The wave equation on static singular space-times

Dissertatio n The w a v e equation on singular space-times angestrebte r ak ademisch er Grad Doktor der Naturwissenschaften (Dr. rer. nat.) V erfasser: Dipl.-Ing. Eb erhard Ma yerhofer Matrik el-Nummer: 9540617 Dissertationsgebiet: Mathematik Betreuer: Prof. Dr. Mic hael Kunzinger Wien, am 1. August 2006 ii iii Abstract. The ﬁrst p art of my thesis lays the found ations to g eneralized Loren tz geometry . The basic algebraic structure of ﬁnite-dimensional mo dules ov er the ring of generalized num- b ers is in vestigated. This includes a new c haracterization of in vertibilit y in the ring of generalized num b ers as wel l as a characteriza tion of free elements inside the n -dimensional mod u le e R n . The index of sy m met ric bilinear forms is introduced; this new concept en - ables a (generalized) p oint wise characteri zation of generalized pseudo Riemannian metrics on smo oth manifolds as introd u ced by M. Kunzinger and R. Steinbauer. It is shown that free submo dules have direct summan d s, how ever e R n turns out n ot to b e semisimple. Applications of these new concepts are a generalized notion of causalit y , the generali zed inv erse Cauch y Sc hw arz inequalit y for time-lik e or n ull ve ctors, c onstructions of pseudo Riemannian metrics as well as generalized energy tensors. The motiv ation for this part of my thesis evolv ed from th e main topic, the wa ve eq uation on singular space-times. The second and main part of m y thesis is devoted to establishing a lo cal existence and uniqueness t h eorem for the wa ve equation on singular space-times. The singular Lorentz metric sub ject to our discussio n is mo d eled within the special algebra on manif olds in the sense of J . F. Colom beau. Inspired b y an approac h to generali zed hyperb olicity of conical-space t imes du e to J. Vic kers and J. Wilson, we succeed in establishing certain energy estima tes, which by a further elaborated equiv alence of energy integrals and S ob olev norms allo w us to prov e existence and uniqueness of lo cal generalized solutions of the w a ve equation with resp ect to a wide class of generalized metrics. The third part of my thesis treats t hree diﬀerent p oint v alue resp. un iqueness ques- tions in al gebras of generalized functions. The ﬁrst one, posed by Mic hael Kunzinger, reads as follo ws: Is the theorem by A lb everio et al., that elements of the so-called p-adic Colom b eau Egorov algebra are determined un iquely on standard p oints, a p-adic scenario? W e answ er this problem by means of a counterexample which shows th at the statemen t in fact does not hold. W e further show that elemen ts of an Egorov algebra of generalized functions on a locally compact ultrametric space allo w a p oint-v alue c haracterization if and only if the metric ind uces th e discrete top ology . Se c ond ly , we pro ve that the ring of generalize d (rea l or complex) num b ers endow ed with th e sharp norm does not admit nested sequences of closed balls to hav e an empty intersection. As an application w e out- line a possible v ersion of the Hahn-Banach Theorem a s well as the ultrametric Ba nach ﬁxed p oin t th eorem. Fi nal l y , we establish th at scaling in va rian t generalized functions on the real line are constant and w e p ro ve sev eral n ew characteri zations of lo cally constant generalized fun ctions. Preface The present thesis r epresents m y research w ork 2004 - 2006 in the ﬁeld o f gener- alized functions carried out under the superv is ion of Professor Mic hael Kunzinger at the F aculty of Mathematics, Universit y of Vienna. All se c tio ns in this b o o k ha ve bee n the basis for scientiﬁc papers. F o r references concerning publication of this material I refer to the arxiv , where all of my submitted pa p er s can b e found, along with up dated information concer ning their publication status. Vienna, F ebruary 200 8 Eb erhard Ma yerhofer v Con ten ts Preface v Chapter 1. Int ro duction 3 1.1. Algebras o f g e neralized functions on manifolds and applica tions in general rela tivity 4 1.2. Uniqueness issues in algebr a s of generalized functions 8 Chapter 2. Algebraic fo unda tions of Colombeau Lorentz g eometry 11 2.1. Preliminaries 12 2.2. Causality and the in verse Cauch y-Sch warz inequa lit y 15 2.3. F urther algebraic prop erties of ﬁnite dimensiona l mo dules o ver the ring of generalized num b ers 24 2.4. Energy tensors and a do minant energy co ndition 25 2.5. Generalized p oint v alue characterizations of genera lized pseudo- Riemannian metrics and of causality of genera lized vector ﬁelds 28 2.6. Appendix. Inv ertibility and strict p ositivity in gener alized function algebras revisited 30 Chapter 3. The wav e equation on singular s pace-times 33 3.1. Preliminaries 33 3.2. Description of the metho d 42 3.3. The assumptions 43 3.4. Equiv alence of energy integrals a nd Sob ole v norms (Part A) 51 3.5. Bounds on initial energies via bounds on initial data (Part B) 56 3.6. Energy inequalities (P art C) 58 3.7. Bounds on energies via b ounds on initial e ne r gies (Part D) 65 3.8. Estimates via a Sob olev e mbedding theorem (Part E) 66 3.9. Existence and uniqueness (Part F) 68 3.10. Dependence on the representativ e of the metric (Part G) 70 3.11. Possible gener alizations 71 Chapter 4. P o int v a lues and uniqueness q uestions in algebras of g eneralized functions 73 4.1. P oint v alue character izations o f ultra metr ic Ego rov algebras 73 4.2. Spherical completeness of the r ing of g e neralized num ber s 77 4.3. Scaling inv a riance in algebras of generalized functions 86 Bibliogra phy 95 1 CHAPTER 1 In tro duction Diﬀerent ial algebras of generalized functions in the sense of J. F. C o lombeau provide a rigor ous setting for tre a ting numerous problems for which a genera l con- cept of m ultiplication of distributions is needed. P opular examples o f such include partial diﬀerential equations w ith singular (in the sense of no n-smo oth, sa y distr i- butional) data or coeﬃcients: A sensible theory m ust admit singular solutions of the latter, therefore it is necessary to in tr o duce a pr o duct of singula r ob jects (in o ur example a singular coe ﬃcient times a s ingular solution); how ever, it is advisable to do this in a co nsistent wa y , meaning that on reasonable function subspa ces of the distr ibutions, the usual p oint-wise pro duct coincides with suc h a pro duct o f singular ob jects, and is ass o ciative and co mmu tative. Many co un terexamples support that on D ′ such a pro duct with v a lues in D ′ cannot exist ( cf. [ 18 ], c hapter 1 ). Let us consider the fo llowing: as sume we w ere given an a s so ciative pro duct ◦ on D ′ and let vp(1 /x ) deno te the pr incipal v alue of 1 /x , then w e w o uld hav e δ = δ ◦ ( x ◦ vp(1 /x )) = ( δ ◦ x ) ◦ vp(1 /x ) = 0 , which is imp ossible, s inc e δ 6 = 0. Apart from certain ”ir regular (intrinsic or extr in- sic) op eratio ns” (cf. [ 36 ]), there are basically t wo wa ys out o f this dilemma: (i) W e co uld res tr ict o urselves to strict subspaces of D ′ which hav e a natura l algebraic structure, for ins ta nce Sobo lev spaces H s ( R n ) for s > n/ 2 , L ∞ loc , C k etc., and (ii) we could try to embed D ′ int o a larg er spa ce G which ca n b e endow e d with the structure of a diﬀerent ial alge br a. Since we wan t to m ultiply distributions unr estrictedly , we shall settle for (ii). First we formulate the des ired pro p erties o f a diﬀer e ntial alg e bra ( G , ◦ , +) con- taining the distributions. Let Ω ⊂ R s op en. W e wish to cons tr uct an asso ciative, commutativ e alge br a ( G , + , ◦ ) such that: (i) There exists a linea r embedding ι : D ′ ֒ → G such that ι (1) is the unit in G . (ii) There exist deriv a tion oper a tors D i : G → G (1 ≤ i ≤ s ), which a re linear and s a tisfy the Leibniz-rule. (iii) D i | D ′ = ∂ ∂ x i (1 ≤ i ≤ s ), that is the deriv a tion op erator s restricted to D ′ are the usual partial deriv ations. (iv) ◦ | C ∞ (Ω) ×C ∞ (Ω) is the point-wise pro duct o f functions. Item (iv) co rresp onds to the ab ove requirement that the new pr o duct should co in- cide with the usual p oint-wise pro duct on a ”rea s onable” subspace of D ′ . Sc hw artz’s famous imp oss ibilit y result ([ 44 ]) states that such an alg ebra, do es not ex ist, if the requirement (iv) is repla ced b y the resp ective r equirement for contin uous functions. 