessarily hyperbolic in any PDE-sense and that theycannot be characterized in general by well-posedness of a Cauchy problem.The fourth section is devoted to extending the Green’s operators to distributional sections.We show that an important analytical result for the causal propagator (the difference of
Mathematics Subject Classification.
Key words and phrases.
Globally hyperbolic Lorentzian manifolds, Green-hyperbolic operators, wave oper-ators, normally hyperbolic operators, Dirac-type operators, Green’s operators, support system, symmetric hyper-bolic system, Cauchy problem, energy estimate, finite propagation speed, locally covariant quantum field theory. the advanced and the retarded Green’s operator), also holds when one replaces smooth bydistributional sections.In the last section we study symmetric hyperbolic systems over globally hyperbolic man-ifolds. We provide detailed proofs of well-posedness of the Cauchy problem, finitenessof the speed of propagation and the existence of Green’s operators. The crucial step inthese investigations is an energy estimate for the solution to such a symmetric hyperbolicsystem. We conclude by observing that a symmetric hyperbolic system can be quantizedin two ways; one yields a bosonic and the other one a fermionic locally covariant quantumfield theory.
Acknowledgments.
It is a great pleasure to thank Klaus Fredenhagen, Ulrich Menne, andMiguel S´anchez for very helpful discussions and an anonymous referee for very inter-esting suggestions. Thanks also go to Sonderforschungsbereich 647 funded by DeutscheForschungsgemeinschaft for financial support.1. G
LOBALLY H YPERBOLIC L ORENTZIAN M ANIFOLDS
We summarize various facts about globally hyperbolic Lorentzian manifolds. For detailsthe reader is referred to one of the classical textbooks [5, 11, 12]. Throughout this article, M will denote a time oriented Lorentzian manifold. We use the convention that the signatureof M is ( − + · · · +) . Note that we do not specify the dimension of M nor do we assumeorientability or connectedness.1.1. Cauchy hypersurfaces.
A subset S ⊂ M is called a Cauchy hypersurface if everyinextensible timelike curve in M meets S exactly once. Any Cauchy hypersurface is atopological submanifold of codimension 1. All Cauchy hypersurfaces of M are homeo-morphic.If M possesses a Cauchy hypersurface then M is called globally hyperbolic . This class ofLorentzian manifolds contains many important examples: Minkowski space, Friedmannmodels, the Schwarzschild model and deSitter spacetime are globally hyperbolic. Bernaland S´anchez proved an important structural result [6, Thm. 1.1]: Any globally hyperbolicLorentzian manifold has a Cauchy temporal function . This is a smooth function t : M → R with past-directed timelike gradient (cid:209) t such that the levels t − ( 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 curvesemanating from x . For any subset A ⊂ M we put J + ( A ) : = S x ∈ A J + ( x ) . If A is closedso is J + ( A ) . We call a subset A ⊂ M strictly past compact if it is closed and there is acompact subset K ⊂ M such that A ⊂ J + ( K ) . If A is strictly past compact so is J + ( A ) because J + ( A ) ⊂ J + ( J + ( K )) = J + ( K ) . A closed subset A ⊂ M is called future compact if A ∩ J + ( x ) is compact for all x ∈ M . For example, if S is a Cauchy hypersurface, then J − ( S ) is future compact.We denote by I + ( x ) the set of all points in M that can be reached by future-directed timelikecurves emanating from x . The set I + ( x ) is the interior of J + ( x ) ; in particular, it is an opensubset of M .Interchanging the roles of future and past, we similarly define J − ( x ) , J − ( A ) , I − ( x ) , strictlyfuture compact and past compact subsets of M . If A ⊂ M is past compact and futurecompact then we call A temporally compact . For any compact subsets K , K ⊂ M theintersection J + ( K ) ∩ J − ( K ) is compact. If A is past compact so is J + ( A ) because J + ( A ) ∩ J − ( x ) = J + ( A ∩ J − ( x )) ∩ J − ( x ) . Similarly, if A is future compact then J − ( A ) is futurecompact too. If A is strictly past compact then it is past compact because A ∩ J − ( x ) ⊂ J + ( K ) ∩ J − ( x ) is compact. 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 ) . REEN-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 0containing the closed cone J − ( ) \ { } . Then A = M \ C is past compact but not strictlypast compact. Indeed, for each x ∈ M , the set J − ( x ) ∩ A = J − ( x ) \ C is compact. But A is not strictly past compact because the intersection of A and spacelike hyperplanes is notcompact, compare Lemma 1.5 below (Fig. 1). b b x J − ( x ) J − ( ) C F IG . 1: Past-compact set which is not strictly past compact
This example also shows that (surjective) Cauchy temporal functions need not be boundedfrom below on past compact sets. However, we have:
Lemma 1.2.
For any closed subset A ⊂ M the following are equivalent:(i) A is past compact;(ii) there exists a smooth spacelike Cauchy hypersurface S ⊂ M such that A ⊂ J + ( S ) ;(iii) there exists a surjective Cauchy temporal function t : M → R which is bounded frombelow on A.Proof. The implication “(iii) ⇒ (ii)” is trivial and the inverse implication is a consequenceof [7, Thm. 1.2]. The implication “(ii) ⇒ (i)” is also trivial because J + ( S ) is past compact.We only need to show “(i) ⇒ (ii)”.Let A be past compact. Then J + ( A ) is also past compact. Moreover, M ′ : = M \ J + ( A ) is an open subset of M with the property J − ( M ′ ) = M ′ . Hence M ′ is globally hyperbolicitself. Let S be a smooth spacelike Cauchy hypersurface in M ′ . Since A ⊂ J + ( A ) ⊂ J + ( S ) it remains to show that S is also a Cauchy hypersurface in M .Let c be an inextensible future-directed timelike curve in M . Once c has entered J + ( A ) itremains in J + ( A ) . Since J + ( A ) is past compact and c is inextensible, c must also meet M ′ .Thus c is the concatenation of an inextensible future-directed timelike curve c in M ′ and a(possibly empty) curve c in J + ( A ) . Since c meets S exactly once, so does c . This showsthat S is a Cauchy hypersurface in M as well. (cid:3) Reversing future and past, we see that a closed subset A ⊂ M is future compact if andonly if A ⊂ J − ( S ) for some Cauchy hypersurface S ⊂ M . This in turn is equivalent tothe existence of a surjective Cauchy temporal function t : M → R which is bounded fromabove on A .Consequently, A is temporally compact if and only if A ⊂ J + ( S ) ∩ J − ( S ) for someCauchy hypersurfaces S , S ⊂ M . Lemma 1.3.
For any past-compact subset A ⊂ M there exists a past-compact subset A ′ ⊂ M such that A is contained in the interior of A ′ . Analogous statements hold for future-compact sets and for temporally compact sets.Proof. Let A ⊂ M be past compact. Choose a Cauchy hypersurface S ⊂ M such that A ⊂ J + ( S ) . Choose a second Cauchy hypersurface S ′ ⊂ I − ( S ) . Then A ′ : = J + ( S ′ ) does thejob. (cid:3) For A ⊂ M we write J ( A ) : = J + ( A ) ∪ J − ( A ) . We call A spacially compact if A is closedand there exists a compact subset K ⊂ M with A ⊂ J ( K ) . We have the following analog toLemma 1.2: CHRISTIAN B ¨AR
Lemma 1.4.
For any closed subset A ⊂ M the following holds:(i) A is strictly past compact if and only if A ⊂ J + ( K S ) for some compact subset K S ofsome smooth spacelike Cauchy hypersurface S ⊂ M;(ii) A is strictly future compact if and only if A ⊂ J − ( K S ) for some compact subset K S ofsome smooth spacelike Cauchy hypersurface S ⊂ M;(iii) A is spacially compact if and only if A ⊂ J ( K S ) for some compact subset K S of any Cauchy hypersurface S ⊂ M.Proof.
One direction in (i) is trivial: if A ⊂ J + ( K S ) , then A is strictly past compact bydefinition. Conversely, let A be strictly past compact and let K ⊂ M be a compact subsetsuch that A ⊂ J + ( K ) . Then choose a smooth spacelike Cauchy hypersurface S ⊂ M suchthat K ⊂ J + ( S ) and put K S : = S ∩ J − ( K ) . Then K S is compact and A ⊂ J + ( K ) ⊂ J + ( J + ( S ) ∩ J − ( K )) = J + ( S ∩ J − ( K )) = J + ( K S ) . The proof of (ii) is analogous. As to (iii), if A is spacially compact and S ⊂ M a Cauchyhypersurface, then K S : = S ∩ J ( K ) does the job. (cid:3) We have the following diagram of implications of possible properties of a closed subset of M : compact n v ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ( ❩❩❩❩❩❩❩❩❩❩ ❩❩❩❩❩❩❩❩❩❩ strictly past compact ( ❩❩❩❩❩❩ ❩❩❩❩❩❩ (cid:11) (cid:19) strictly future compact n v ❞❞❞❞❞❞❞❞❞❞❞❞ (cid:11) (cid:19) spacially compactpast compact future compacttemporally compact h p ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ . ❞❞❞❞❞❞❞❞ ❞❞❞❞❞❞❞❞ D IAGRAM Possible properties of closed subsets
Spacially compact manifolds.
None of the reverse implications in the diagram holdsin general. In a special case however, the diagram simplifies considerably, see Remark 1.8.The terminology “spacially compact” is justified by the following lemma:
Lemma 1.5.
Let A ⊂ M be spacially compact and let S ⊂ M be a Cauchy hypersurface.Then A ∩ S is compact.Proof. For any x ∈ M the intersection J − ( x ) ∩ J + ( S ) is compact by Lemma 40 in [12,p. 423]. Thus J − ( x ) ∩ S is compact as well. Let K ⊂ M be compact with A ⊂ J ( K ) .The sets I − ( x ) where x ∈ M form an open cover of M . Hence there are finitely manypoints x , . . . , x n such that K ⊂ S ni = I − ( x i ) . Then we have J − ( K ) ⊂ S ni = J − ( x i ) . Hence S ∩ J − ( K ) ⊂ S ni = ( S ∩ J − ( x i )) is compact.Similarly, one shows that S ∩ J + ( K ) is compact. Thus S ∩ A ⊂ S ∩ J ( K ) is compact aswell. (cid:3) Recall that since all Cauchy hypersurfaces are homeomorphic they are all compact or allnoncompact.
Lemma 1.6.
The globally hyperbolic manifold M is spacially compact if and only if it hascompact Cauchy hypersurfaces.Proof.
If the Cauchy hypersurfaces are compact, let S be one of them. Then M = J ( S ) ,hence M is spacially compact.Conversely, if M is spacially compact, then Lemma 1.5 with A = M shows that the Cauchyhypersurfaces are compact. (cid:3) Lemma 1.7.
Let M be globally hyperbolic and spacially compact. Let A ⊂ M be closed.Then the following are equivalent:
REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 5 (i) A is strictly past compact;(ii) A is past compact;(iii) some Cauchy temporal function t : M → R attains its minimum on A;(iv) all Cauchy temporal functions t : M → R attain their minima on A.Proof. Since the Cauchy hypersurfaces of M are compact, Lemmas 1.2 and 1.4 show“(i) ⇔ (ii)”. The implication “(iv) ⇒ (iii)” is clear. To show “(i) ⇒ (iv)” let A ⊂ J + ( K ) for some compact subset K ⊂ M and let t be a Cauchy temporal function. Choose T larger than the infimum of t on A . Since A ∩ J − ( t − ( T )) is contained in the compact set J + ( K ) ∩ t − (( − ¥ , T ) = J + ( K ) ∩ J − ( t − ( T )) , the function t attains its minimum t on thisset. On the rest of A , the values of t are even larger than T , hence t is the minimum of t on all of A .As to “(iii) ⇒ (ii)”, let t : M → R be a Cauchy temporal function which attains its minimumon A . By composing with an orientation-preserving diffeomorphism t ( M ) → R , we mayw.l.o.g. assume that t is surjective. Now Lemma 1.2 shows that A is past compact. (cid:3) Remark 1.8. If M is spacially compact, then every closed subset of A ⊂ M is spa-cially compact. Moreover, if A is temporally compact, then any Cauchy temporal func-tion t : M → R attains its maximum s + and its minimum s − by Lemma 1.7. Thus A ⊂ t − ([ s − , s + ]) ≈ S × [ s − , s + ] where S = t − ( s − ) is a Cauchy hypersurface. Since S is compact, so is A .Summarizing, Diagram 1 of implications for closed subsets simplifies as follows for spa-cially compact M :strictly past compact K S (cid:11) (cid:19) compact k s + K S (cid:11) (cid:19) strictly future compact K S (cid:11) (cid:19) past compact temporally compact k s + future compactD IAGRAM Closed subsets of a spacially compact manifold
Duality.
We will need the following duality result:
Lemma 1.9.
Let M be globally hyperbolic and let A ⊂ M be closed. Then the followingholds:(i) A is past compact if and only if A ∩ B is compact for all strictly future compact setsB;(ii) A is future compact if and only if A ∩ B is compact for all strictly past compact setsB;(iii) A is temporally compact if and only if A ∩ B is compact for all spacially compact setsB;(iv) A is strictly past compact if and only if A ∩ B is compact for all future compact setsB;(v) A is strictly future compact if and only if A ∩ B is compact for all past compact setsB;(vi) A is spacially compact if and only if A ∩ B is compact for all temporally compact setsB.Proof. (a) We show (i). If A ∩ B is compact for every strictly future compact B , then, inparticular, A ∩ J − ( x ) is compact for every x ∈ M . Hence A is past compact.Conversely, let A be past compact and B be strictly future compact. Then A ⊂ J + ( S ) and B ⊂ J − ( K ) for some Cauchy hypersurface S ⊂ M and some compact subset K ⊂ M . Thus A ∩ B ⊂ J + ( S ) ∩ J − ( K ) , hence A ∩ B is contained in a compact set, hence compact itself.(b) The proof of (ii) is analogous. As to (iii), if A ∩ B is compact for every spacially compact B , then, in particular, A ∩ J + ( x ) and A ∩ J − ( x ) are compact for every x ∈ M . Hence A istemporally compact. CHRISTIAN B ¨AR
Conversely, let A be temporally compact and B be spacially compact. We choose a compact K ⊂ M with B ⊂ J ( K ) . By (i), A ∩ J − ( K ) is compact and by (ii), A ∩ J + ( K ) is compact.Thus A ∩ B ⊂ A ∩ J ( K ) = ( A ∩ J + ( K )) ∪ ( A ∩ J − ( K )) is compact.(c) We show (iv). By (ii) the intersection of a strictly past compact set and a future compactset is compact. Now assume A is not strictly past compact. We have to find a futurecompact set B such that A ∩ B is noncompact. Let K ⊂ K ⊂ K ⊂ · · · ⊂ M be an exhaustionby compact subsets. We choose the exhaustion such that every compact subset of M iscontained in K j for sufficiently large j . Since A is not strictly past compact there exists x j ∈ A \ J + ( K j ) for every j . The set B : = { x , x , x , . . . } is not compact because otherwise,for sufficiently large j , we would have B ⊂ K j ⊂ J + ( K j ) contradicting the choice of the x i . But B is future compact. Namely, let x ∈ M . Then x ∈ K j for j large and therefore B ∩ J + ( x ) ⊂ B ∩ J + ( K j ) ⊂ { x , . . . , x j − } is finite, hence compact. Now A ∩ B = B is notcompact which is what we wanted to show.(d) The proof of (v) is analogous. As to (vi), we know already by (iii) that the intersectionof a temporally compact and a spacially compact set is always compact. If A is not spaciallycompact, then the same construction as in the proof of (iv) with J + ( K j ) replaced by J ( K j ) yields a noncompact set B ⊂ A which is temporally compact. This concludes the proof. (cid:3) Causal compatibility.
An open subset N of a time oriented Lorentzian manifold M is a time oriented Lorentzian manifold itself. We call N causally compatible if J ± N ( x ) = J ± M ( x ) ∩ N for all x ∈ N . In other words, any two points in N which can be connected bycausal curve in M can also be connected by causal curve that stays in N .2. T HE F UNCTION S PACES
Throughout this section, let M denote a globally hyperbolic Lorentzian manifold. In par-ticular, M carries a time-orientation and an induced volume element which we denote bydV. Moreover, let E → M be a (real or complex, finite dimensional) vector bundle.2.1. Smooth sections.
We denote the space of smooth sections of E by C ¥ ( M , E ) . Anyconnection (cid:209) on E induces, together with the Levi-Civita connection on T ∗ M , a connectionon T ∗ M ⊗ ℓ ⊗ E for any ℓ ∈ N . For any f ∈ C ¥ ( M , E ) , the ℓ th covariant derivative (cid:209) ℓ f : = (cid:209) · · · (cid:209)(cid:209) f is a smooth section of T ∗ M ⊗ ℓ ⊗ E .For any compact subset K ⊂ M , any m ∈ N , any connection (cid:209) on E and any auxiliarynorms | · | on T ∗ M ⊗ ℓ ⊗ E we define the semi-norm k f k K , m , (cid:209) , |·| : = max ℓ = , ··· , m max x ∈ K | (cid:209) ℓ f ( x ) | for f ∈ C ¥ ( M , E ) . By compactness of K , different choices of (cid:209) and | · | lead to equivalentsemi-norms. For this reason, we may suppress (cid:209) and | · | in the notation and write k f k K , m instead of k f k K , m , (cid:209) , |·| . This family of semi-norms is separating and turns C ¥ ( M , E ) into alocally convex topological vector space. If we choose a sequence K ⊂ K ⊂ K ⊂ · · · ⊂ M of compact subsets with S ¥ i = K i = M and such that each K i is contained in the interiorof K i + , then the countable subfamily k · k K i , i of semi-norms is equivalent to the originalfamily. Hence C ¥ ( M , E ) is metrizable. An Arzel`a-Ascoli argument shows that C ¥ ( M , E ) is complete. Thus C ¥ ( M , E ) is a Fr´echet space. A sequence of sections converges in C ¥ ( M , E ) if and only if the sections and all their (higher) derivatives converge locallyuniformly.2.2. Support systems.
For a closed subset A ⊂ M denote by C ¥ A ( M , E ) the space of allsmooth sections f of E with supp f ⊂ A . Then C ¥ A ( M , E ) is a closed subspace of C ¥ ( M , E ) and hence a Fr´echet space in its own right. Moreover, if A ⊂ A then C ¥ A ( M , E ) is a closedsubspace of C ¥ A ( M , E ) .We denote by C M the set of all closed subsets of M . Definition 2.1.
A subset A ⊂ C M is called a support system on M if the following holds: REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 7 (i) For any A , A ′ ∈ A we have A ∪ A ′ ∈ A ;(ii) For any A ∈ A there is an A ′ ∈ A such that A is contained in the interior of A ′ ;(iii) If A ∈ A and A ′ ⊂ A is a closed subset, then A ′ ∈ A .The first condition implies that A is a direct system with respect to inclusion. The thirdcondition is harmless; if A satisfies (i) and (ii), then adding all closed subsets of themembers of A to A will give a support system.Given a support system on M we obtain the direct system { C ¥ A ( M , E ) } A ∈ A of subspaces of C ¥ ( M , E ) and denote by C ¥ A ( M , E ) its direct limit as a locally convex topological vectorspace. As a vector subspace, C ¥ A ( M , E ) is simply S A ∈ A C ¥ A ( M , E ) . A convex subset O ⊂ C ¥ A ( M , E ) is open if and only if O ∩ C ¥ A ( M , E ) is open for all A ∈ A . Note that C ¥ A ( M , E ) is not a closed subspace of C ¥ ( M , E ) in general. Definition 2.2.
We call a support system essentially countable if there is a sequence A , A , A , . . . ∈ A such that each A j ⊂ A j + and for any A ∈ A there exists a j with A ⊂ A j . Such a sequence A ⊂ A ⊂ A ⊂ · · · is called a basic chain of A . Lemma 2.3.
Let A ⊂ C M be an essentially countable support system on M. If V ⊂ C ¥ A ( M , E ) is a bounded subset then there exists an A ∈ A such that V ⊂ C ¥ A ( M , E ) . Inparticular, for any convergent sequence f j ∈ C ¥ A ( M , E ) there exists an A ∈ A such thatf j ∈ C ¥ A ( M , E ) for all j. This shows that a sequence ( f j ) converges in C ¥ A ( M , E ) if and only if there exists an A ∈ A such that f j ∈ C ¥ A ( M , E ) for all j and ( f j ) converges in C ¥ A ( M , E ) . Proof of Lemma 2.3.
Consider a basic chain A ⊂ A ⊂ A ⊂ . . . . Let V ⊂ C ¥ A ( M , E ) be asubset not contained in any C ¥ A j ( M , E ) . We have to show that V is not bounded. Pick points x j ∈ M \ A j and sections f j ∈ V with f j ( x j ) =
0. Define the convex set W : = (cid:26) f ∈ C ¥ A ( M , E ) (cid:12)(cid:12)(cid:12)(cid:12) | f ( x j ) | < | f j ( x j ) | j for all j (cid:27) . Each A ∈ A contains only finitely many x j . Thus W ∩ C ¥ A ( M , E ) = { f ∈ C ¥ A ( M , E ) |k f k { x j } , < | f j ( x j ) | / j } is open in C ¥ A ( M , E ) . Therefore W is an open neighborhood of0 in C ¥ A ( M , E ) .For any T > T · W = { f ∈ C ¥ A ( M , E ) | | f ( x j ) | < Tj | f j ( x j ) | for all j } and hence f j / ∈ TW for j > T . Thus V is not contained in any TW and is therefore not bounded. (cid:3) Example 2.4.
The system A = C M of all closed subsets is an essentially countable supportsystem on M . A basic chain is given by the constant sequence M ⊂ M ⊂ M ⊂ · · · . Clearly, C ¥ C M ( M , E ) = C ¥ ( M , E ) . Example 2.5.