3 4 1. INTR ODUCTION Nevertheless J. F. Colombea u succes s fully constructed diﬀerential a lgebras ( G , + , ◦ ) satisfying (i)–(iv) ([ 9, 10 ]). Meanwhile there are a num b er of such algebr as of gen- eralized functions. F or a general construction scheme, cf. [ 18 ]. In the follo wing subsection we explain how the so-called sp e cial ve rsion on op en sets of R n is con- structed. W e then may in tro duce the s pec ia l alg e bra on manifolds and we shall discuss its relev ance for a pplications in general relativity . The chapter will end with an in tro ductio n to point-v alue concepts in algebras of g eneralized functions. Colombeau’s sp ecial algebra. L et Ω be an op en subset of R d . The s o-called sp e cial Algebr a 1 due to J. F. C o lombeau is giv en by the quotien t G (Ω) := E M (Ω) / N (Ω) , where the (r ing of ) modera te functions E M (Ω) resp. the ring of negligible elements (being an ideal in E M (Ω)) are giv en by E M (Ω) := { ( u ε ) ε ∈ C ∞ (Ω) (0 , 1] |∀ K ⊂⊂ Ω ∀ α ∃ N sup x ∈ K | ∂ α u ε ( x ) | = O ( ε − N ) } N (Ω) := { ( u ε ) ε ∈ C ∞ (Ω) (0 , 1] |∀ K ⊂ ⊂ Ω ∀ α ∀ m sup x ∈ K | ∂ α u ε ( x ) | = O ( ε m ) } . The alg e br aic op erations (+ , ◦ ) as well as (par tial) diﬀerentiation, comp osition of functions e tc. are meant to b e p erformed co mpo nent -wise o n the lev el of represen- tatives; the tra nsfer to the quotient G (Ω) is then well deﬁned (cf. the comprehensive presentation in the ﬁrst chapter of [ 18 ]). Once a Sch wartz molliﬁer ρ on R d with all momen ts v anishing has b een c hos en, the spa ce of compactly suppor ted d istri- butions may b e embedded into G (Ω) via conv olution; an embedding of all of D ′ (Ω) int o our algebra is achieved via a partition of unity using sheaf theoretic arguments, therefore b eing not canonica l. 1.1. Alg ebras of generalized functions on manifolds and applications in general relativit y The aim o f this section is to review the basics of the sp ecial alg ebra on manifolds X as w ell as the deﬁnitions of g eneralized sections of vector bundles with base space X and we recall the deﬁnition of generalized pseudo-Riema nnian metrics. At the end of the section w e motiv ate the use of diﬀeren tia l algebras for applications in r elativity , in pa rticular f or the wav e equation on singular space-times which is treated in the present bo o k. 1.1.1. The sp ecial algebra on m anifolds. Similarly as in section 1 o ne may deﬁne alg ebras of generalized functions on ma nifo lds . W e star t ﬁrst by in- tro ducing the sp ecial alg ebra on manifolds in a co ordinate independent way as in [ 28 ]. How ever, for t wo r easons we shall la ter tr anslate the deﬁnitions into resp ective deﬁnitions in terms of co o rdinate expressions: F or for the sake of clarity and sim- plicit y , but also for the following purp ose: In chapter 3 we s hall p erform estimates in a co o r dinate patch in or der to derive a (lo cal) existence result for the Cauc h y problem of the wa ve equa tion in a g eneralized setting. 1 In the literature the special algebra is often denoted b y G s (with the aim to di s tinguish it from other Colomebau algebras), ho wev er, since we only work in the sp ecial algebra we shall omit the i ndex s throughout. 1.1. GENERALIZED FUNCTIONS AND APPLICA TIONS TO RE LA TIVITY 5 The material present ed here stems from t he original sources [ 28, 30 ]. F or a comprehensive pres e n tation we refer to the–meanwhile standar d re fer ence on gen- eralized function a lgebras – [ 18 ]. Moreover, for further w o rks in geo metr y based on Colombea u’s ideas we refer to ([ 20, 25, 27, 29, 30, 32, 33 ]). F or what follows in this section, X shall deno te a paracompa ct, s mo oth Haus- dorﬀ ma nifo ld o f dimension n and by P ( X ) w e de no te the space of linear diﬀerential op erator s on X . The spe c ial alg ebra of generalized functions on X is constructed as the quotient G ( X ) := E M ( X ) / N ( X ), where the ring of moder ate (resp. negligible) functions is giv en by E M ( X ) := { ( u ε ) ε ∈ ( C ∞ ( X )) I | ∀ K ⊂⊂ X ∀ P ∈ P ( X ) ∃ N ∈ N : sup x ∈ K | P u ε | = O ( ε − N ) ( ε → 0) (1.1) resp. N ( X ) := { ( u ε ) ε ∈ ( C ∞ ( X )) I | ∀ K ⊂⊂ X ∀ P ∈ P ( X ) ∀ m ∈ N : sup x ∈ K | P u ε | = O ( ε m ) ( ε → 0) . (1.2) The C ∞ sections of a v ector bundle ( E , X , π ) with base space X we denote by ( E , X , π ). Moreov er , let P ( X , E ) b e the space of linear par tial diﬀerential op era tors acting on Γ( X, E ). The G ( X ) mo dule of genera lized sections Γ G ( X, E ) of a vector bundle ( E , X , π ) on X is deﬁned simila rly as (the algebra of ge ne r alized functions on X ) ab ov e, in that we use asymptotic estimates with r e sp e ct to the norm induced by some a rbitrary Riemannian metric on the resp ective ﬁb ers , that is, w e deﬁne the quotient Γ G ( X, E ) := Γ E M ( X, E ) / Γ N ( X, E ) , where the ring (resp. ideal) of modera te (resp. ne g ligible) nets of sections is given by Γ E M ( X, E ) := { ( u ε ) ε ∈ (Γ( X , E )) I | ∀ K ⊂⊂ X ∀ P ∈ P ( X, E ) ∃ N ∈ N : sup x ∈ K k P u ε k = O ( ε N ) ( ε → 0) (1.3) resp. Γ N ( X, E ) := { ( u ε ) ε ∈ (Γ( X, E )) I | ∀ K ⊂⊂ X ∀ P ∈ P ( X, E ) ∀ m ∈ N : sup x ∈ K k P u ε k = O ( ε m ) ( ε → 0) . (1.4) In this b o ok we shall deal with gene r alized sections o f the tensor bundle T r s ( X ) ov er X , this we denote by G r s ( X ) := Γ G ( X, T r s ( X )) . Elements o f the latter we call gener alize d t ensors of typ e ( r , s ). W e end this section by tra nslating the globa l description of genera liz ed v e ctor bundles in ter ms of co- ordinate expressions. F ollowing the notation of [ 30 ], we denote by ( V , Ψ ) a vector bundle chart over a chart ( V , ψ ) of the bas e X . With R n ′ , the t ypica l ﬁbre, we can write: Ψ : π − 1 ( V ) → ψ ( V ) × R n ′ , z 7→ ( ψ ( p ) , ψ 1 ( z ) , . . . , ψ n ′ ( z )) . Let no w s ∈ Γ G ( X, E ). Then the lo cal expressions of s , s i = Ψ i ◦ s ◦ ψ − 1 lie in G ( ψ ( V )). 6 1. INTR ODUCTION An equiv alen t ” lo cal deﬁnition” of generaliz ed vector bundles can be achiev ed by deﬁning moder ate nets ( s ε ) ε of smoo th sections s ε to be such for which the lo cal expressions s i ε = Ψ i ◦ s ε ◦ ψ − 1 are mo dera te, that is ( s i ε ) ε ∈ E M ( ψ ( V )). The notion negligible is deﬁned completely similar. The pro o f of this fact can b e a chiev ed by using Peetre’s theorem (cf. [ 18 ], p. 289 ). 1.1.2. Generalized pseudo-Riemanni an ge o metry . W e b egin with recall- ing the following characterizatio n of non-degenerateness o f s ymmetric (generalized) tensor ﬁelds of type (0,2 ) on X ([ 31 ], Theorem 3. 1). F or a characterization of inv ertibility o f genera lized functions w e refer to Pr op osition 2. 1 of [ 31 ] and for a further characterization w e refer to the appendix of c hapter 2 (namely The o rem 2.46). Theorem 1. 1. L et g ∈ G 0 2 ( X ) . The fol lowing ar e e quivalent: (i) F or e ach chart ( V α , ψ α ) and e ach e x ∈ ( ψ α ( V α )) ∼ c the map g α ( e x ) : e R n × e R n → e R is symmetric and non-de gener ate. (ii) g : G 0 1 ( X ) × G 0 1 ( X ) → G ( X ) is symmetric and det( g ) is invertible in G ( X ) . (iii) det g is invertible in G ( X ) and for e ach r elatively c omp act op en set V ⊂ X ther e exists a r epr esentative ( g ε ) ε of g and ε 0 > 0 such that g ε | V is a smo oth pseudo-Ri emannian metric for al l ε < ε 0 . F urthermo re, the index of g ∈ G 0 2 ( X ) is introduced in the following well deﬁned wa y (cf. Deﬁnition 3. 2 and Prop osition 3. 3 in [ 31 ]): Deﬁnition 1.2. Let g ∈ G 0 2 ( X ) satisfy one (hence all) of the equiv alent conditions in Theorem 1.1. If there e xists some j ∈ N with the pr op erty that for each r elatively compact o pen set V ⊂ X there exists a r epresentativ e ( g ε ) ε of g as in Theorem 1.1 (iii) suc h for each ε < ε 0 the index o f g ε is equals j we sa y g has index j . Such symmetric 2- fo rms we call generalized pseudo- Riemannian metrics on X . W e shall work in g eneralized space- times. The s e are pair s ( M , g ), where M is an orientable pa r acompact four dimensional s mo oth manifold and g is a symmetric generalized (0,2) tensor with in vertible det g (cf. Theorem 1.1) and index ν = 1. In chapter 2 we develop algebraic foundatio ns of ge ne r alized Lorentz geometry; here the emphasis lies o n consider ing Lorentz metrics from a generalized p oint of view and to develop causality notions in the generaliz e d context. In the subsequent chapter 3 w e us e the so found new concepts to deﬁne and work with space-time symmetries, namely (smo o th) time-like Killing vector ﬁelds ξ with resp ect to a generalized metric g (cf. Deﬁnition 3.15 and the subsequent elab o ration). W e e nd this section with rev iewing the notio n o f genera lized connections and curv ature ([ 30 ], section 5). A g eneralized connection ˆ D is a mapping G 1 0 ( M ) × G 1 0 ( M ) → G 1 0 ( M ) s atisfying (for the notion of g e ne r alized Lie deriv a tive, cf. [ 30 ]) (i) ˆ D ξ η is e R –linear in η , (ii) ˆ D ξ η is G ( M )–linea r in ξ and (iii) ˆ D ξ ( uη ) = u ˆ D ξ η + ξ ( u ) η for a ll u in G ( M ). In ana logy with the standard pseudo-Riemannian geometr y , the connec- tion is unique provided the f ollowing additional conditions are satisﬁed (cf. [ 30 ], Theorem 5.2 ). F or arbitr a ry ξ , η , ζ ∈ G 1 0 ( M ) we hav e: 1.1. GENERALIZED FUNCTIONS AND APPLICA TIONS TO RE LA TIVITY 7 (iv) [ ξ , η ] = ˆ D ξ η − ˆ D η ξ a nd (v) ξ g ( η , ζ ) = g ( D ξ η , ζ ) + g ( η , D ξ ζ ). In terms of co o rdinate expressio ns, the connec tion can be written do wn by means of ”generalized” Chr istoﬀel s y m bo ls: Assume we ar e given a c hart ( V α , ψ α ) on M with coordina tes x i ( i = 1 , . . . , 4). The Christoﬀel symbols ar e generalize d functions Γ k ij ∈ G ( V α ) deﬁned b y ˆ D ∂ i ∂ j = Γ k ij ∂ k , 1 ≤ i, j ≤ n. 1.1.3. Generalized function concepts in general relativity. Even though suﬃcient motiv atio n to study the Cauch y problem of the wa ve equation on a space-time whose metric is of lo wer diﬀ erentiabilit y may emerge f rom a purely mathematical in terest, our original motiv ation actually stems fro m ph ysics. The aim of this section is to answ er the fo llowing t wo q uestions: ” Wh y do w e in tend to solve the wa ve equation on a singular space- time” and, ”Why do we emplo y generalized function algebras for this matter ?”