Let A = c where c is the set of all compact subsets of M . A basic chain canbe constructed as follows: Provide M with a complete Riemannian metric g . Fix a point x ∈ M . Now let A j be the closed ball centered at x with radius j with respect to g .Then C ¥ c ( M , E ) is the space of compactly supported smooth sections, also called test sec-tions . Example 2.6.
Let A = sc be the set of all spacially compact subsets of M . If K ⊂ K ⊂ K ⊂ · · · is a basic chain of c , then J ( K ) ⊂ J ( K ) ⊂ J ( K ) ⊂ · · · is a basic chain of sc .Hence sc is essentially countable.Now C ¥ sc ( M , E ) is the space of smooth sections with spacially compact support. Recallthat a sequence ( f j ) converges in C ¥ sc ( M , E ) if and only if there exists a compact subset K ⊂ M such that supp ( f j ) ⊂ J ( K ) for all j and ( f j ) converges locally uniformly with allderivatives. CHRISTIAN B ¨AR
Example 2.7.
Let A = spc be the set of all strictly past compact subsets of M . As in theprevious example we see that spc is essentially countable. Now C ¥ spc ( M , E ) is the space ofsmooth sections with strictly past-compact support.Similarly, one can define the space C ¥ s f c ( M , E ) of smooth sections with strictly future-compact support. Example 2.8.
Let A = pc be the set of all past-compact subsets. If M is spacially compactthen pc = spc by Lemma 1.7 but in general pc is strictly larger than spc . We obtain thespace C ¥ pc ( M , E ) of smooth sections with past-compact support.In general, the support system pc is not essentially countable. The following example wascommunicated to me by Miguel S´anchez. Let M be the ( + ) -dimensional Minkowskispace. Let A ⊂ A ⊂ A ⊂ · · · ⊂ M be a chain of past-compact subsets. Look at the “future-diverging” sequence of points ( n , ) ∈ M and choose points p n ∈ M \ ( A n ∪ J − ( n , )) .By construction, A : = { p , p , p , . . . } is not contained in any A n but A is past compact.Namely, let x ∈ M . Then there exists an n such that x ∈ J − ( n , ) . Now J − ( x ) ∩ A ⊂ J − ( n , ) ∩ A is finite and hence compact. Thus no chain in pc captures all elements of pc ,so pc is not essentially countable. Example 2.9.
A similar discussion as in the previous example yields the space C ¥ f c ( M , E ) of smooth sections with future-compact support and the space C ¥ tc ( M , E ) of smooth sectionswith temporally compact support. Both support systems are not essentially countable ingeneral. But again, if M is spacially compact, they are because then f c = s f c and tc = c by Remark 1.8.If A ⊂ A ′ , then C ¥ A ( M , E ) ⊂ C ¥ A ′ ( M , E ) and the inclusion map is continuous. Hence weobtain the following diagram of continuously embedded spaces: C ¥ spc ( M , E ) (cid:31) (cid:127) / / v(cid:22) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙ C ¥ pc ( M , E ) s(cid:19) & & ▼▼▼▼▼▼▼▼▼ C ¥ c ( M , E ) * (cid:10) ♣♣♣♣♣♣♣♣♣ t(cid:20) & & ◆◆◆◆◆◆◆◆◆◆ (cid:31) (cid:127) / / C ¥ tc ( M , E ) ( (cid:8) ❦❦❦❦❦❦ ❧❧❧❧❧❧ v(cid:22) ❙❙❙❙❙❙ ) ) ❘❘❘❘❘❘ C ¥ sc ( M , E ) (cid:31) (cid:127) / / C ¥ ( M , E ) C ¥ s f c ( M , E ) (cid:31) (cid:127) / / ( (cid:8) ❦❦❦❦❦❦❦❦❦❦❦❦❦ C ¥ f c ( M , E ) +(cid:11) qqqqqqqqqq D IAGRAM Smooth sections with various support properties
All embeddings in Diagram 3 have dense image. Namely, we have
Lemma 2.10.
Let A be a support system on M such that c ⊂ A , i.e., each compact set iscontained in A . Then C ¥ c ( M , E ) is a dense subspace of C ¥ A ( M , E ) .Proof. Let f ∈ C ¥ A ( M , E ) and let O be a convex open neighborhood of f in C ¥ A ( M , E ) .Let A ∈ A with f ∈ C ¥ A ( M , E ) . Since O ∩ C ¥ A ( M , E ) is open in C ¥ A ( M , E ) there exists an e > k · k K , m such that { g ∈ C ¥ A ( M , E ) | k f − g k K , m < e } ⊂ O ∩ C ¥ A ( M , E ) . Pick a cutoff function c ∈ C ¥ c ( M , R ) with c ≡ K . Then g : = c · f ∈ C ¥ c ( M , E ) and k f − g k K , m =
0. Thus g ∈ O ∩ C ¥ A ( M , E ) . (cid:3) Note that M \ I − ( n , ) is not past compact so that A n ∪ J − ( n , ) cannot be all of M . Compare with Example 1.1however. REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 9
Distributional sections.
Now denote the dual bundle of E → M by E ∗ → M . Thecanonical pairing E ∗ ⊗ E → R is denoted by h· , ·i . A locally integrable section f of E canbe considered as a continuous linear functional on C ¥ c ( M , E ∗ ) by f [ j ] = R M h j , f i dV. Wedenote the topological dual space of C ¥ c ( M , E ∗ ) by D ′ ( M , E ) . The elements of D ′ ( M , E ) ,i.e., the continuous linear functionals on C ¥ c ( M , E ∗ ) , are called distributional sections of E .It is well known that a distributional section of E has compact support if and only if it ex-tends to a continuous linear functional on C ¥ ( M , E ∗ ) . We denote the space of distributionalsections of E with compact support by D ′ c ( M , E ) .More generally, for any closed subset A ⊂ M we denote by D ′ A ( M , E ) the space of all distri-butional sections of E whose support is contained in A . Likewise, for any support system A on M , we denote by D ′ A ( M , E ) the space of all distributional sections of E whose supportis an element of A . Again, we have the continuous embedding C ¥ A ( M , E ) ֒ → D ′ A ( M , E ) defined by f [ j ] = R M h j , f i dV for any f ∈ C ¥ A ( M , E ) and any test section j ∈ C ¥ c ( M , E ∗ ) .2.4. Duality.
We now characterize the topological dual spaces of the other spaces in Dia-gram 3 (with E replaced by E ∗ ). We will show that the support of a distributional section iscontained in a support system if and only if it extends to test sections having their supportin a dual support system. Definition 2.11.
Two support systems A and B be on M are said to be in duality if forany C ∈ C M :(i) C ∈ A if and only if C ∩ B is compact for all B ∈ B ;(ii) C ∈ B if and only if C ∩ A is compact for all A ∈ A . Example 2.12.
Here are some examples of support systems A and B in duality. The lastcolumn contains a justification of this fact. A B why? C M c obvious pc s f c Lemma 1.9 (i) and (v) f c spc
Lemma 1.9 (ii) and (iv) tc sc
Lemma 1.9 (iii) and (vi)T
ABLE Support systems in duality
Lemma 2.13.
Let A and B be two support systems on M in duality. Then a distributionalsection f ∈ D ′ ( M , E ) has support contained in A if and only if f extends to a continuouslinear functional on C ¥ B ( M , E ∗ ) .Proof. (a) Suppose first that supp f ∈ A . Let B ∈ B . Since supp f ∩ B is compact there isa cutoff function c ∈ C ¥ c ( M , R ) with c ≡ f ∩ B . We extend f to a linear functional F B on C ¥ B ( M , E ∗ ) by F B [ j ] : = f [ cj ] . This extension is independent of the choice of c because for another choice c ′ , f and cj − c ′ j have disjoint supports. If j j → C ¥ B ( M , E ∗ ) , then cj j → C ¥ c ( M , E ∗ ) andhence F B [ j j ] = f [ cj j ] →
0. Thus F B is continuous.Doing this for every B ∈ B we obtain an extension F of f to a linear functional on C ¥ B ( M , E ∗ ) with F B being the restriction of F to C ¥ B ( M , E ∗ ) . Continuity of F holds be-cause each F B is continuous.(b) Conversely, assume that f extends to a continuous linear functional F on C ¥ B ( M , E ∗ ) .We check that supp f ∈ A by showing that supp f ∩ B is compact for every B ∈ B .Let B ∈ B . Choose B ′ ∈ B such that B is contained in the interior of B ′ . Since therestriction F B ′ of F to C ¥ B ′ ( M , E ∗ ) is linear and continuous, there exists a seminorm k · k K , m and a constant C > | F B ′ [ j ] | ≤ C · k j k K , m for all j ∈ C ¥ B ′ ( M , E ∗ ) . In particular, F B ′ [ j ] = ( j ) and K are disjoint. Claim: B ∩ ( M \ K ) ⊂ M \ supp ( F ) .Namely, let x ∈ B ∩ ( M \ K ) . Then x lies in the interior of B ′ . Hence there is an openneighborhood U of x entirely contained in B ′ . Since x / ∈ K we may assume that U and K are disjoint. Now we know that for all j ∈ C ¥ c ( M , E ∗ ) with supp ( j ) ⊂ U we have F [ j ] = x / ∈ supp ( F ) . ✓ The claim implies supp ( F ) ⊂ ( M \ B ) ∪ K and hence supp ( F ) ∩ B ⊂ K . Therefore theintersection supp ( F ) ∩ B is compact. (cid:3) Remark 2.14.
Observe that for the proof of Lemma 2.13 we only need (i) in Defini-tion 2.11 but not (ii).Dualizing Diagram 3, Table 1 and Lemma 2.13 yield the following diagram of continuousembeddings of several spaces of distributions, characterized by different support proper-ties: D ′ f c ( M , E ) kK x x qqqqqqqqq D ′ s f c ( M , E ) ? _ o o hH ❦❦❦❦❦❦ u u ❦❦❦❦❦❦ D ′ ( M , E ) D ′ sc ( M , E ) ? _ o o D ′ tc ( M , E ) i i ❙❙❙❙❙❙❙❙❙❙❙❙❙ hH u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ D ′ c ( M , E ) ? _ o o f f ◆◆◆◆◆◆◆◆◆ jJ x x ♣♣♣♣♣♣♣♣♣♣ D ′ pc ( M , E ) f f ▼▼▼▼▼▼▼▼▼▼ D ′ spc ( M , E ) ? _ o o ❙❙❙❙❙❙❙ i i ❙❙❙❙❙❙ D IAGRAM Distributional sections with various support properties
Convergence of distributions.
Let A be one of the support systems C , pc , f c , tc , sc , spc , s f c , or c . Let B be the dual support system as in Table 1. The continuity propertyof a distributional section f ∈ D ′ A ( M , E ) means that for any B ∈ B the restriction of f to C ¥ B ( M , E ∗ ) is continuous. In other words, for any sequence of smooth sections j j withsupport contained in B which converge locally uniformly with their derivatives to some j ∈ C ¥ B ( M , E ∗ ) , we must have f [ j j ] → f [ j ] .Our distribution spaces are always equipped with the weak*-topology. This means thata sequence f j ∈ D ′ A ( M , E ) converges if and only if f j [ j ] converges for every fixed j ∈ C ¥ B ( M , E ∗ ) .We have the following analog to Lemma 2.10: Lemma 2.15.
Let A be one of the support systems C , pc, f c, tc, sc, spc, s f c, or c. ThenC ¥ c ( M , E ) is a dense subspace of D ′ A ( M , E ) .Proof. Let B be the dual support system to A as in Table 1. Let u ∈ D ′ A ( M , E ) . Put A : = supp ( u ) , hence u ∈ D ′ A ( M , E ) . It is well known that C ¥ c ( M , E ) is dense in D ′ ( M , E ) .Hence there is a sequence u j ∈ C ¥ c ( M , E ) with u j → u in D ′ ( M , E ) .Choose A ′ ∈ A such that A is contained in the interior of A ′ . Let c ∈ C ¥ ( M , R ) be afunction such that c ≡ A and supp c ⊂ A ′ .Let j ∈ C ¥ B ( M , E ∗ ) where B ∈ B . Since A ′ ∩ B is compact, the section cj has compactsupport. Therefore ( c u j )[ j ] = u j [ cj ] → u [ cj ] = ( c u )[ j ] = u [ j ] . Thus the compactly supported sections c u j converge to u in D ′ A ( M , E ) . (cid:3)
3. P
ROPERTIES OF G REEN -H YPERBOLIC O PERATORS
Green’s operators and Green hyperbolic operators.
Let E , E → M be vectorbundles over a globally hyperbolic manifold. Let P : C ¥ ( M , E ) → C ¥ ( M , E ) be a lineardifferential operator. Differential operators do not increase supports and yield continuousmaps P : C ¥ A ( M , E ) → C ¥ A ( M , E ) for any support system A . REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 11
There is a unique linear differential operator t P : C ¥ ( M , E ∗ ) → C ¥ ( M , E ∗ ) characterized by(1) Z M h j , P f i dV = Z M h t P j , f i dVfor all f ∈ C ¥ ( M , E ) and j ∈ C ¥ ( M , E ∗ ) such that supp f ∩ supp ( j ) is compact. Hereagain, h· , ·i denotes the canonical pairing of E ∗ i and E i . The operator t P is called the for-mally dual operator of P . Definition 3.1. An advanced Green’s operator of P is a linear map G + : C ¥ c ( M , E ) → C ¥ ( M , E ) such that(i) G + P f = f for all f ∈ C ¥ c ( M , E ) ;(ii) PG + f = f for all f ∈ C ¥ c ( M , E ) ;(iii) supp ( G + f ) ⊂ J + ( supp f ) for all f ∈ C ¥ c ( M , E ) .A linear map G − : C ¥ c ( M , E ) → C ¥ ( M , E ) is called a retarded Green’s operator of P if(i), (ii) hold and(iii)’ supp ( G − f ) ⊂ J − ( supp f ) holds for every f ∈ C ¥ c ( M , E ) . Definition 3.2.
The operator P is be called Green hyperbolic if P and t P have advancedand retarded Green’s operators.We will see in Corollary 3.12 that uniqueness of the Green’s operators comes for free. Example 3.3.
The most prominent examples of Green-hyperbolic operators are wave op-erators , also called normally hyperbolic operators . They are second-order differentialoperators P whose principal symbol is given by the Lorentzian metric. Locally they takethe form P = (cid:229) i j g i j ( x ) ¶ ¶ x i ¶ x j + (cid:229) j B j ( x ) ¶¶ x j + C ( x ) where g i j denote the components of the inverse metric tensor, and B j and C are matrix-valued coefficients depending smoothly on x .The class of wave operators contains the d’Alembert operator P = ✷ , the Klein-Gordonoperator P = ✷ + m , and the Klein-Gordon operator with a potential, P = ✷ + V . In thesecases, the operator acts on functions, i.e., the underlying vector bundles E and E aresimply trivial line bundles.On any vector bundle E one may choose a connection (cid:209) and put P = tr ( (cid:209) ) to obtain a waveoperator C ¥ ( M , E ) → C ¥ ( M , E ) . If E = L k T ∗ M is the bundle of k -forms, then P = d d + d d is a wave operator where d denotes the exterior differential and d the codifferential.It is shown in [3, Cor. 3.4.3] that wave operators have Green’s operators. Since the for-mally dual operator of a wave operator is again a wave operator, wave operators are Greenhyperbolic. Example 3.4.
Let us consider a concrete special case of Example 3.3. Let M = R be2-dimensional Minkowski space. We denote a generic point of M by ( t , x ) . Let P = ✷ = − ¶ ¶ t + ¶ ¶ x be the d’Alembert operator. Then one checks by explicit calculation that ( G + f )( t , x ) = − Z J − ( t , x ) f ( t , x ) d x d t = − Z t − ¥ (cid:18) Z x + t − t x + t − t f ( t , x ) d x (cid:19) d t yields an advanced Green’s operator for ✷ . Replacing J − ( t , x ) by J + ( t , x ) we get a retardedGreen’s operator. In other words, the integral kernel of G + is − times the characteristicfunction of { ( t , x , t , x ) | ( t , x ) ∈ J − ( t , x ) } = { ( t , x , t , x ) | ( x − x ) ≤ ( t − t ) , t ≤ t } ⊂ M × M . For the d’Alembert operator on higher-dimensional Minkowski space the integral kernelof G ± is no longer an L ¥ -function but is given by the so-called Riesz distributions, see [3,Sec. 1.2]. Example 3.5.
Let E = T ∗ M and m >
0. Then P = d d + m is the Proca operator . Now˜ P : = d d + d d + m is a wave operator and hence has Green’s operators ˜ G ± . One can checkthat G ± : = ( m − d d + id ) ◦ ˜ G ± are Green’s operators of P , compare [2, Sec. 2.4]. Similarly,one gets Green’s operators for t P . Thus the Proca operator is not a wave operator but it isGreen hyperbolic.3.2. Restrictions to subregions.
Green hyperbolicity persists under restriction to suitablesubregions of the manifold M . Lemma 3.6.
Let M be globally hyperbolic and let N ⊂ M be an open subset which iscausally compatible and globally hyperbolic. Then the restriction of P to N is again Greenhyperbolic.Proof.
We construct an advanced Green’s operator for the restriction P | N of P to N . Theconstruction of the retarded Green’s operator and the ones for t P are analogous. Denote byext : C ¥ c ( N , E | N ) → C ¥ c ( M , E ) the extension-by-zero operator and by res : C ¥ ( M , E ) → C ¥ ( N , E | N ) the restriction operator. Let G + : C ¥ c ( M , E ) → C ¥ ( M , E ) be the advancedGreen’s operator of P . We claim that G N + : = res ◦ G + ◦ ext : C ¥ c ( N , E | N ) → C ¥ ( N , E | N ) is an advanced Green’s operator of P | N . Since differential operators commute with restric-tions and extensions we easily check for f ∈ C ¥ c ( N , E i | N ) : P | N ( G N + f ) = res ◦ P ◦ G + ◦ ext f = res ◦ ext f = f and G N + ( P | N f ) = res ◦ G + ◦ ext ◦ res ◦ P ◦ ext f = res ◦ G + ◦ P ◦ ext f = res ◦ ext f = f . This shows (i) and (ii) in Definition 3.1. As to (iii) we seesupp ( G N + f ) = supp ( res ◦ G + ◦ ext f ) = supp ( G + ◦ ext f ) ∩ N ⊂ J + M ( supp ( ext f )) ∩ N = J + M ( supp f ) ∩ N = J + N ( supp f ) . In the last equality we used that N is causally compatible. (cid:3) Definition 3.7.
Let G ± be advanced and retarded Green’s operators of P . Then G : = G + − G − : C ¥ c ( M , E ) → C ¥ ( M , E ) is called the causal propagator .3.3. Extensions of Green’s operators.
From (iii) and (iii)’ in Definition 3.1 we see thatthe Green’s operators of P give rise to linear maps G + : C ¥ c ( M , E ) → C ¥ spc ( M , E ) , G − : C ¥ c ( M , E ) → C ¥ s f c ( M , E ) , G : C ¥ c ( M , E ) → C ¥ sc ( M , E ) . Theorem 3.8.
There are unique linear extensionsG + : C ¥ pc ( M , E ) → C ¥ pc ( M , E ) and G − : C ¥ f c ( M , E ) → C ¥ f c ( M , E ) of G + and G − respectively, such that(i) G + P f = f for all f ∈ C ¥ pc ( M , E ) ;(ii) PG + f = f for all f ∈ C ¥ pc ( M , E ) ;(iii) supp ( G + f ) ⊂ J + ( supp f ) for all f ∈ C ¥ pc ( M , E ) ;and similarly for G − . REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 13
Proof.
We only consider G + , the proof for G − being analogous.(a) Let f ∈ C ¥ pc ( M . E ) . Given x ∈ M we define ( G + f )( x ) as follows: Since J − ( x ) ∩ supp f is compact we can choose a cutoff function c ∈ C ¥ c ( M , R ) with c ≡ J − ( x ) ∩ supp f . Now we put(2) ( G + f )( x ) : = ( G + ( c f ))( x ) . (b) The definition in (2) is independent of the choice of c . Namely, let c ′ be anothersuch cutoff function. It suffices to show x / ∈ supp ( G + (( c − c ′ ) f )) . If x ∈ supp ( G + (( c − c ′ ) f )) ⊂ J + ( supp (( c − c ′ ) f )) then there would be a causal curve from supp (( c − c ′ ) f ) to x . Hence supp (( c − c ′ ) f ) ∩ J − ( x ) would be nonempty. On the other hand,supp (( c − c ′ ) f ) ∩ J − ( x ) = supp ( c − c ′ ) ∩ supp f ∩ J − ( x ) ⊂ supp ( c − c ′ ) ∩ { c ≡ c ′ ≡ } = /0 , a contradiction.(c) The section G + f is smooth. Namely, a cutoff function c for x ∈ M also works for all x ′ ∈ J − ( x ) simply because J − ( x ′ ) ⊂ J − ( x ) . In particular, on the open set I − ( x ) we have G + f = G + ( c f ) for a fixed c . Hence G + f is smooth on I − ( x ) . Since any point in M iscontained in I − ( x ) for some x , G + f is smooth on M .(d) The operator G + is linear. The only issue here is additivity. Let f , f ∈ C ¥ pc ( M , E ) .Then supp ( f ) ∩ J − ( x ) and supp ( f ) ∩ J − ( x ) are both compact and we may choosethe cutoff function c such that c ≡ ( f ) ∩ J − ( x ) andsupp ( f ) ∩ J − ( x ) . Then c ≡ ( f + f ) ∩ J − ( x ) and we get ( G + ( f + f ))( x ) = ( G + ( c f + c f ))( x )= ( G + ( c f )( x ) + ( G + ( c f ))( x )= ( G + f )( x ) + ( G + f ))( x ) . (e) Let x ∈ M and c a cutoff function which is identically ≡ f ∩ J − ( x ) . In particular, we may choose c ≡ x . Then ( PG + f )( x ) = ( PG + ( c f ))( x ) = ( c f )( x ) = f ( x ) . This shows (ii). Moreover, ( G + P f )( x ) = ( G + ( c · P f ))( x )= ( G + P ( c f ))( x ) + ( G + ([ c , P ] f ))( x )= f ( x ) + ( G + ([ c , P ] f ))( x ) . In order to prove (i) we have to show x / ∈ supp ( G + ([ c , P ] f )) . The coefficients of thedifferential operator [ c , P ] vanish where c ≡
1, hence in particular on supp f ∩ J − ( x ) . Nowwe find supp ( G + ([ c , P ] f )) ⊂ J + ( supp ([ c , P ] f )) ⊂ J + ( supp f \ J − ( x )) ⊂ J + ( supp f ) \ { x } and therefore x / ∈ supp ( G + ([ c , P ] f )) .(f) As to (iii) we see for f ∈ C ¥ pc ( M , E ) supp ( G + f ) ⊂ [ c supp ( G + ( c f )) ⊂ [ c J + ( supp ( c f )) ⊂ J + ( supp f ) . Here the union is taken over all c ∈ C ¥ c ( M , R ) .(g) Since the causal future of a past-compact set is again past compact, (iii) shows that G + maps sections with past-compact support to sections with past-compact support. Now (i) and (ii) show that P considered as an operator C ¥ pc ( M , E ) → C ¥ pc ( M , E ) is bijective andthat G + is its inverse. In particular, G + is uniquely determined. (cid:3) Corollary 3.9.