. The ﬁeld o f genera l re lativity is a non-linear theor y , in the sense that the curv ature depends non- line a rly on th e metric and its der iv atives. This res ults in several problems when one comes to consider the concept of sing ularities in space- times: (i) Firstly , from a mathematical p oint of view, an immediate pro blem whe n a singular space–time is mo deled by means of a distributional metric, is: In the c o ordinate formula f or the Christoﬀel symbols (hence in the formula for t he curv ature), products of the metric co e ﬃcie n ts and their deriv atives o ccur , and a (distributional) meaning has to be given to the latter. As outlined ab ov e, this is not alw ays p ossible in the framework of distributions, b ecause they for m a line a r theory ( cf. the dis c us sion at the b eginning of the chapter). (ii) The s econd natural obs ta cle is the diﬃculty of distinguishing ”strong” singularities fro m ”weak” singularities. Sing ularities were originally de- ﬁned as endp oints of inco mplete geo desics , which could not be extended such that the diﬀer e n tiability o f the resulting spa ce-time remained C 2 − (cf. Hawking and Ellis, [ 21 ]). The class of singularities deﬁned in this manner unfortunately includes b oth genuine gravitational singula rities such a s Sch warzschild a nd ”weak er” singularities as in conical space- times, impulsive gra vitational wav es and shell crossing singula rities. A recent idea put for ward by C. J. S. Clarke in ([ 7 ]) supp orts a new con- cept of ”w eak” singularities: A singula rity in a space-time should only be co nsidered es sential if it disrupts the evolution of linear test ﬁelds. According to this idea, Clarke ca lls a space-times g eneralized hyperb olic, if the Cauc h y problem for the scalar wav e equation is well p osed, and then shows that spa ce-times with loc a lly integrable curv ature ar e in this class. Vic kers and Wilson are the ﬁrst authors who apply Clar ke’s c o ncepts by show- ing that conical space-times ar e genera lized hyperb olic (cf. [ 49 ]; in the context of generalized function algebras this is called G –generalized hyperb olic). T o fur- ther overcome obstacle (i) in a mathematically rigorous w ay , they reformulate the Cauch y pro blem in the full Colombeau alg ebra. Fina lly , they show that the r e - sulting gener alized solution is asso cia ted with a dis tr ibutional so lution (this ca n b e 8 1. INTR ODUCTION done b y considering weak limits with r esp ect to the smo othing parameter ε , cf. the deﬁnitions given in sec tion 4.3.1 .4). W e shall fo llow Vick er s’ and Wilson’s a pproach and try to genera lize their result to a wide range o f g eneralized space-times in chapter 3. How ever, it should b e noted that contrary to [ 49 ], we work in the sp e cial algebr a exclusively . Moreover, the techn ique we are using (based on certain energy in teg rals a nd Sob o le v norms) lies somewhere b etw een Hawking and Ellis’ metho d ([ 21 ]) and Vick er s ’ and Wilson’s . F or mo re information on th e use of g eneralized function alg ebras in relativity we refer to the recent review [ 46 ] on this topic by R. Steinbauer and J. Vic kers as w ell as J. Vic kers’s article ([ 17 ] pp. 275–2 90) and the in tro duction to [ 49 ]. F or relativistic applicatio ns in the fr amework o f Colombeau’s theory , see [ 8, 18, 19 ]. 1.2. Uniqueness i ssues i n algebras of gene rali zed functions The la st chapter of the present w ork consis ts of three diﬀerent problems which we hav e summar ized under the title ”p oint v alues and uniqueness questions in algebras of generalized functions”. Even though the problems are quite diﬀerent, they all have to do with the basic question: ”given tw o genera lized functions f , g , how can we dec ide if f = g ?” It is clear that w e ca n reduce this to the problem of determining whether a gener alized function h v a nishes ident ically . Before we come to a p o ssible answer oﬀered by M. Kunzinger and M. Ober g uggenberg er in [ 38 ] in form of a ”uniqueness test” via ev aluation of gener alized functions on so-called compactly supp orted p oints, we motiv ate the problem from the distributional p oint of view. By deﬁnition, a distribution w ∈ D ′ is zero if the test with arbitrary test functions φ yields h w, φ i = 0 . The question, r e formulated in the context of the sp ecial alg ebra, reads, ”is the embedded o b ject ι ( w ) ∈ G iden tically zero?”. How ever, since the key idea o f embedding distributions into G is regular ization of the la tter, we shall leave aside the em bedding and a nswer this q uestion for regular iz ed nets of distributions in terms of the following characteriza tion: Theorem 1. 3. L et u ∈ D ′ ( R n ) and let ρ ∈ D ( R n ) b e a standar d mol liﬁer, tha t is, with R ρ ( x ) dx n = 1 and let ρ ε ( x ) := 1 ε n ρ ( x ε ) . The fol lowing ar e e quivalent: (i) u = 0 i n D ′ ( R n ) . (ii) F or e ach c omp actly supp orte d net ( x ε ) ε ∈ ( R n ) (0 , 1] we have ( u ∗ ρ ε )( x ε ) → 0 if ε → 0 . Proof. The implication (i) ⇒ (ii) is obvious, since for each ε > 0 and each x ∈ R n , ρ ε ∗ u ( x ) = h u ( y ) , ρ ε ( x − y ε ) i = 0. T o show the conv erse direction, a ssume u 6 = 0 but that (ii ) holds. Then there exists φ ∈ D ( R n ) suc h that h u, φ i 6 = 0. It follows that there e x ists a po sitive consta nt C 1 and an index ε 0 ∈ (0 , 1] such that for each ε < ε 0 we hav e (1.5)     Z ( u ∗ ρ ε ) φ dx n     ≥ C 1 . Therefore there exist a sequence ε k → 0 in (0 , 1], a c o mpactly supported sequence x ε k ∈ R n and a positive num b er C such that for each k ≥ 1 we have (1.6) | u ∗ ρ ε k ( x ε k ) | ≥ C. 1.2. UNIQUENESS ISSUES 9 Indeed, if we a ssume the co nt rary , then for ea ch set K ⊂⊂ R n we would hav e sup x ∈ K | u ∗ ρ ε | → 0 whenev er ε → 0. Fix K such that supp φ ⊆ K . Then w e ha ve     Z ( u ∗ ρ ε ) φ dx n     ≤ v ol( K ) k φ k ∞ k u ∗ ρ ε k K, ∞ → 0 whenever ε → 0, a c ontradiction to (1 .5). Finally deﬁne ( x ε ) ε as follows: x ε := x ε k whenever ε ∈ ( ε k +1 , ε k ] ( k ≥ 1 ) and x ε := x ε 1 when ε ∈ ( x 1 , 1]. By (1.6) w e ha ve a contradiction to our as sumption. Therefore u = 0 and we are do ne.  It is further evident that, in the ab ove characterizatio n, (ii) canno t be replaced by the condition F or e ach x ∈ R n we have ( u ∗ ρ ε )( x ) → 0 whenever ε → 0 . T o see this, take a sta ndard molliﬁer ρ with suppor t supp ρ = [0 , 1]. Then for each x there exists an index ε 0 such that ρ ε ( x ) = 0 f or each ε < ε 0 . But for ε → 0 w e hav e ρ ε → δ in D ′ . W e go on now by showing how these ideas are ela b o rated in the context of the sp ecial alg ebra: 1.2.0.1. The gener alize d p oint values c onc ept. Gener a lized functions can be ev aluated at standard points. T o be more precis e, let us intro duce the ring o f generalized num b er s e R , deﬁned b y the quo tient e R := E M / N , where the ring of mo dera te num b ers E M := { ( x ε ) ε ∈ R (0 , 1] : ∃ N : | x ε | = O ( ε − N ) } . Similarly the ideal of negligible num b ers N in E M is given by N := { ( x ε ) ε ∈ R (0 , 1] : ∀ m : | x ε | = O ( ε m ) } . Let e R c denote the set of compactly supp orted elements o f e R , that is: x c lies in e R c if and only if th ere exists a compact set K ⊆ R suc h that for one (hence any) representative ( x ε ) ε there exists an index ε 0 such that for all ε < ε 0 we have x ε ∈ K . It can eas ily b e shown that ev aluation of generalized functions f on compactly supp o rted genera lized points makes perfect sense in the fo llowing wa y: let ( f ε ) ε be a r epresentativ e of f ∈ G ( R ), then e f ( x c ) := ( f ε ( x ε )) ε + N ∈ e R yields a well deﬁned generalized num ber . W e denote by e f : e R c → e R the ab ov e map induced by the ge ne r alized function f . By a standa rd p oint x we shall mea n an element of e R which admits a constant representative, i. e. x = ( α ) ε + N for a certain real n umber α . M. Kunzinger and M. O be r guggenber ger sho w in ([ 38 ]) that it does not s uﬃce to know the v alues of generalized functions at standard po ints in o rder to determine them uniquely . F urthermo re, the following analo g of Theore m 1.3 holds: Theorem 1. 