There are no nontrivial solutions f ∈ C ¥ ( M , E ) of the differential equationP f = with past-compact or future-compact support. For any g ∈ C ¥ pc ( M , E ) or g ∈ C ¥ f c ( M , E ) there exists a unique f ∈ C ¥ ( M , E ) solving P f = g and such that supp ( f ) ⊂ J + ( supp ( g )) or supp ( f ) ⊂ J − ( supp ( g )) , respectively. (cid:3) Since the causal future of a strictly past-compact set is again strictly past compact we canrestrict G + to smooth sections with strictly past-compact support and we get Corollary 3.10.
There are unique linear extensions ˜ G + : C ¥ spc ( M , E ) → C ¥ spc ( M , E ) and ˜ G − : C ¥ s f c ( M , E ) → C ¥ s f c ( M , E ) of G + and G − respectively, such that(i) ˜ G + P f = f for all f ∈ C ¥ spc ( M , E ) ;(ii) P ˜ G + f = f for all f ∈ C ¥ spc ( M , E ) ;(iii) supp ( ˜ G + f ) ⊂ J + ( supp f ) for all f ∈ C ¥ spc ( M , E ) ;and similarly for ˜ G − . (cid:3) Uniqueness and continuity of Green’s operators.
The extension of Green’s op-erators to sections with past-compact support will now be used to show continuity anduniqueness of the Green’s operators.
Corollary 3.11.
The Green’s operators G ± : C ¥ c ( M , E ) → C ¥ ( M , E ) as well as the ex-tensions ˜ G + : C ¥ spc ( M , E ) → C ¥ spc ( M , E ) , ˜ G − : C ¥ s f c ( M , E ) → C ¥ s f c ( M , E ) , G + : C ¥ pc ( M , E ) → C ¥ pc ( M , E ) , G − : C ¥ f c ( M , E ) → C ¥ f c ( M , E ) are continuous.Proof. The operator G + : C ¥ pc ( M , E ) → C ¥ pc ( M , E ) is the inverse of P when consideredas an operator C ¥ pc ( M , E ) → C ¥ pc ( M , E ) . If A ∈ pc , then also J + ( A ) ∈ pc . Now G + mapssections with support in J + ( A ) to sections with support in J + ( J + ( A )) = J + ( A ) . Hence P yields a bijective linear operator C ¥ J + ( A ) ( M , E ) → C ¥ J + ( A ) ( M , E ) with inverse given by therestriction of G + to C ¥ J + ( A ) ( M , E ) . By the open mapping theorem for Fr´echet spaces [15,Cor. 1, p. 172], G + is continuous as a map C ¥ J + ( A ) ( M , E ) → C ¥ J + ( A ) ( M , E ) . Since we havethe continuous embeddings C ¥ A ( M , E ) ⊂ C ¥ J + ( A ) ( M , E ) and C ¥ J + ( A ) ( M , E ) ⊂ C ¥ pc ( M , E ) ,the operator G + is also continuous as a map C ¥ A ( M , E ) → C ¥ pc ( M , E ) . Since this holds forany A ∈ pc , we conclude that G + : C ¥ pc ( M , E ) → C ¥ pc ( M , E ) is continuous.A similar argument shows that ˜ G + : C ¥ spc ( M , E ) → C ¥ spc ( M , E ) is continuous. Using thecontinuous embeddings C ¥ c ( M , E ) ⊂ C ¥ spc ( M , E ) and C ¥ spc ( M , E ) ⊂ C ¥ ( M , E ) we seethat the Green’s operator G + is continuous. The same reasoning proves the claim for G − ,˜ G − , and G − . (cid:3) Corollary 3.12.
The Green’s operators of a Green-hyperbolic operator are unique.Proof.
The advanced Green’s operator G + is a restriction of the operator G + which isuniquely determined by P (as the inverse of P : C ¥ pc ( M , E ) → C ¥ pc ( M , E ) ) and similarly for G − . (cid:3) REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 15
Composition of Green-hyperbolic operators.
We now show that the compositionas well as “square roots” of Green-hyperbolic operators and again Green hyperbolic.
Corollary 3.13.
Let P : C ¥ ( M , E ) → C ¥ ( M , E ) and P : C ¥ ( M , E ) → C ¥ ( M , E ) beGreen hyperbolic. Then P ◦ P : C ¥ ( M , E ) → C ¥ ( M , E ) is Green hyperbolic.Proof. Denote the Green’s operators of P i by G i ± . We obtain an advanced Green’s operatorof P ◦ P by composing the following maps: C ¥ c ( M , E ) ֒ → C ¥ pc ( M , E ) G + −−→ C ¥ pc ( M , E ) G + −−→ C ¥ pc ( M , E ) ֒ → C ¥ ( M , E ) and similarly for the retarded Green’s operator. (cid:3) Example 3.14.
Let P = ✷ = ¶ ¶ t − ¶ ¶ x t + ¶ ¶ x be the square of the d’Alembert op-erator on 2-dimensional Minkowski space M = { ( t , x ) ∈ R } . In Example 3.4 be haveseen that the integral kernel of the Green’s operator G ✷ + is given by ( G + f )( t , x ) = − R J − ( t , x ) f ( t , x ) d x d t . Hence P has the Green’s operator ( G + f )( t , x ) = (( G ✷ + ) f )( t , x ) = Z J − ( t , x ) Z J − ( t , x ) f ( s , y ) d t d x ds dy = Z M Area ( J − ( t , x ) ∩ J + ( s , y )) f ( s , y ) ds dy . The integral kernel Area ( J − ( t , x ) ∩ J + ( s , y )) of G + is a continuous function in this case.There is a very useful partial inverse to Corollary 3.13. Corollary 3.15.
Let P : C ¥ ( M , E ) → C ¥ ( M , E ) be a differential operator such that P isGreen hyperbolic. Then P itself is Green hyperbolic.Proof. Theorem 3.8 applied to P tells us that P maps C ¥ pc ( M , E ) bijectively onto itself.Hence P itself also maps C ¥ pc ( M , E ) bijectively onto itself. Let G + denote the composition C ¥ c ( M , E ) ֒ → C ¥ pc ( M , E ) P − −−→ C ¥ pc ( M , E ) ֒ → C ¥ ( M , E ) . Then G + obviously satisfies (i) and(ii) in Definition 3.1.As to (iii), let f ∈ C ¥ c ( M , E ) . Put A : = J + ( supp f ) ∈ pc . Again by Theorem 3.8, P maps C ¥ A ( M , E ) bijectively onto itself. Hence so does P which implies that G + maps C ¥ A ( M , E ) bijectively onto itself. In particular, supp ( G + f ) ⊂ A = J + ( supp f ) .The arguments for G − and for t P are analogous. (cid:3) Example 3.16.
A differential operator P of first order is said to be of Dirac type if P isa wave operator. Since wave operators are Green hyperbolic, Corollary 3.15 tells us thatDirac-type operators are Green hyperbolic too. Examples are the classical Dirac operatoracting on sections of the spinor bundle E = SM (see [4] for details) or, more generally,on sections of a twisted spinor bundle E = SM ⊗ F where F is any “coefficient bundle”equipped with a connection.Particular examples are the Euler operator P = i ( d − d ) on E = L k L k T ∗ M and, in dimen-sion dim ( M ) =
4, the
Buchdahl operators on SM ⊗ S ⊙ k + M . See [2, Sec. 2.5] for details.If the vector bundles E , E → M carry possibly indefinite but nondegenerate fiber metrics h· , ·i , then the formally adjoint operator P ∗ is characterized by(3) Z M h g , P f i dV = Z M h P ∗ g , f i dVfor all f ∈ C ¥ ( M , E ) and g ∈ C ¥ ( M , E ) with supp f ∩ supp g compact. This definition issimilar to that of the formally dual operator in (1). In (3) the brackets h· , ·i denote fibermetrics while in (1) they denote the canonical pairing. Direct sum of Green-hyperbolic operators.
The direct sum of two Green-hyperbolic operators is again Green hyperbolic.
Lemma 3.17.
Let P : C ¥ ( M , E ) → C ¥ ( M , E ) and Q : C ¥ ( M , E ′ ) → C ¥ ( M , E ′ ) be Greenhyperbolic. Then the operator (cid:18) P Q (cid:19) : C ¥ ( M , E ⊕ E ′ ) → C ¥ ( M , E ⊕ E ′ ) is also Green hyperbolic.Proof. If G ± and G ′± are the Green’s operators for P and Q respectively, then (cid:18) G ± G ′± (cid:19) yields Green’s operators for (cid:18) P Q (cid:19) . (cid:3) Remark 3.18.
The simple construction in Lemma 3.17 shows that Green hyperbolic-ity cannot be read off the principal symbol of the operator. For instance, P could be awave operator and Q a Dirac-type operator. Then the total Green-hyperbolic operator inLemma 3.17 is of second order and the principal symbol does not see Q and thereforecannot recognize Q as a Green hyperbolic operator.For similar reasons, it is not clear how to characterize Green hyperbolicity in terms ofwell-posedness of a Cauchy problem in general.Now we get the following variation of Corollary 3.15 for operators acting on sections oftwo different bundles: Corollary 3.19.