4. L et f ∈ G ( R ) . The fol lowing ar e e qu ivalent: (i) f = 0 in G ( R ) , 10 1. INTR ODUCTION (ii) ∀ x c ∈ e R c : e f ( x c ) = 0 . Note that a similar statemen t holds in Egor ov algebras (cf. the ﬁnal remark in [ 38 ]). In the ﬁrst se c tion of chapter 4 we show that also in p -adic Eg orov algebras such a characterization holds and that ev alua tion at s tandard p o in ts do es not suﬃce to determine e lement s of such algebras uniquely . In section 4.2 we elab orate a topolo gical q uestion in the ring of generalized n um ber s e R e ndow ed with the so-called sharp topo logy . Finally , in the end of chapter 4, w e apply some new diﬀerential ca lc ulus on e R due to Aragona ([ 4 ]) for showing t hat the only scaling inv ar iant functions o n the r eal line are the cons ta nt s. CHAPTER 2 Algebraic found ations of Colom b eau Loren tz geometry In the course of chapter 3 w e sha ll establish a lo cal e xistence a nd uniqueness theorem for the Cauch y pr oblem of the w av e equation in a genera lized context. The consideratio ns we had to under take to achiev e this result show ed that a gener alized concept of ca usality might b e useful t o descr ibe scenarios in a no n- smo oth space- time without always having to deal merely with the standar d concepts comp onent- wise o n the lev el of repr esentativ es . How ever, also from a purely theore tical p oint of view, the n eed of a such a concept beco mes clear: the non-standard aspect in Colombeau theory , which gives r ise to a description of o b jects no t p oint-wise but on so-called generalized p oints (cf. [ 38 ] and c hapter 4). This has b een taken up in the recen t and initial w ork by M. Kunzinger and R . Steinbauer o n gener alized pseudo-Riemannian geometry ([ 31 ]), on which we ba se our considerations (cf. the assumptions on the metric in s e c tion 3.3.2), but it has not yet bee n in vestigated to a wide extent. F o r instance, in vertibilit y of generalized functions has b een character- ized (cf. [ 31 ], Prop os itio n 2. 1) and a llow ed a notable characteriza tion o f sy mmetr ic generalized non-degenera te (0 , 2) forms (cf. Theorem 1.1). But so far there has not bee n given a character iz a tion of generalize d pseudo-Riema nnian metrics h in ter ms of bilinear forms e h stemming fro m ev aluation of h at compactly supported points (on the r esp ective ma nifo ld). The main aim of this c hapter, therefore, is to describe and discuss so me ele- men tary questions of gener alized pseudo-Riemannian geometry under the a sp ect of generalized po int s. Our prog ram is as follows: In tro ducing the index o f a sy mmetr ic bilinear form on the n - dimensional mo dule e R n ov er the gener alized num b ers e R en- ables us to deﬁne the appropriate no tio n of a bilinear form of Lorentz signature . W e can ther efore pr o p ose a notion o f causality in this co n text. The general s tatement of t he in verse Ca uch y-Sch w artz inequality is then giv en. W e further s how that a dominant energy condition in the sense of Hawking and E llis for g eneralized energy tensors (suc h as also indirectly as sumed in [ 4 9 ]) is s atisﬁed. W e also answ er the algebraic question: ”Do es any submo dule in e R n hav e a dir ect summand?”: F or free submo dules, the answer is po sitive and is b asically due to a new c haracteriza tion of free elements in e R n . In general, how ever, direct summands do not exis t: e R n is not semisimple. In the end of the c ha pter w e present a new c haracteriza tion o f inv ertibility in algebras of generalize d functions. Finally , w e wan t to p oint o ut that the po sitivity issues o n the ring of generalized n umbers t reated here hav e links to pap ers by M. O b er guggenber ger et al. ([ 22, 35 ]). 11 12 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y 2.1. Preli minaries Let I := (0 , 1] ⊆ R , and let K d enote R resp. C . The ring of generalized nu m ber s o ver K is constructed in the following wa y: Giv en the r ing of mo der a te nets o f n um ber s E ( K ) := { ( x ε ) ε ∈ K I | ∃ m : | x ε | = O ( ε m ) ( ε → 0) } and, similar ly , the ideal of negligible nets in E ( K ) which are of the form N ( K ) := { ( x ε ) ε ∈ K I | ∀ m : | x ε | = O ( ε m ) ( ε → 0) } , w e may deﬁne the gener alized num b er s a s the factor ring e K := E M ( K ) / N ( K ). An elemen t α ∈ e K is called strictly p ositive if it lies in e R (this means that for an y represe n tative ( α ε ) ε = (Re( α ε )) ε + i (Im( α ε )) ε we hav e (Im( α ε )) ε ∈ N ( R ) ) and if α has a represe ntative ( α ε ) ε such that there exists m ≥ 0 such that Re( α ε ) ≥ ε m for each ε ∈ I = (0 , 1], w e shall wr ite α > 0 . Clearly any strictly p ositive num ber is invertible. β ∈ e R is calle d strictly neg ative, if − β > 0 . Note that a generalized nu m ber u is stric tly positive precisely when it is inv er tible (due to [ 31 ] Prop os ition 2. 2 this means that u is strictly non-zero) and p ositive (i. e., u has a r epresentativ e ( u ε ) ε which is greater or equals zero for each ε ∈ I ). In the app endix to this chapter a new and so mewhat sur pr ising characterization of inv ertibility and strict p ositivity in the frame of the special alg ebra construction is presented. Let A ⊂ I , then the characteristic function χ A ∈ e R is g iven b y the class o f ( χ ε ) ε , wher e χ ε := ( 1 , if ε ∈ A 0 , otherwise . Whenever e R n is involv ed, w e co ns ider it as a n e R –mo dule of dimension n ≥ 1 . Clearly the la tter c an b e identiﬁed with E M ( R n ) / N ( R n ), but we will not often use this fact subsequently . Finally , we denote by e R n 2 := M n ( e R ) the ring o f n × n matrices ov er e R . A matrix A is ca lled or thogonal, if U U t = I in e R n 2 and det U = 1 in e R . Clearly , ther e are t wo diﬀerent wa ys to in tro duce e R n 2 : Remark 2 .1. Denote b y E M ( M n ( R )) the ring of mo derate nets of n × n matri- ces ov er R , a subring of M n ( R ) I . Similarly let N ( M n ( R )) denote the ideal o f negligible nets of re al n × n matrices. There is a ring isomorphis m ϕ : e R n 2 → E M ( M n ( R )) / N ( M n ( R )). F or the conv e nie nce o f the reader we rep ea t Lemma 2. 6 from [ 31 ]: Lemma 2. 2. L et A ∈ e R n 2 . The fol lowing ar e e quivalent: (i) A is non-de gener ate, that is, ξ ∈ e R n , ξ t Aη = 0 f or e ach η ∈ e R n implies ξ = 0 . (ii) A : e R n → e R n is inje ctive. (iii) A : e R n → e R n is bije ctive. (iv) det A is invertible in e R . Note that the equiv alence of (i)–(iii) and (iv) res ults fr om th e fact t hat in e R any nonz e ro non-inv ertible element is a zero-div is or. Since we deal with sy mmetric matrices throug hout, we start by giving a ba sic characterization of symmetry o f generalized matric es: Lemma 2. 3. L et A ∈ e R n 2 . The fol lowing ar e e quivalent: (i) A is symmetric, that is A = A t in e R n 2 . 2.1. PRELIMINARIES 13 (ii) Ther e ex ist s a symmetric re pr esentative ( A ε ) ε := (( a ε ij ) ij ) ε of A . Proof. Since (ii) ⇒ (i) is clear, we o nly need to show (i) ⇒ (ii). Let ((¯ a ε ij ) ij ) ε a repr e sentativ e of A . Symmetrizing yields the desired representative ( a ε ij ) ε := (¯ a ε ij ) ε + (¯ a ε j i ) ε 2 of A . This follows from the fact that for each pair ( i, j ) ∈ { 1 , . . . , n } 2 of indices one has (¯ a ε ij ) ε − (¯ a ε j i ) ε ∈ N ( R ) due to the symmetry of A .  Denote by k k F the F r ob enius nor m on M n ( C ). In or der to prepare a notion of eigenv alues for symmetric matrices, we rep eat a numeric result given in [ 47 ] (Theorem 5. 2): Theorem 2. 4 . L et A ∈ M n ( C ) b e a Hermitian matrix with eigenvalues λ 1 ≥ · · · ≥ λ n . Denote by e A a non-H ermitian p ertu rb ation of A , i. e., E = e A − A is not Hermitian. We further c al l the eigenval ues of e A (which might b e c omplex) µ k + iν k (1 ≤ k ≤ n ) wher e µ 1 ≥ · · · ≥ µ n . In this notation, we have v u u t n X k =1 | ( µ k + iν k ) − λ k | 2 ≤ √ 2 k E k F . Deﬁnition 2 . 5. Let A ∈ e R n 2 be a symmetric matr ix and let ( A ε ) ε be a n arbitr ary representative of A . Let for an y ε ∈ I , θ k,ε := µ k,ε + iν k,ε (1 ≤ k ≤ n ) b e the eigenv a lues of A ε ordered by the size of the real parts, i. e., µ 1 ,ε ≥ · · · ≥ µ n,ε . The generalized eig env alues θ k ∈ e C (1 ≤ k ≤ n ) of A are deﬁned as the classes ( θ k,ε ) ε + N ( C ). Lemma 2.6. L et A ∈ e R n 2 b e a symmetric matrix. Then t he ei genvalues λ k (1 ≤ k ≤ n ) of A as intr o duc e d in Deﬁnition 2.5 ar e wel l deﬁne d elements of e R . F ur- thermor e, ther e exists an ortho gonal U ∈ e R n 2 such that (2.1) U AU t = diag ( λ 1 , . . . , λ n ) . We c al l λ i (1 ≤ i ≤ n ) the e igenvalues of A . A is non-de gener ate if and only if al l gener alize d eigenvalues ar e invertible. Before we prove the lemma, we note that throughout the chapter w e shall omit the term ”genera lized” (eigen v alues) and w e shall call the generalized n um ber s constructed in the ab ov e wa y simply ”eig env alues ” (of a g eneralized s ymmetric matrix). Proof. Due to Lemma 2.3 w e may cho ose a symmetric repr esentativ e ( A ε ) ε = (( a ε ij ) ij ) ε ∈ E M ( M n ( R )) of A . F or an y ε , denote b y λ 1 ,ε ≥ · · · ≥ λ n,ε the resp. (real) eigenv a lue s of ( a ε ij ) ij ordered b y size. F or any i ∈ { 1 , . . . , n } , deﬁne λ i := ( λ i,ε ) ε + N ( R ) ∈ e R . F or the w ell- deﬁnedness of the eigen v alues of A , we only need to show that for any other (not necess a rily symmetric ) r epresentativ e of A , the r esp. net of eigenv alues lies in the same clas s of E M ( C ); note that the us e of complex nu m ber s is indisp ensable here. Let ( e A ε ) ε = (( e a ε ij ) ij ) ε be a nother representativ e of A . Denote by µ k,ε + i ν k + ε the eig env a lue s of e A ε for a n y ε ∈ I s uch that the real 14 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y parts a re o rdered b y size, i. e., µ 1 ,ε ≥ · · · ≥ µ n,ε . Denote by ( E ε ) ε := ( e A ε ) ε − ( A ε ) ε . Due to The o rem 2.4 w e hav e for eac h ε ∈ I : (2.2) v u u t n X k =1 | ( µ k,ε + iν k,ε ) − λ k,ε | 2 ≤ √ 2 k E ε k F . Since ( E ε ) ε ∈ N ( M n ( R )), (2 .2) implies for any k ∈ { 1 , . . . , n } a nd any m , | ( µ k,ε + iν k,ε ) − λ k,ε | = O ( ε m ) ( ε → 0) which means that the resp. eigen v alues of ( A ε ) ε and of ( e A ε ) ε in the abov e o rder belo ng to the s a me class in E M ( C ). In particular they yield the same elements of e R . The preceding argument and L e mma 2.3 s how that without loss of generality we may construct the eigenv alues of A b y means of a s ymmetric representative ( A ε ) ε = (( a ε ij ) ij ) ε ∈ E M ( M n ( R )). F or such a choice we hav e for any ε a n or thogonal matrix U ε such that U ε A ε U t ε = diag( λ 1 ,ε , . . . , λ n,ε ) , λ 1 ,ε ≥ · · · ≥ λ n,ε . Declaring U as the class of ( U ε ) ε ∈ E M ( M n ( R )) yields the pro of of the second claim, since o rthogona lit y for any U ε implies orthogonality of U in M n ( e R ). Finally , decomp osition (2.1) gives, by applying the multip lication theorem for determinan ts and the or thogonality of U , det A = Q n i =1 λ i . This shows in conjunction with Lemma 2 .2 that inv ertibility of all eigenv alues is a s uﬃcie nt a nd necessary condition for the non-degenerateness of A and w e are done.  Remark 2.7. A remark on the notion eigenv alue of a genera lized symmetric matr ix A ∈ e R n 2 is in order: Since for any eig env alue λ of A we hav e det( A − λ I ) = det( U ( A − λ I ) U t ) = det(( U AU t ) − λ I ) = 0, Lemma 2.2 implies that A − λ I : e R n → e R n is not injectiv e. Howev er , aga in by the same lemma, det( A − λ I ) = 0 is no t necessary for A − λ I to b e not injectiv e, and a θ ∈ e R for which A − θ I is not injectiv e need not be an eigenv alue of A . More explicitly , w e give tw o exa mples o f possible scenarios here: (i) Let ∀ i ∈ { 1 , . . . , n } : λ i 6 = 0 a nd for some i let λ i be a zero divisor . Then bes ides A − λ i ( i = 1 , . . . , n ), also A : e R n → e R n fails to be injectiv e . (ii) ”Mixing ” representativ es of λ i , λ j ( i 6 = j ) migh t g ive rise to gener - alized n um ber s θ ∈ e R , θ 6 = λ j ∀ j ∈ { 1 , . . . , n } for which A − θ I is not injective as w ell. Consider for the sake of simplicity the matr ix D := diag(1 , − 1) ∈ M 2 ( R ). A rota tion U ϕ :=  cos( ϕ ) sin( ϕ ) − sin( ϕ ) cos( ϕ )  yields b y ma trix multiplication U ϕ D U t ϕ =  cos(2 ϕ ) − sin(2 ϕ ) − sin(2 ϕ ) − co s(2 ϕ )  . The choice o f ϕ = π / 2 therefo re switches the order of the entries of D , i. e., U π / 2 D U t π / 2 = diag( − 1 , 1 ). Deﬁne U, λ a s the classes o f ( U ε ) ε , ( λ ε ) ε deﬁned by U ε := ( I : ε ∈ I ∩ Q U π / 2 : else , 2.2. CA US ALITY AND THE INVERSE CAUC HY-SCHW ARZ INEQUALITY 15 λ ε := ( 1 : ε ∈ I ∩ Q − 1 else , further deﬁne µ ∈ e R by µ + λ = 0. Then we hav e for A := [( D ) ε ]: U D U t = diag( λ, µ ) . Therefore as shown a bove, D − λ I , D − µ I ar e not injectiv e considere d as maps e R n → e R n . But neither λ , nor µ are eigenv alues of D . Deﬁnition 2.8. Let A ∈ e R n 2 . W e deno te by ν + ( A ) (resp. ν − ( A )) the num b er of str ictly positive (res p. str ic tly nega tive) eigenv alues, co un ting multiplicit y . F ur- thermore, if ν + ( A ) + ν − ( A ) = n , we s imply write ν ( A ) := ν − ( A ). If A is symmetr ic and ν ( A ) = 0, we call A a p ositive deﬁnite symmetric matr ix. If A is symmetric and ν + ( A ) + ν − ( A ) = n and ν ( A ) = 1, w e sa y A is a symmetric L -matrix. The f ollowing corollary shows that f or a symmetric no n-degenera te matrix in e R n 2 counting n strictly po sitive resp. nega tive eigenv alues is eq uiv alent to having a (symmetric) representative for which any ε -comp onent has the same num b er (total n ) of positive resp. nega tive rea l e igenv a lues. W e sk ip the pro of. Corollary 2.9. L et A ∈ e R n 2 b e symmetric and non-de gener ate and j ∈ { 1 , . . . , n } . The following ar e e quivalent: (i) ν + ( A ) + ν − ( A ) = n , ν ( A ) = j . (ii) F or e ach symmetric r epr esentative ( A ε ) ε of A ther e exists some ε 0 ∈ I such that for any ε < ε 0 we have for the eigenvalues λ 1 ,ε ≥ · · · ≥ λ n,ε of A ε : λ 1 ,ε , . . . , λ n − j,ε > 0 , λ n − j +1 ,ε , . . . , λ n,ε < 0 . 2.2. Causali t y and the in verse Cauc hy-Sc h warz inequalit y In a free mo dule over a c ommut ative ring R 6 = { 0 } , a n y t wo bas es hav e the same car dinality . Therefore, any free module M n of dimensio n n ≥ 1 (i. e., with a basis ha ving n element s) is isomor phic to R n considered as mo dule ov er R (whic h is free, since it ha s the canonica l basis ). As a c onsequence we may conﬁne ourselves to consider ing the mo dule e R n ov er e R and its submo dules. W e fur ther a ssume that from now on n , the dimension o f e R n , is greater than 1. It is quite natural to start with an appropriate version o f the Steinitz exchange lemma: Prop ositi o n 2.1 0 . L et B = { v 1 , . . . , v n } b e a b asis for e R n . Le t w = λ 1 v 1 + · · · + λ n v n ∈ e R n such that for some j (1 ≤ j ≤ n ) , λ j is not a zer o di visor. Then, also B ′ := { v 1 , . . . , v j − 1 , w , v j +1 , . . . , v n } is a b asis for e R n . Proof. Without lo ss o f ge ne r ality we may as sume j = 1, that is λ 1 is in vert- ible. W e we ha v e to show that B ′ := { w , v 2 , . . . , v n } is a ba sis fo r e R n . Assume we are given a vector v = P n i =1 µ i v i ∈ e R n , µ i ∈ e R . Since λ 1 is inv ertible, we may write v 1 = 1 λ 1 w − λ 2 λ 1 v 2 − · · · − λ n λ 1 v n . Thus we ﬁnd v = µ 1 λ 1 w + P n k =2 ( µ k − µ 1 λ k λ 1 ) v k , which proves that B ′ spans e R n . It remains to prove linear indep endence of B ′ : Assume that for µ, µ 2 , . . . , µ n ∈ e R w e hav e µw + µ 2 v 2 + · · · + µ n v n = 0. Inser ting w = P n i =1 λ i v i yields µλ 1 v 1 + ( µλ 2 + µ 2 ) v 2 + · · · + ( µλ n + µ n ) v n = 0 and since B is a bas is, it follows that µλ 1 = µλ 2 + µ 2 = · · · = µλ n µ n = 0. Now, since λ 1 is 16 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y inv ertible, it follows that µ = 0. Therefor e µ 2 = · · · = µ n = 0 which proves that w, v 1 , . . . , v n are linea rly indep endent, and B ′ is a ba sis.  Deﬁnition 2.11. Let b : e R n × e R n → e R be a symmetric bilinear form on e R n . Let j ∈ N 0 . If for some basis B := { e 1 , . . . , e n } of e R n we ha ve ν (( b ( e i , e j )) ij ) = j we call j the index of b . If j = 0 we say that b is p o s itive deﬁnite a nd if j = 1 w e call b a s y mmetric bilinear form of Lo rentz sig na ture. Note that a s in the classical setting, there is no no tion of ’eig env alue s ’ of a symmetric bilinea r form, s ince a c hange o f co o rdinates that is not induced b y a n orthogo nal matrix need not co nserve the eig env alue s of the original co eﬃcient ma- trix. W e a r e obliged to show that the no tion ab ov e is w ell deﬁned. The main argument is Sylv ester’s inertia law (cf. [ 1 3 ], pp. 306): Prop ositi o n 2.1 2. The index of a biline ar form b on f R n as intr o duc e d in Deﬁnition 2.11 is wel l deﬁne d. Proof. Let B , B ′ be ba ses of f R n and let A b e a matr ix descr ibing a linea r map which maps B onto B ′ (this ma p is uniquely determined in the sense that it only depends on the o rder of the basis v ectors of th e resp. ba ses). Let B b e the co eﬃcient matrix of the given bilinear form b and let further k := ν ( B ). The change of bases results in a ’generalized’ equiv alence transformation of the form B 7→ T := A t B A, T being the co eﬃcien t matrix of h with resp ect to B ′ . W e o nly need to sho w that ν ( B ) = ν ( T ). Since the index o f a matrix is well deﬁned (and this again follows from Lemma 2.6, where it is proved that the eigenv alues of a symmetric genera lized matrix are well deﬁned), it is suﬃcient to show tha t for one (hence any) symmetric representative ( T ε ) ε of T there exis ts an ε 0 ∈ I suc h that for each ε < ε 0 we hav e λ 1 ,ε > 0 , . . . , λ n − k,ε > 0 , λ n − k +1 ,ε < 0 , . . . , λ n − k,ε < 0 , where ( λ i,ε ) ε ( i = 1 , . . . , n ) are the ordered eigenv alues of ( T ε ) ε . T o this end, let ( B ε ) ε be a s ymmetric represe ntative of B , and deﬁne b y ( T ε ) ε a representativ e of T comp onent-wise via T ε := A t ε B ε A ε . Clearly ( T ε ) ε is symmetric. F or each ε let λ 1 ,ε ≥ · · · ≥ λ n,ε be the ordered eig en- v alues of T ε and let µ 1 ,ε ≥ · · · ≥ µ n,ε be the ordered eigen v a lues of B ε . Since A and B are non-degenerate, there exists some ε 0 ∈ I and an integer m 0 such that for each ε < ε 0 and for e ach i = 1 , . . . , n we hav e | λ i,ε | ≥ ε m 0 and | µ i,ε | ≥ ε m 0 . F urthermo re due to our as s umption k = ν ( B ), therefore taking in to account the comp onent-wise or de r of the eig env alues µ i,ε , for each ε < ε 0 we hav e: µ i,ε ≥ ε m 0 ( i = 1 , . . . , n − k ) a nd µ i,ε ≤ − ε m 0 ( i = n − k + 1 , . . . , n ) . As a co ns equence of Sylv ester’s inertia law we ther efore have for ea ch ε < ε 0 : λ i,ε ≥ ε m 0 ( i = 1 , . . . , n − k ) and λ i,ε ≤ − ε m 0 ( i = n − k + 1 , . . . , n ) , since fo r ea ch ε < ε 0 the n um ber of p ositive resp. negative eigenv alues of B ε resp. T ε coincides. W e hav e thereby shown that ν ( T ) = k and we are done.  2.2. CA US ALITY AND THE INVERSE CAUC HY-SCHW ARZ INEQUALITY 17 Deﬁnition 2.13. Let b : e R n × e R n → e R be a symmetric bilinear form on e R n . A basis B := { e 1 , . . . , e k } of e R n is ca lled an o rthogona l ba sis with r esp ect to b if b ( e i , e j ) = 0 whenever i 6 = j . Corollary 2.14. A ny symmetric biline ar form b on e R n admits an ortho gonal b asis. Proof. Let B := { v 1 , . . . , v n } b e s ome bas is of e R n , then the co eﬃcien t matrix A := ( b ( v i , v j )) ij ∈ e R n 2 is symmetric. Due to Lemma 2.6, there is an orthogonal matrix U ∈ e R n 2 and generalized n umbers θ i (1 ≤ i ≤ n ) (the so-ca lled eigen v a l- ues) such that U AU t = diag ( θ 1 , . . . , θ n ). Therefore the (clearly non-degener ate) matrix U induces a mapping e R n → e R n which ma ps B on to some basis B ′ which is orthogo nal.  Deﬁnition 2.15. Let λ 1 , . . . , λ k ∈ e R ( k ≥ 1). Then the span of λ i (1 ≤ i ≤ k ) is denoted by h{ λ 1 , . . . , λ n }i . W e no w int ro duce a notion of ca usality in o ur framework: Deﬁnition 2.16. Let g b e a symmetric bilinear form of Lorentzian signature on e R n . Then we call u ∈ e R n (i) time-like, if g ( u, u ) < 0, (ii) null, if u = 0 or u is free and g ( u, u ) = 0, (iii) space- like, if g ( u, u ) > 0 . F urthermo re, we say tw o time-like v ectors u, v ha ve the sa me time-orientation when- ever g ( u, v ) < 0. Note that there exis t elements in e R n which are neither time-like, nor n ull, nor space-like. The next statemen t provides a characterization of free elements in e R n . W e shall rep ea tedly make use of it in the sequel. Theorem 2. 17. L et v b e an element of e R n . Then the fol lowing a r e e quivalent: (i) F or any p ositive deﬁnite symmetric biline ar form h on e R n we have h ( v , v ) > 0 (ii) The c o eﬃcients of v w ith r esp e ct t o some (henc e any) b asis sp an e R . (iii) v is fr e e. (iv) The c o eﬃcients v i ( i = 1 , . . . , n ) of v with r esp e ct to some (henc e any) b a- sis of e R n satisfy the fol lowing: F or any choi c e of r epr esentatives ( v i ε ) ε (1 ≤ i ≤ n ) of v i ther e exists some ε 0 ∈ I such that for e ach ε < ε 0 we have max i =1 ,...,n | v i ε | > 0 . (v) F or e ach r epr esentative ( v ε ) ε ∈ E M ( R n ) of v t her e exists some ε 0 ∈ I such that for e ach ε < ε 0 we have v ε 6 = 0 in R n . (vi) Ther e exists a b asis of e R n such that the ﬁrst c o eﬃcient v i of v is st rictly non-zer o. (vii) v c an b e extende d to a b asis o f e R n . 18 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y (viii) L et v i ( i = 1 , . . . , n ) denote the c o eﬃcients of v with re sp e ct to some arbitr ary b asis of e R n . Then we have k v e k := n X i =1 ( v i ) 2 ! 1 / 2 > 0 . Proof. W e pro ceed by establishing the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (i), further the equiv a lence (i) ⇔ (viii) as well as (iv ) ⇔ (viii) and (iv) ⇔ (v) and end with the proo f of (iv) ⇒ (vi) ⇒ (vii) ⇒ (viii) ⇒ (iv). If v = 0 the equiv alences are trivia l. W e shall therefore assume v 6 = 0. (i) ⇒ (ii): Let ( h ij ) ij be the co e ﬃcient matrix of h with resp ect to some ﬁxed basis B of e R n . Then λ := P 1 ≤ i,j ≤ n h ij v i v j = h ( v , v ) > 0, in particular λ is invertible and P j ( P i h ij v i λ ) v j = 1 which s hows that h{ v 1 , . . . , v n }i = e R . Since the choice of the basis was ar bitrary , (ii) is shown. (ii) ⇒ (iii): W e assume h{ v 1 , . . . , v n }i = e R but tha t there exists some λ 6 = 0 : λv = 0, that is, ∀ i : 1 ≤ i ≤ n : λv i = 0. Since the co eﬃcients of v span e R , there exist µ 1 , . . . , µ n such that λ = P n i =1 µ i v i . It follows that λ 2 = P n i =1 µ i ( λv i ) = 0 but this is imp ossible, since e R co n tains no nilp otent elements. (iii) ⇒ (i): Due to Lemma 2.6 we may assume that we have chosen a basis such that the co eﬃcien t matrix with r esp ect to the la tter is in diag onal form, i. e., ( h ij ) ij = diag( λ 1 , . . . , λ n ) with λ i > 0 (1 ≤ i ≤ n ). W e hav e to show that h ( v , v ) = P n i =1 λ i ( v i ) 2 > 0. Since there exists ε 0 ∈ I such that for a ll r epresentativ es of λ 1 , . . . , λ n , v 1 , . . . , v n we hav e for ε < ε 0 that γ ε := λ 1 ε ( v 1 ε ) 2 + · · · + λ nε ( v n ε ) 2 ≥ 0, h ( v , v ) 6 > 0 would imply that there exis ts a zero sequence ε k → 0 ( k → 0) such tha t γ ε k < ε k . This implies that h ( v , v ) is a zero divisor a nd it means that all summands share a sim ultaneous zero divisor, i. e., ∃ µ 6 = 0 ∀ i ∈ { 1 , . . . , n } : µλ i ( v i ) 2 = 0. Since v was free, this is a con tradiction and w e ha ve shown that (i) holds. The equiv alence (i) ⇔ (viii) is evident. W e pr o ceed by establishing the equiv alence (iv) ⇔ (viii). First, assume (viii) holds, and let ( v i ε ) ε (1 ≤ i ≤ n ) b e arbitr a ry representatives of v i ( i = 1 , . . . , n ). Then n X i =1 ( v i ε ) 2 ! ε is a representativ e of ( k v e k ) 2 as w ell, and since k v e k is strictly pos itive, there exists some m 0 and s o me ε 0 ∈ I s uch that ∀ ε < ε 0 : n X i =1 ( v i ε ) 2 > ε m 0 . This immediately implies (iv). In o rder to see the conv erse direction, w e proceed indirectly . Assume (viii) does not hold, that is, w e assume there exist repr e senta- tives ( v i ε ) ε of v i for i = 1 , . . . , n such tha t for some sequence ε k → 0 ( k → ∞ ) w e hav e for eac h k > 0 that n X i =1 ( v i ε k ) 2 < ε k k . Therefore one may even constr uct r epresentativ es ( e v i ε ) ε for v i ( i = 1 , . . . , n ) suc h that for each k > 0 and each i ∈ { 1 , . . . , n } we have e v i ε k = 0. It is now eviden t that ( e v i ε ) ε violate condition (iv) and w e are done with (iv) ⇔ (viii). (iv) ⇔ (v) is 2.2. CA US ALITY AND THE INVERSE CAUC HY-SCHW ARZ INEQUALITY 19 evident. So we ﬁnish the pro of by showing (iv) ⇒ (vi) ⇒ (vii) ⇒ (iv) Clear ly (vii) ⇒ (iv). T o s ee (iv) ⇒ (vi) w e ﬁrst observe that the condition (iv) implies that there exists some m 0 such that for suitable re presentativ e s ( v i ε ) ε of v i ( i = 1 , . . . , n ) we hav e for eac h ε ∈ I max i =1 ,...,n | v i ε | > ε m 0 , i. e., ∀ ε ∈ I ∃ i ( ε ) ∈ { 1 , . . . , n } : | v i ( ε ) ε | > ε m 0 . W e may view ( v ε ) ε := (( v 1 ε , . . . , v n ε ) t ) ε ∈ E M ( R n ) as a repres ent ative of v in E M ( R n ) / N ( R n ). Denote for each ε ∈ I by A ε the representing matrix of the linear map R n → R n that merely p er m utes the i ( ε ) th. canonical co or dinate of R n with the ﬁrst one. Deﬁne A : f R n → f R n the bijectiv e linear map with representing matrix A := ( A ε ) ε + E M ( M n ( R )) . What is eviden t now from our construction, is: The ﬁrst co eﬃcient of e v := Av = ( A ε v ε ) ε + E M ( R n ) is strictly nonzero a nd we hav e shown (vi). Finally we verify (vi) ⇒ (vii). Let { e i | 1 ≤ i ≤ n } denote the canonical basis o f e R n . Poin t (vi) ensures the existence of a bijectiv e linear map A on e R n such that the ﬁrst c o eﬃcien t ¯ v 1 of ¯ v = ( ¯ v 1 , . . . , ¯ v n ) t := Av is strictly non-zero; applying P rop ositio n 2.10 yields a nother basis { ¯ v, e 2 , . . . , e n } of e R n . Since A is bijectiv e, { v = A − 1 ¯ v , A − 1 e 2 , . . . , A − 1 e n } is a basis of e R n as w ell and we a re done.  W e may a dd a no n-trivial ex ample of a free vector to the ab ove characterization: Example 2 . 