Let P : C ¥ ( M , E ) → C ¥ ( M , E ) be a differential operator and let E andE carry nondegenerate fiber metrics. Let P ∗ : C ¥ ( M , E ) → C ¥ ( M , E ) be the formallyadjoint operator.If P ∗ P and PP ∗ are Green hyperbolic, then P and P ∗ are Green hyperbolic too.Proof. Consider the operator P : C ¥ ( M , E ⊕ E ) → C ¥ ( M , E ⊕ E ) defined by P = (cid:18) P ∗ P (cid:19) . Since P ∗ P are PP ∗ are Green hyperbolic so is P = (cid:18) P ∗ P PP ∗ (cid:19) . By Corollary 3.15, P is Green hyperbolic. Let G ± = (cid:18) G ± G ± G ± G ± (cid:19) be the Green’s operators of P . Then one easily sees that G ± are Green’s operators for P and G ± for P ∗ . (cid:3) Example 3.20. If M is even dimensional, then the spinor bundle splits into “chirality sub-bundles” SM = S + M ⊕ S − M . The twisted Dirac operators in Example 3.16 interchangethese bundles and we get operators P : C ¥ ( M , S + M ⊗ F ) → C ¥ ( M , S − M ⊗ F ) . By Corol-lary 3.19, they are Green hyperbolic too.3.7. Green’s operators of the dual operator.
Next we show that the Green’s operatorsof the dual operator are the duals of the Green’s operators. The roles of “advanced” and“retarded” get interchanged.
REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 17
Lemma 3.21.
Let P : C ¥ ( M , E ) → C ¥ ( M , E ) be Green hyperbolic. Denote the Green’soperators of P by G ± and the ones of t P by G ∗± . Then Z M h ˜ G ∗− j , f i dV = Z M h j , G + f i dV holds for all j ∈ C ¥ s f c ( M , E ∗ ) and f ∈ C ¥ pc ( M , E ) . Similarly, Z M h ˜ G ∗ + j , f i dV = Z M h j , G − f i dV holds for all j ∈ C ¥ spc ( M , E ∗ ) and f ∈ C ¥ f c ( M , E ) .Proof. By (ii) in Theorem 3.8 we have Z M h ˜ G ∗− j , f i dV = Z M h ˜ G ∗− j , P ( G + f ) i dV = Z M h t P ( ˜ G ∗− j ) , G + f i dV = Z M h j , G + f i dV . The integration by parts is justified because the intersection supp ( ˜ G ∗− j ) ∩ supp ( G + f ) ofa strictly future-compact set and a past-compact set is compact. The second assertion isanalogous. (cid:3) The causal propagator.
The following theorem contains important informationabout the solution theory of Green-hyperbolic operators. It was proved in [2, Thm. 3.5],compare also Theorem 4.3.
Theorem 3.22.
Let G be the causal propagator of the Green-hyperbolic operator P : C ¥ ( M , E ) → C ¥ ( M , E ) . Then (4) { } → C ¥ c ( M , E ) P −→ C ¥ c ( M , E ) G −→ C ¥ sc ( M , E ) P −→ C ¥ sc ( M , E ) is an exact sequence. (cid:3)
4. G
REEN - HYPERBOLIC O PERATORS A CTING ON D ISTRIBUTIONAL S ECTIONS
Green’s operators acting on distributional sections.
We extend any differentialoperator P : C ¥ ( M , E ) → C ¥ ( M , E ) as usual to distributional sections by taking thedual map of t P : C ¥ c ( M , E ∗ ) → C ¥ c ( M , E ∗ ) thus giving rise to a continuous linear map P : D ′ ( M , E ) → D ′ ( M , E ) . Lemma 4.1.
The Green’s operators G + : C ¥ pc ( M , E ) → C ¥ pc ( M , E ) and G − : C ¥ f c ( M , E ) → C ¥ f c ( M , E ) extend uniquely to continuous operators b G + : D ′ pc ( M , E ) → D ′ pc ( M , E ) and b G − : D ′ f c ( M , E ) → D ′ f c ( M , E ) , respectively. Moreover(i) b G + P f = f holds for all f ∈ D ′ pc ( M , E ) ;(ii) P b G + f = f holds for all f ∈ D ′ pc ( M , E ) ;(iii) supp ( b G + f ) ⊂ J + ( supp f ) holds for all f ∈ D ′ pc ( M , E ) ;and similarly for b G − .Proof. Recall from Lemma 2.13 and Table 1 that D ′ pc ( M , E i ) can be identified with thedual space of C ¥ s f c ( M , E ∗ i ) . Let b G + be the dual map of ˜ G ∗− : C ¥ s f c ( M , E ∗ ) → C ¥ s f c ( M , E ∗ ) where G ∗− is the retarded Green’s operator of t P . By Lemma 3.21, b G + is an extension of G + . The extension is unique because C ¥ c ( M , E ) is dense in D ′ pc ( M , E ) by Lemma 2.15. Dualizing (i) and (ii) for t P and G ∗− in Corollary 3.10 we get (i) and (ii) as asserted. Asto (iii) let f ∈ D ′ pc ( M , E ) and let j ∈ C ¥ c ( M , E ∗ ) be a test section such that J + ( supp f ) ∩ supp ( j ) = /0. Then supp f ∩ J − ( supp ( j )) = /0 and therefore ( b G + f )[ j ] = f [ G ∗− j ] = . Thus supp ( b G + f ) ⊂ J + ( supp f ) . (cid:3) Summarizing Theorem 3.8, Corollary 3.10 and Lemma 4.1 we get the following diagramof continuous extensions of the Green’s operator G + of P : C ¥ c ( M , E ) (cid:31) (cid:127) / / G + ( ( C ¥ spc ( M , E ) ˜ G + / / _(cid:127) (cid:15) (cid:15) C ¥ spc ( M , E ) _(cid:127) (cid:15) (cid:15) (cid:31) (cid:127) / / C ¥ ( M , E ) C ¥ c ( M , E ) (cid:31) (cid:127) / / _(cid:127) (cid:15) (cid:15) C ¥ pc ( M , E ) G + / / _(cid:127) (cid:15) (cid:15) C ¥ pc ( M , E ) (cid:31) (cid:127) / / _(cid:127) (cid:15) (cid:15) C ¥ ( M , E ) _(cid:127) (cid:15) (cid:15) D ′ c ( M , E ) (cid:31) (cid:127) / / D ′ pc ( M , E ) b G + / / D ′ pc ( M , E ) (cid:31) (cid:127) / / D ′ ( M , E ) D IAGRAM Extensions of the advanced Green’s operator
By (iii) in Lemma 4.1, b G + also restricts to an operator D ′ spc ( M , E ) → D ′ spc ( M , E ) .Corollary 3.9 holds also for distributional sections: Corollary 4.2.
There are no nontrivial distributional solutions f ∈ D ′ ( M , E ) of thedifferential equation P f = with past-compact or future-compact support. For anyg ∈ D ′ pc ( M , E ) or g ∈ D ′ f c ( M , E ) there exists a unique f ∈ D ′ ( M , E ) solving P f = gand such that supp ( f ) ⊂ J + ( supp ( g )) or supp ( f ) ⊂ J − ( supp ( g )) , respectively. (cid:3) The causal propagator.
Using the restriction of b G + to an operator D ′ c ( M , E ) → D ′ spc ( M , E ) ֒ → D ′ sc ( M , E ) and b G − : D ′ c ( M , E ) → D ′ sc ( M , E ) we obtain an extension ofthe causal propagator G : C ¥ c ( M , E ) → C ¥ sc ( M , E ) to distributions: b G : = b G + − b G − : D ′ c ( M , E ) → D ′ sc ( M , E ) . Now we get the analog to Theorem 3.22.
Theorem 4.3.
The sequence (5) { } → D ′ c ( M , E ) P −→ D ′ c ( M , E ) b G −→ D ′ sc ( M , E ) P −→ D ′ sc ( M , E ) is exact. (cid:3) Proof.
It is clear from (i) and (ii) in Lemma 4.1 that P b G = b GP = D ′ c ( M , S ) , hence (5)is a complex.In Corollary 4.2 we have seen that P is injective on D ′ pc ( M , E ) . Hence P is injective on D ′ c ( M , E ) and the complex is exact at D ′ c ( M , E ) .Let f ∈ D ′ c ( M , E ) with b G f =
0, i.e., b G + f = b G − f . We put g : = b G + f = b G − f ∈ D ′ ( M , S ) and we see that supp ( g ) = supp ( b G + f ) ∩ supp ( b G − f ) ⊂ J + ( supp ( f )) ∩ J − ( supp ( f )) . Since J + ( supp ( f )) ∩ J − ( supp ( f )) is compact, g ∈ D ′ c ( M , E ) . From Pg = P b G + f = f we see that f ∈ P ( D ′ c ( M , E )) . This shows exactness at D ′ c ( M , E ) .It remains to show that any f ∈ D ′ sc ( M , E ) with P f = f = b Gg for some g ∈ D ′ c ( M , E ) . Using a cutoff function decompose f as f = f + − f − where supp ( f ± ) ⊂ J ± ( K ) where K is a suitable compact subset of M . Then g : = P f + = P f − satisfies supp ( g ) ⊂ REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 19 J + ( K ) ∩ J − ( K ) . Thus g ∈ D ′ c ( M , E ) . We check that b G + g = f + . Namely, for all j ∈ C ¥ c ( M , E ∗ ) we have by the definition of b G + , b G + P f + [ j ] = P f + [ G ∗− j ] = f + [ t PG ∗− j ] = f + [ j ] . The second equality is justified because supp ( f + ) ∩ supp ( G ∗− j ) ⊂ J + ( K ) ∩ J − ( supp ( j )) is compact. Similarly, one shows b G − g = f − . Now b Gg = b G + g − b G − g = f + − f − = f whichconcludes the proof. (cid:3)
5. S
YMMETRIC H YPERBOLIC S YSTEMS
Definition and example.
Now we consider an important class of operators of firstorder on Lorentzian manifolds, the symmetric hyperbolic systems. We will show that theCauchy problem for such operators is well posed on globally hyperbolic manifolds. Wewill deduce that they are Green hyperbolic so that the results of the previous sections apply.For an approach based on the framework of hyperfunctions see [13].For a linear first-order operator P : C ¥ ( M , E ) → C ¥ ( M , F ) the principal symbol s P : T ∗ M ⊗ E → F can be characterized by P ( f u ) = f Pu + s P ( d f ) u where u ∈ C ¥ ( M , E ) and f ∈ C ¥ ( M , R ) . Definition 5.1.
Let M be a time oriented Lorentzian manifold. Let E → M be a real orcomplex vector bundle with a (possibly indefinite) nondegenerate sesquilinear fiber metric h· , ·i . A linear differential operator P : C ¥ ( M , E ) → C ¥ ( M , E ) of first order is called a symmetric hyperbolic system over M if the following holds for every x ∈ M :(i) The principal symbol s P ( x ) : E x → E x is symmetric or Hermitian with respect to h· , ·i for every x ∈ T ∗ x M ;(ii) For every future-directed timelike covector t ∈ T ∗ x M , the bilinear form h s P ( t ) · , ·i on E x is positive definite.The first condition relates the principal symbol of P to the fiber metric on E , the second re-lates it to the Lorentzian metric on M . The Lorentzian metric enters only via its conformalclass because this suffices to specify the causal types of (co)vectors. Example 5.2.