18. F o r n > 1 , let λ i ∈ e R (1 ≤ i ≤ n ) have the following prop erties (i) λ 2 i = λ i ∀ i ∈ { 1 , . . . , n } (ii) λ i λ j = 0 ∀ i 6 = j (iii) h{ λ 1 , . . . , λ n }i = e R This c ho ice of zero divisors in e R is possible (idemp otent elemen ts in e R are thor- oughly discussed in [ 5 ], pp. 2221–22 2 4). Now, let B = { e 1 , . . . , e n } b e th e canon- ical basis of e R n . Theorem 2.17 (iii) implies that v := P n i =1 ( − 1) ( i +1)( n +1) λ i e i is free. F urthermore let γ ∈ Σ n be the cy clic p ermutation which sends { 1 , . . . , n } to { n, 1 , . . . , n − 1 } . Clearly the sign o f γ is p os itive if and o nly if n is o dd. Deﬁne n vec- tors v j (1 ≤ j ≤ n ) by v 1 := v , and such that v j is g iven by v j := P n k =1 λ γ j − 1 ( k ) e k whenever j > 1. Let A be the matrix ha ving the v j ’s as c olumn vectors. T he n det A = n X l =1 λ n l = n X l =1 λ l . Due to pr o p erties (i,iii), det A is in vertible. Ther efore, B ′ := { v , v 2 , . . . , v n } is a basis of e R n , to o. The reader is invited to chec k further equiv alent pro pe rties of v according to Theorem 2.17. Since any symmetric bilinear form admits an orthogona l basis due to Cor ollary 2.14 we fur ther co nclude by mea ns of Theorem 2.17: Corollary 2 .19. L et b b e a symmetric biline ar form on e R n . Then the fol lowing ar e e quivalent: (i) F or any fr e e v ∈ e R n , b ( v , v ) > 0 . (ii) b is p ositive deﬁnite. 20 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y F or showing further alg ebraic prope rties of e R n (cf. section 2.3.1), also the fol- lowing lemma will b e c r ucial: Lemma 2.20. L et h b e a p ositive deﬁnite symmetric biline ar form. Then we have the fol lowing: (i) ∀ v ∈ e R n : h ( v , v ) ≥ 0 and h ( v , v ) = 0 ⇔ v = 0 . (ii) L et m b e a fr e e submo dule of e R n . Then h is a p ositive deﬁnite symmetric biline ar form on m . Proof. First, we verify (i): Let v i (1 ≤ i ≤ n ) be the co eﬃcien ts o f v with resp ect to some orthogona l basis B for h . Then we can write h ( v, v ) = P n i =1 λ i ( v i ) 2 with λ i strictly p ositive for ea ch i ∈ { 1 , . . . , n } . T hus h ( v, v ) ≥ 0, a nd h ( v, v ) = 0 implies ∀ i ∈ { 1 . . . n } : v i = 0, i. e., v = 0 . This ﬁnishes the proof of pa r t (i). In order to sho w (ii) we ﬁrst no tice that b y deﬁnition, any free s ubmo dule a dmits a basis. Let B m := { w 1 , . . . , w k } be suc h for m and deno te by h m the restriction of h to m . Then, due to Theor em 2.1 7 (i), we hav e for all 1 ≤ i ≤ k , h m ( w i , w i ) > 0. Let A := ( h m ( w i , w j )) ij be the co eﬃcient matrix of h m with resp ect to B m . Since h m is symmetric, so is the matrix A and th us, due to Lemma 2.6 there is an orthogona l matrix U ∈ e R k 2 and ther e are ge ne r alized num bers λ i (1 ≤ i ≤ k ) such that U AU t = diag ( λ 1 , . . . , λ k ) whic h implies that the (orthogonal, th us non-degenera te) U maps B m on an or thogonal basis B := { e 1 , . . . , e k } of m with resp ect to h m and again by Theorem 2.17 (i) we hav e λ i > 0 (1 ≤ i ≤ k ). By Deﬁnition 2.11, h m is also p ositive deﬁnite on m and w e are do ne.  Since any time- like or space-like vector is free, w e further ha ve as a co nsequence of Theorem 2.17: Prop ositi o n 2.21. Supp ose we ar e given a biline ar form of L or en tzian signatur e on e R n and let u ∈ e R n \ { 0 } b e time-like, nul l or sp ac e-like. Then u c an b e extende d to a b asis of e R n . In the case of a time-like vector we know a s p eciﬁc ba s is in which the ﬁrst co ordinate is in v ertible: Remark 2.2 2. Suppo se we are given a bilinea r form b of Lorentzian signa ture on e R n , let u b e a time-like vector. Due to the deﬁnition o f g we may suppo se that we hav e a basis so that the scala r pr o duct of u takes the form g ( u, u ) = − λ 1 ( u 1 ) 2 + λ 2 ( u 2 ) 2 · · · + λ n ( u n ) 2 . with λ i strictly pos itive for eac h i = 1 , . . . , n . Since g ( u, u ) < 0, w e see that the ﬁrst co or dinate u 1 of u m ust be strictly non-ze r o. It is worth men tioning that a n a nalogue of the well known criter ion of p ositive deﬁniteness of matrices in M n ( R ) holds in our setting: Lemma 2 . 23. L et A ∈ e R n 2 b e symmetric. If the determinants of al l princip al subminors of A (that ar e the su bmatric es A ( k ) := ( a ij ) 1 ≤ i,j ≤ k (1 ≤ k ≤ n ) ) ar e strictly p ositive, then A is p ositive deﬁnite. Proof. Cho o se a s y mmetric representativ e ( A ε ) ε of A (cf. Lemma 2.3). Clearly the assumption det A ( k ) > 0 (1 ≤ k ≤ n ) implies that ∃ ε 0 ∃ m ∀ k : 1 ≤ k ≤ n ∀ ε < ε 0 : det A ( k ) ε ≥ ε m , that is, for ea ch suﬃciently small ε , A ε is a p ositive 2.2. CA US ALITY AND THE INVERSE CAUC HY-SCHW ARZ INEQUALITY 21 deﬁnite sy mmetric matrix due to a well known cr iterion in linear algebra . F urther- more det A ( n ) = det A > 0 implies A is non-degene r ate which ﬁnally shows that A is p ositive deﬁnite.  Before w e go o n w e note tha t t yp e changing of tensors o n e R n by means of a non-degenera te symmetric bilinea r form g clea rly is pos sible. Moreov er , given a (generalized) metric g ∈ G 0 2 ( X ) on a ma nifold X (cf. s e ction 1.1.2), lowering (resp. raising) indices of generalized tenso r ﬁelds on X (r esp. tensors on e R n ) is compatible with ev aluatio n on compactly supp or ted gener alized po int s (which actually yields the resp. ob ject on e R n ). This basica lly follows fro m Prop ositio n 3.9 ([ 31 ]) combined with Theorem 3.1 ([ 31 ]). As usual we write the covector asso ciated to ξ ∈ e R n in abstract index notation as ξ a := g ab ξ b . W e call ξ i ( i = 1 , . . . , n ) the cov ariant comp onents of ξ . The following technical lemma is requir e d in the sequel: Lemma 2.2 4. L et u, v ∈ e R n such that u is fr e e and u t v = 0 . Then for e ach r epr esentative ( u ε ) ε of u t her e exists a r epr esentative ( v ε ) ε of v such that for e ach ε ∈ I we have u t ε v ε = 0 . Proof. Let ( u ε ) ε , ( ˆ v ε ) ε be representativ es o f u, v resp ectively . Then there exists ( n ε ) ε ∈ N suc h that ( u t ε ) ε ( ˆ v ε ) ε = ( n ε ) ε . By Theor e m 2.17 (iv) w e conclude ∃ ε 0 ∃ m 0 ∀ ε < ε 0 ∃ j ( ε ) : | u j ( ε ) ε | ≥ ε m 0 . Therefore w e may deﬁne a new representativ e ( v ε ) ε of v in the follo wing wa y : F or ε ≥ ε 0 we set v ε := 0, otherwise w e deﬁne v ε := ( ˆ v j ε , j 6 = j ( ε ) ˆ v j ( ε ) ε − n ε u j ( ε ) ε otherwise and clear ly we have u t ε v ε = 0 for ea ch ε ∈ I .  The following result in the style of [ 14 ] (Lemma 3.1.1, p. 74 ) prepares the in v erse Cauch y-Schw arz inequality in our framework. W e f ollow the bo o k of F riedlander which helps us to ca lculate the determinant of the co eﬃcien t matrix of a symmetric bilinear form, w hich then turns out to be strictly po sitive, thus inv ertible. This is equiv alent to non-degenerateness of the bilinear for m (cf. Lemma 2 .2): Prop ositi o n 2. 25. L et g b e a symmetric biline ar form of L or entzian s ignatu r e. If u ∈ e R n is time-like, t hen u ⊥ is an n − 1 dimensional submo dule of e R n and g | u ⊥ × u ⊥ is p ositive deﬁnite. Proof. Due to Prop osition 2.21 we c a n choose a basis of e R n such that Π := h{ u }i is spanned b y the ﬁrst vector, i. e., Π = { ξ ∈ e R n | ξ A = 0 , A = 2 , . . . , n } . Consequently we hav e h ξ , ξ i| Π × Π = g 11 ( ξ 1 ) 2 , 22 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y and g 11 = h u, u i < 0. If η ∈ Π ′ := u ⊥ , then h ξ , η i = ξ i η i , hence the cov ariant comp onent η 1 m ust v a nish (set ξ := u , i. e., h ξ , η i = h u , η i = η 1 = 0). Therefore we hav e (2.3) h η , θ i| Π ′ × Π ′ = g AB η A θ B . Our ﬁrst observ a tion is t hat u ⊥ is a free ( n − 1 dimensiona l) submodule with the basis ξ (2) , . . . , ξ ( n ) given in terms of the chosen c o ordinates ab ov e via ξ j ( k ) := g ij δ k i , k = 2 , . . . , n (cf. (2.4) b elow, these are precise ly the n − 1 row v ectors there!) Due to the matrix m ultiplication (2.4)     1 0 . . . 