Let M = R n + and denote generic elements of M by x = ( x , x , . . . , x n ) . Weprovide M with the Minkowski metric g = − ( dx ) + ( dx ) + . . . + ( dx n ) . The coordinatefunction t = x / c : M → R is a Cauchy temporal function; here c is a positive constant tobe thought of as the speed of light.Let E be the trivial real or complex vector bundle of rank N over M and let h· , ·i denote thestandard Euclidean scalar product on the fibers of E , canonically identified with K N where K = R or K = C . Any linear differential operator P : C ¥ ( M , E ) → C ¥ ( M , E ) of first orderis of the form P = A ( x ) ¶¶ t + n (cid:229) j = A j ( x ) ¶¶ x j + B ( x ) where the coefficients A j and B are N × N -matrices depending smoothly on x . Condition (i)in Definition 5.1 means that all matrices A j ( x ) are symmetric if K = R and Hermitian if K = C . Condition (ii) with t = dt means that A ( x ) is positive definite. Thus P is asymmetric hyperbolic system in the usual PDE sense, see e.g. [1, Def. 2.11]. But (ii) saysmore than that; it means that A ( x ) dominates A ( x ) , . . . , A n ( x ) in the following sense: Thecovector t = dt + (cid:229) nj = a j dx j is timelike if and only if (cid:229) nj = a j < c − . Thus the matrix s P ( t ) = A ( x ) + n (cid:229) j = a j A j ( x ) must be positive definite whenever (cid:229) nj = a j < c − . Corollary 5.4 below will tell us thatwaves u solving the equation Pu = c . Many examples important in mathematical physics can be found in [10, App. A].Given a first order operator P which is not symmetric hyperbolic, one can still try to finda fiberwise invertible endomorphism field A ∈ C ¥ ( M , Hom ( E , E )) such that Q = A ◦ P issymmetric hyperbolic. Then the analytic results below apply to Q and hence yield anal-ogous results for P as well. Finding such an endomorphism field is an algebraic problemwhich is treated e.g. in [14].5.2. The energy estimate.
The following energy estimate will be crucial for controllingthe support of solutions to symmetric hyperbolic systems. It will establish finiteness of thespeed of propagation and uniqueness of solutions to the Cauchy problem.Let M be globally hyperbolic and let t : M → R be a Cauchy temporal function. We write S s : = t − ( s ) and S xs : = J − ( x ) ∩ S s for x ∈ M . The scalar product h· , ·i : = p b h s P ( dt ) · , ·i ispositive definite. Here the smooth positive function b : M → R is chosen for normalization,more precisely, the Lorentzian metric on M is given by g = − b dt + g t where each g s isthe induced Riemannian metric on S s . Let dA s be the volume density of S s . We denote thenorm corresponding to h· , ·i by | · | . Theorem 5.3 (Energy estimate) . Let M be globally hyperbolic, let P be a symmetric hy-perbolic system over M and let t : M → R be a Cauchy temporal function. For each x ∈ Mand each t ∈ t ( M ) there exists a constant C > such that Z S xt | u | dA t ≤ (cid:20) C Z t t Z S xs | Pu | dA s ds + Z S xt | u | dA t (cid:21) e C ( t − t ) holds for each u ∈ C ¥ ( M , E ) and for all t ≥ t .Proof. Denote the dimension of M by n +
1. Without loss of generality, we assume that M is oriented; if M is nonorientable replace the ( n + ) - and n -forms occurring below bydensities or, alternatively, work on the orientation covering of M .Let vol be the volume form of M . We define the n -form w on M by w : = n (cid:229) j = Re ( h s P ( b ∗ j ) u , u i ) b j y vol . Here b , . . . , b n denotes a local tangent frame, b ∗ , . . . , b ∗ n the dual basis, and y denotes theinsertion of a tangent vector into the first slot of a form. It is easily checked that w doesnot depend on the choice of b , . . . , b n . For the sake of brevity, we write(6) f : = Pu . We choose a metric connection (cid:209) on E . The symbol (cid:209) will also be used for the Levi-Civita connection on T M . Since the first-order operator (cid:229) nj = s P ( b ∗ j ) (cid:209) b j has the sameprincipal symbol as P , it differs from P only by a zero-order term. Thus there exists B ∈ C ¥ ( M , Hom ( E , E )) such that(7) P = n (cid:229) j = s P ( b ∗ j ) (cid:209) b j − B . To simplify the computation of the exterior differential of w , we assume that the localtangent frame is synchronous at the point under consideration, i.e., (cid:209) b j = [ b j , b k ] vanish at that point. Then we get REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 21 at that point d w ( b , . . . , b n ) = n (cid:229) k = ( − ) k ¶ b k ( w ( b , . . . , b b k , . . . , b n ))= n (cid:229) k = ( − ) k ¶ b k (cid:18) n (cid:229) j = Re ( h s P ( b ∗ j ) u , u i ) vol ( b j , b , . . . , b b k , . . . , b n ) (cid:19) = Re n (cid:229) j = ¶ b j ( h s P ( b ∗ j ) u , u i ) vol ( b , . . . , b n ) and thus d w = Re n (cid:229) j = ¶ b j ( h s P ( b ∗ j ) u , u i ) vol . We put e B : = (cid:229) nj = (cid:209) b j s P ( b ∗ j ) ∈ C ¥ ( M , Hom ( E , E )) . Using the symmetry of the principalsymbol, (6), and (7) we get n (cid:229) j = ¶ b j ( h s P ( b ∗ j ) u , u i ) = h e Bu , u i + n (cid:229) j = [ h s P ( b ∗ j ) (cid:209) b j u , u i + h s P ( b ∗ j ) u , (cid:209) b j u i ]= h e Bu , u i + h ( P + B ) u , u i + h u , ( P + B ) u i = h ( e B + B ) u , u i + h u , Bu i + h f , u i + h u , f i and hence d w = Re ( h ( e B + B ) u , u i + h f , u i ) vol . Thus we have for any compact K ⊂ M Z K d w = Z K Re ( h ( e B + B ) u , u i + h f , u i ) vol ≤ Z K ( C | u | + C | f | | u | ) vol ≤ C Z K ( | u | + | f | ) volwith constants C , C , C depending on P and K but not on u and f . We apply this to K = J − ( x ) ∩ t − ([ t , t ]) where [ t , t ] is a compact subinterval of the image of t (Fig. 2). x S t S t J − ( x ) K F IG . 2: Integration domain in the energy estimate
By the Fubini theorem,(8) Z K d w ≤ C Z t t Z S xs ( | u | + | f | ) dA s ds . The boundary ¶ J − ( x ) is a Lipschitz hypersurface (see [11, p. 187] or [12, pp. 413–415]).The Stokes’ theorem for manifolds with Lipschitz boundary [9, p. 209] yields(9) Z K d w = Z ¶ K w = Z S xt w − Z S xt w + Z Y w where Y = ( ¶ J − ( x )) ∩ t − ([ t , t ]) . Choosing b = p b dt and b , . . . , b n tangent to S s , wesee that(10) Z S xs w = Z S xs h s P ( p b dt ) u , u i dA s = Z S xs | u | dA s . The boundary ¶ J − ( x ) is ruled by the past-directed lightlike geodesics emanating from x .Thus at each differentiable point y ∈ ¶ J − ( x ) the tangent space T y ¶ J − ( x ) contains a lightlikevector but no timelike vectors. We choose a positively oriented generalized orthonormaltangent basis b , b , . . . , b n of T y M in such a way that b is future-directed timelike and b + b , b , . . . , b n is a oriented basis of T y ¶ J − ( x ) . Then w ( b + b , b , . . . , b n ) = n (cid:229) j = Re ( h s P ( b ∗ j ) u , u i ) vol ( b j , b + b , b , . . . , b n )= Re h s P ( b ∗ ) u , u i − Re h s P ( b ∗ ) u , u i = Re h s P ( b ∗ − b ∗ ) u , u i . Since h s P ( t ) · , ·i is positive definite for each future-directed timelike covector, it is, bycontinuity, still positive semidefinite for each future-directed causal covector. Now b ∗ − b ∗ is future-directed lightlike. Therefore w ( b + b , b , . . . , b n ) = h s P ( b ∗ − b ∗ ) u , u i ≥ . This implies(11) Z Y w ≥ . Combining (8), (9), (10), and (11) we find Z S xt | u | dA t − Z S xt | u | dA t ≤ C Z t t Z S xs ( | u | + | f | ) dA s ds . In other words, the function h ( s ) = R S xs | u | dA s satisfies the integral inequality h ( t ) ≤ a ( t ) + C Z t t h ( s ) ds for all t ≥ t where a ( t ) = C R t t R S xs | f | dA s ds + h ( t ) . Gr¨onwall’s lemma gives h ( t ) ≤ a ( t ) e C ( t − t ) which is the claim. (cid:3) Finite speed of propagation.
We deduce that a “wave” governed by a symmetrichyperbolic system can propagate with the speed of light at most (Fig. 3).
Corollary 5.4 (Finite propagation speed) . Let M be globally hyperbolic, let S ⊂ M be asmooth spacelike Cauchy hypersurface and let P be a symmetric hyperbolic system overM. Let u ∈ C ¥ ( M , E ) and put u : = u | S and f : = Pu. Then (12) supp ( u ) ∩ J ± ( S ) ⊂ J ± (( supp f ∩ J ± ( S )) ∪ supp u ) . S J + ( S ) supp ( u ) supp f F IG . Finite propagation speed
REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 23
In particular, supp ( u ) ⊂ J ( supp f ∪ supp ( u )) . Proof.
One can choose a Cauchy temporal function in such a way that S = S where again S s = t − ( s ) , see [7, Thm. 1.2 (B)]. Let x ∈ J + ( S ) . Assume x ∈ M \ J + (( supp f ∩ J + ( S )) ∪ supp ( u )) . This means that there is no future-directed causal curve starting in supp f ∪ supp u , entirely contained in J + ( S ) , which terminates at x . In other words, there is no past-directed causal curve starting at x , entirely contained in J + ( S ) , which terminates in supp f ∪ supp u . Hence J − ( x ) ∩ J + ( S ) does not intersect supp f ∪ supp ( u ) . By Theorem 5.3, u vanishes on J − ( x ) ∩ J + ( S ) , in particular u ( x ) =
0. This proves (12) for J + .The case x ∈ J − ( S ) can be reduced to the previous case by time reversal. For the supportof u we deducesupp u ⊂ J + (( supp f ∩ J + ( S )) ∪ supp u ) ∪ J − (( supp f ∩ J − ( S )) ∪ supp u ) ⊂ J + ( supp f ∪ supp u ) ∪ J − ( supp f ∪ supp u )= J ( supp f ∪ supp u ) . (cid:3) Uniqueness of solutions to the Cauchy problem.
As a consequence we obtainuniqueness for the Cauchy problem.
Corollary 5.5 (Uniqueness for the Cauchy problem) . Let M be globally hyperbolic, let S ⊂ M be a smooth spacelike Cauchy hypersurface and let P be a symmetric hyperbolicsystem over M. Given f ∈ C ¥ ( M , E ) and u ∈ C ¥ ( S , E ) there is at most one solutionu ∈ C ¥ ( M , E ) to the Cauchy problem (13) ( Pu = f , u | S = u . Proof.
By linearity, we only need to consider the case f = u =
0. In this case,Corollary 5.4 shows supp u ⊂ J ( /0 ) = /0, hence u = (cid:3) Existence of solutions to the Cauchy problem.
Existence of solutions is obtained bygluing together local solutions. The latter exist due to standard PDE theory. A uniquenessand existence proof for local solutions to quasilinear hyperbolic systems was also sketchedin [10, App. B]. It should be noted that global hyperbolicity of the underlying manifold isstill crucial. It is needed in order to have several compactness properties used in the proofof Theorem 5.6.
Theorem 5.6 (Existence for the Cauchy problem) . Let M be globally hyperbolic, let S ⊂ Mbe a smooth spacelike Cauchy hypersurface and let P be a symmetric hyperbolic systemover M. For any f ∈ C ¥ ( M , E ) and any u ∈ C ¥ ( S , E ) there is a solution u ∈ C ¥ ( M , E ) tothe Cauchy problem (13) .Proof. (a) We first assume that u and f have compact supports. We reduce the existencestatement to standard PDE theory. Choose a Cauchy temporal function t : M → R with S = S . Write t ( M ) = ( t − , t + ) where − ¥ ≤ t − < < t + ≤ ¥ . Put t ∗ : = sup { t ∈ [ , t + ] | there exists a C ¥ -solution u to (13) on t − ([ , t )) } . We have to show t ∗ = t + . Assume t ∗ < t + . For each t < t ∗ we have a solution of (13)on t − ([ , t )) . By uniqueness, the solutions for different t ’s coincide on their commondomain. Thus we have a solution u on t − ([ , t ∗ )) . Put K : = supp ( u ) ∪ supp f . We coverthe compact set J ( K ) ∩ S t ∗ by finitely many causally compatible, globally hyperbolic co-ordinate charts U , . . . , U N over which the vector bundle E is trivial. Choose e > U ∪ · · · ∪ U N still contains J ( K ) ∩ S t for each t ∈ [ t ∗ − e , t ∗ + e ] .Choose y j ∈ C ¥ c ( M , R ) such that supp y j ⊂ U j and(14) y + · · · + y N ≡ J ( K ) ∩ t − ([ t ∗ − e , t ∗ + e ]) . In local coordinates and with respect to a local trivialization of E , the operator P is asymmetric hyperbolic system in the classical PDE sense so that we can find local solutions u j ∈ C ¥ ( U j , E ) of the Cauchy problem ( Pu j = y j f , u j | U j ∩ S t ∗− e = y j u | U j ∩ S t ∗− e , see e.g. [1, Thm. 7.11]. By Corollary 5.5, supp u j ⊂ J ( supp y j ) . Since supp y j is a compactsubset of U j , there exists an e j > J ( supp y j ) ∩ t − ([ t ∗ − e j , t ∗ + e j ]) ⊂ U j . Thuswe can extend u j by zero to a smooth section, again denoted by u j , defined on t − ([ t ∗ − e j , t ∗ + e j ]) . For e : = min { e , e , . . . , e N } , v : = u + · · · + u N is a smooth section defined on t − ([ t ∗ − e , t ∗ + e ]) . Now v | S t ∗− e = ( y + . . . + y N ) · u | S t ∗− e = u | S t ∗− e because supp u ⊂ J ( K ) so that (14) applies. Moreover, on t − ([ t ∗ − e , t ∗ + e ]) , Pv = Pu + . . . + Pu N = ( y + . . . + y N ) · f = f because supp f ⊂ K ⊂ J ( K ) . Thus u and v solve the same Cauchy problem on t − ([ t ∗ − e , t ∗ )) and hence coincide in this region. Therefore v extends u smoothly to a solutionof (13) on t − ([ , t ∗ + e ]) which contradicts the maximality of t ∗ . This shows t ∗ = t + .Similarly, one extends the solution to t − (( t − , ]) , hence to all of M .(b) Now we drop the assumption that u and f have compact supports. Let x ∈ M . Withoutloss of generality assume x ∈ J + ( S ) . Since M is globally hyperbolic, J − ( x ) ∩ J + ( S ) iscompact. We choose a cutoff function c ∈ C ¥ c ( M , R ) with c ≡ J − ( x ) ∩ J + ( S ) . Define u ( x ) : = u x ( x ) where u x is the solution of the Cauchy problem ( Pu x = c f , u x | S = c u . The solution u x exists by part (a) of the proof.We claim that, in a neighborhood of x , the solution u x does not depend on the choice ofcutoff function c . Namely, let ˜ c be another such cutoff function and ˜ u x the correspondingsolution. Then v : = u x − ˜ u x solves the Cauchy problem ( Pv = ( c − ˜ c ) f , v | S = ( c − ˜ c ) u . Since c − ˜ c vanishes on a neighborhood of J − ( x ) ∩ J + ( S ) , v must vanish in a neighborhoodof x by Corollary 5.5. ✓ In particular, u is a smooth section which coincides with u x in a neighborhood of x . Thuswe have ( Pu )( x ) = ( Pu x )( x ) = c ( x ) f ( x ) = f ( x ) and u ( x ) = u x ( x ) = c ( x ) u ( x ) = u ( x ) if x ∈ S . Hence u solves the Cauchy problem (13). (cid:3) Stability for the Cauchy problem.
We conclude the discussion of the Cauchy prob-lem for symmetric hyperbolic systems by showing stability. This means that the solutionsdepend continuously on the data. Note that if u and f have compact supports, then thesolution u of the Cauchy problem (13) has spacially compact support by Corollary 5.5. Proposition 5.7 (Stability of the Cauchy problem) . Let P be a symmetric hyperbolic sys-tem over the globally hyperbolic manifold M. Let S ⊂ M be a smooth spacelike Cauchyhypersurface.Then the map C ¥ c ( M , E ) × C ¥ c ( S , E ) → C ¥ sc ( M , E ) mapping ( f , u ) to the solution u of theCauchy problem (13) is continuous. REEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES 25
Proof.
The map P : C ¥ ( M , E ) → C ¥ ( M , E ) × C ¥ ( S , E ) , u ( Pu , u | S ) , is linear and con-tinuous. Fix a compact subset A ⊂ M . Then C ¥ A ( M , E ) × C ¥ A ∩ S ( S , E ) is a closed subsetof C ¥ ( M , E ) × C ¥ ( S , E ) and thus V A : = P − ( C ¥ A ( M , E ) × C ¥ A ∩ S ( S , E )) is a closed subsetof C ¥ ( M , E ) . In particular, C ¥ A ( M , E ) × C ¥ A ∩ S ( S , E ) and V A are Fr´echet spaces. By Corol-lary 5.5 and Theorem 5.6, P maps V A bijectively onto C ¥ A ( M , E ) × C ¥ A ∩ S ( S , E ) . The openmapping theorem for Fr´echet spaces tells us that ( P | V A ) − : C ¥ A ( M , E ) × C ¥ A ∩ S ( S , E ) → V A is continuous. Now V A ⊂ C ¥ ( M , E ) and C ¥ J ( A ) ( M , E ) ⊂ C ¥ ( M , E ) carry the relative topolo-gies and V A ⊂ C ¥ J ( A ) ( M , E ) by Corollary 5.4. Thus the embeddings V A ֒ → C ¥ J ( A ) ( M , E ) ֒ → C ¥ sc ( M , E ) are continuous. Hence the solution operator for the Cauchy problem yields acontinuous map C ¥ A ( M , E ) × C ¥ A ∩ S ( S , E ) → C ¥ sc ( M , E ) for every compact A ⊂ M . There-fore it is continuous as a map C ¥ c ( M , E ) × C ¥ c ( S , E ) → C ¥ sc ( M , E ) . (cid:3) Remark 5.8.
Corollary 5.5, Theorem 5.6 and Proposition 5.7 are often summarized bysaying that the Cauchy problem (13) is well posed .5.7.
Green-hyperbolicity of symmetric hyperbolic systems.
Finally, we show that sym-metric hyperbolic systems over globally hyperbolic manifolds are Green hyperbolic.
Theorem 5.9.
Symmetric hyperbolic systems over globally hyperbolic manifolds areGreen hyperbolic.Proof.
Let P be a symmetric hyperbolic system over the globally hyperbolic manifold M .Let t : M → R be a Cauchy temporal function. Put I : = t ( M ) .We construct an advanced Green’s operator for P . Let f ∈ C ¥ c ( M , E ) . Choose t ∈ I suchthat K : = supp f ⊂ I + ( S t ) . We solve the Cauchy problem Pu = f with initial condition u | S t =
0. Now put G + f : = u .This definition does not depend on the particular choice of t . Namely, let t < t be twovalues in I such that K ⊂ I + ( S t i ) . Then the solution of Pu = f and u | S t = t − ([ t , t ]) because of (12). Hence it coincides with the solution of the Cauchy problem Pu = f with initial condition u | S t = G + : C ¥ c ( M , E ) → C ¥ ( M , E ) such that PG + f = f for every f ∈ C ¥ c ( M , E ) . If f = Pv for some v ∈ C ¥ c ( M , E ) , then u = v is the unique solution to theCauchy problem Pu = f with u | S t =
0. This shows G + Pv = v for every v ∈ C ¥ c ( M , E ) .By (12), supp ( G + f ) ⊂ J + ( supp f ) . Hence G + is an advanced Green’s operator. A retardedGreen’s operator is constructed similarly by choosing t ∈ I such that K ⊂ I − ( S t ) .Finally, − t P is again a symmetric hyperbolic system and therefore has Green’s operators.Thus t P has Green’s operators and P is Green hyperbolic. (cid:3) Locally covariant quantum field theory.
In [2, Thm. 3.10] we showed that Green-hyperbolic operators always give rise to bosonic locally covariant quantum field theoriesin the sense of [8]. Fermionic locally covariant quantum field theories are much harderto construct. In [2, Thm. 3.20] it was shown that a construction is possible for formallyselfadjoint Green-hyperbolic operators of first order if they are of positive type , see [2,Def. 3.12]. One easily sees that if Q is formally selfadjoint, then Q is of positive type ifand only if P = iQ is a symmetric hyperbolic system. In [2] we assumed that Q is Greenhyperbolic. Here we have shown that this is actually automatic.It is remarkable that formally selfadjoint symmetric hyperbolic systems give rise to both,bosonic and fermionic quantum field theories. This shows that there is no spin-statisticstheorem on the level of observable algebras. One has to complement the observables bysuitable states as in [16].In [2] it was shown that some but not all Dirac-type operators are of positive type. Theclassical Dirac operator acting on spinor fields is of positive type. In contrast, Buchdahloperators which describe higher spin fields are not. Therefore the theory of symmetrichyperbolic systems does not apply to Buchdahl operators; nevertheless, they are Greenhyperbolic. R EFERENCES[1] S. A
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