0 g 21 g 22 . . . g 2 n . . . . . . . . . . . . g n 1 g n 2 . . . g nn     ( g ij ) =  g 11 ∗ 0 I n − 1  ev aluation o f the deter mina nt s yields det g AB det g ij = g 11 . And it follows from det g ij < 0 , g 11 < 0 that det g AB > 0 which in particular shows that g AB is a non-degenerate s ymmetric matr ix, g | u ⊥ × u ⊥ therefore b eing a non- degenerate symmetric bilinear form on an n − 1 dimensional free submo dule. What is left to prov e is p os itive de ﬁnitenes s o f g AB . W e claim that for each u ∈ v ⊥ , g ( v , v ) ≥ 0. In conjunction with the fact that g | u ⊥ is non-degener ate, it follows that g ( v , v ) > 0 for an y free v ∈ u ⊥ (this can b e seen b y using a suita ble basis for u ⊥ which diago nalizes g | u ⊥ × u ⊥ , cf. Corollary 2.19) a nd we are done. T o show the sub claim we hav e to underg o an ε -wise a rgument. Let ( u ε ) ε ∈ E M ( R n ) b e a represen tative of u and let (( g ε ij ) ij ) ε ∈ E M ( M n ( R )) b e a symmetric representatives of ( g ij ) ij , where ( g ij ) ij is the co eﬃcient matrix of g with resp ect to t he canonical basis of e R n . F o r eac h ε w e denote by g ε the symmetric bilinear form induced by ( g ε ij ) ij , tha t is, the latter shall b e the co eﬃcien t matr ix of g ε with resp ect to the canonical ba sis of R n . First we show tha t (2.5) u ⊥ = { ( v ε ) ε ∈ E M ( R n ) : ∀ ε > 0 : v ε ∈ u ⊥ ε } + N ( R n ) , Since the inclus ion relatio n ⊇ is clear, we o nly need to s how that ⊆ holds. T o this end, pick v ∈ u ⊥ . Then g ( u , v ) = g ij u i v j = 0 and the latter implies that for each representative ( ˆ v ε ) ε of v there exists ( n ε ) ε ∈ N suc h that ( g ε ij u i ε ˆ v j ε ) ε = ( n ε ) ε . W e may interpret ( g ε ij u i ε )( j = 1 , . . . , n ) as the representativ es of the co eﬃcients of a vector w with co ordinates w j := g ij u i , and w is free, since u is free and g is non- degenerate. Therefor e we may e mploy Lemma 2.24 which yields a repr esentativ e ( v j ε ) ε of v such that ( g ε ij u i ε v j ε ) ε = 0 . This precisely means tha t there e xists a repre s ent ative ( v ε ) ε of v suc h that for each ε we have v ε ∈ u ⊥ ε . W e ha ve thus ﬁnished the pro o f of identit y (2.5). T o ﬁnish the pro o f o f the cla im, that is g ( v , v ) ≥ 0, we pick a representativ e ( v ε ) ε of v and an ε 0 ∈ I suc h that for each ε < ε 0 we hav e (i) each g ε is of Lorentzian signature 2.2. CA US ALITY AND THE INVERSE CAUC HY-SCHW ARZ INEQUALITY 23 (ii) u ε is time-like (iii) v ε ∈ u ⊥ ε . Note tha t this c hoic e is p ossible due to (2.5). F urther , by the resp. classic re s ult o f Lorentz geometry (cf. [ 14 ], Lemma 3 . 1. 1) we have g ε ( v ε , v ε ) ≥ 0 unless v ε = 0. Since ( g ε ij v i ε v j ε ) ε is a representativ e of g ( v , v ) w e hav e ac hieved the sub claim.  Corollary 2.2 6. L et u ∈ e R n b e time-lik e. Then u ⊥ := { v ∈ e R n : h u, v i = 0 } is a submo dule of e R n and e R n = h{ u }i ⊕ u ⊥ . Proof. The ﬁrst statement is ob vious. F or v ∈ e R n , deﬁne the o r thogonal pro jection of v o nto h{ u }i as P u ( v ) := h u,v i h u,u i u . Then one sees that v = P u ( v ) + ( v − P u ( v )) ∈ h{ u }i + u ⊥ . Finally , ass ume e R n 6 = h{ u }i ⊕ u ⊥ , i. e ., ∃ ξ 6 = 0 , ξ ∈ h{ u }i ∩ u ⊥ . It follows h ξ , ξ i ≤ 0 and due to the preceding prop osition ξ ∈ u ⊥ implies h ξ , ξ i ≥ 0. Since we ha v e a partial or dering ≤ , this is imp ossible unless h ξ , ξ i = 0. How ever b y Lemma 2.2 0 (i) we hav e ξ = 0. This con tr adicts our a ssumption and pro v es that e R n is the direct sum of u and its orthogo nal co mplemen t.  The following statement on the Ca uch y–Sch w arz inequa lit y is a crucial result in generalized Lo rentz Geometry . It slightly d iﬀers from the classica l res ult as is shown in Example 2.28. How ever it s eems to coincide with the cla s sical inequality in ph ysically relev ant cas es, since algebraic complications which mainly a r ise from the existence of zero diviso r in our sca lar ring o f gene r alized num b er s, presumably are not inherent in the latter. Our pr o of follows the lines of the pro of of the analogous classic statement in O ’Neill’s b o ok ([ 39 ], chapter 5, P rop osition 30 , pp. 1 44): Theorem 2.27. (Inverse Cauchy –Schwarz ine quality) L et u, v ∈ e R n b e time-like ve ctors. Then (i) h u, v i 2 ≥ h u, u ih v , v i , and (ii) e quality in (i) hold s if u, v ar e line arly dep endent over e R ∗ , the units in e R . (iii) If u, v ar e line arly indep endent, t hen h u, v i 2 > h u, u ih v , v i . Proof. In what follows, we keep the notation of the pre c e ding corollar y . Due to Corolla ry 2.26, we may deco mpo se u in a unique way v = au + w with a ∈ e R , w ∈ u ⊥ . Since u is time-like, h v , v i = a 2 h u, u i + h w, w i < 0 . Then (2.6) h u, v i 2 = a 2 h u, u i 2 = ( h v, v i − h w, w i ) h u, u i ≥ h u, u ih v , v i since h w , w i ≥ 0 and this prov es (i). In order to prove (ii), assume u , v are linear ly depe nden t ov er e R ∗ , that is, there exist λ, µ , b o th units in e R such that λu + µv = 0. Then u = − µ λ v and equality in (ii) follows. Pro of of (iii): Assume now, that u, v are linear ly independent. W e show that this implies that w is free. F or the sake o f simplicity we assume without loss of g enerality that h u, u i = h v , v i = − 1 a nd we choose a basis B = { e 1 , . . . , e n } with e 1 = u due to Prop ositio n 2.21. Then with r e s pe c t to the new bas is we can wr ite u = (1 , 0 , . . . , 0) t , 24 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y v = ( v 1 , . . . , v n ) t , w = v − P u ( v ) = ( v 1 − ( − g ( v , e 1 )) , v 2 , . . . , v n ) t = (0 , w 2 , . . . , w n ) t . Assume ∃ λ 6 = 0 : λw = 0, then ( λv 1 ) u + λv = λv 1 e 1 − λg ( v , e 1 ) e 1 = λv 1 e 1 − λv 1 e 1 = 0 which implies that u, v are linear ly dependent. This contradicts the assumption in (iii). Th us w indeed is fr ee. Applying Theor em 2 .17 y ields h w , w i > 0. A gla nce at (2.6) shows tha t the proo f of (iii) is ﬁnished.  The follo wing ex ample indica tes what happ ens when in 2 .27 (ii) linear dep en- dence ov er the units in e R is replaced by linea r dep endence o ver e R : Example 2.28. L et λ ∈ e R b e an idemp otent zer o divisor, and write α := [( ε ) ε ] . L et η = dia g( − 1 , 1 . . . , 1) b e the Minkowski met ric. Deﬁne u = (1 , 0 , . . . , 0) t , v = (1 , λα, 0 , . . . , 0) t . Cle arly h u, u i = − 1 , h v , v i = − 1 + λ 2 α 2 < 0 Bu t h u, v i 2 = 1 6 = h u, u ih v , v i = − ( − 1 + λ 2 α 2 ) = 1 − λ 2 α 2 . However, also t he strict re lation fails, i. e., h u, v i 2 6 > h u, u ih v , v i , sinc e λ is a z er o divisor. 2.3. F urther a lgebraic prop erties of ﬁnite di mensio nal mo dules o v er the ring of generalized num b ers This section is dev oted to a discussion o f direct summands o f s ubmo dules inside e R n . The ques tion ﬁrst in volves free submo dules of arbitrary dimension. Howev er , we establish a g eneraliza tio n of Theore m 2 .17 (vii) not only with res p ect to the dimension of the submo dule; the direct summand we construct is also a n orthogo - nal complemen t with resp ect to a g iven p ositive deﬁnite symmetric bilinear form. Having established this in 2.3.1, w e subsequently s how that e R n is not s emisimple, i. e., non-free submo dules in o ur module do not admit direct summands. 2.3.1. Direct s ummands o f free submo dules. The existence of p o s itive bilinear for ms on e R n ensures the existence o f direct summands o f free submodules of e R n : Theorem 2. 29. Any fr e e submo dule m of e R n has a dir e ct summand. Proof. Denote b y m the free submodule in question with dim m = k , let h be a po sitive deﬁnite symmetric bilinea r for m o n m and h m its restr iction to m . Now, due to Lemma 2.2 0 (ii), h m is a p ositive deﬁnite symmetric bilinear fo rm. In particular, ther e exis ts an orthogo na l basis B m := { e 1 , . . . , e k } of m with re sp e ct to h m . W e further ma y as sume that the latter o ne is or thonormal. Denote b y P m the orthogo nal pro jectio n o n m whic h due to the orthog onality of B m may b e wr itten in the for m P m : e R n → m , v 7→ k X i =1 h v , e i i e i . Finally , w e sho w m ⊥ = k er P m : m ⊥ = { v ∈ e R n | ∀ u ∈ m : h ( v , u ) = 0 } = = { v ∈ e R n | ∀ i = 1 , . . . , k : h ( v , e i ) = 0 } = = { v ∈ e R n | P m ( v ) = 0 } = ker P m . 2.4. ENERG Y TENSORS AND A DOMINANT ENER GY CONDITION 25 Where b oth of the last equalities are due to the deﬁnition of P m and the fact that B m is a basis of m . As alw ays in mo dules, m ⊥ = k er P m ⇔ m ⊥ is a direct summand and w e a r e done. An alternative end of this pro of is provided by Lemma 2.20: Since we hav e m + m ⊥ = e R n , we only need to show that this sum is a direct one. But Lemma 2.20 (i) shows that 0 6 = u ∈ m ∩ m ⊥ is absurd, since h is positive deﬁnite.  W e th us hav e also sho wn (cf. Theorem 2.17): Corollary 2 . 30. L et w ∈ e R n b e fr e e and let h b e a p ositive deﬁnite symmetric biline ar form. Then e R n = h{ w }i ⊕ w ⊥ . W e therefore hav e a dded a further equiv alent prop erty to Theorem 2.17. 2.3.2. e R n is not semisi mple. In this sectio n we show that e R n is not se mis im- ple. Reca ll that a mo dule B over a ring R is called simple, if RA 6 = { 0 } and if A contains no non-tr ivial strict submo dules. F or the convenience of the rea der, we recall the follo wing fact on modules (e. g., see [ 23 ], p. 417): Theorem 2.31. The fol lowing c onditions on a nonzer o mo dule A over a ring R ar e e quivalent: (i) A is the sum of a family of simple submo dules. (ii) A is the dir e ct sum of a family of simple submo dules. (iii) F or every nonzer o element a of A , Ra 6 = 0 ; and every submo dule B of A is a dir e ct summand (that is, A = B ⊕ C for some submo dule C . Such a module is called semisimple. Ho wever, pr op erty (i) is violated in e R n ( n ≥ 1): Prop ositi o n 2.32. Every submo dule A 6 = { 0 } in e R n c ontains a strict submo dule. Proof. Let u ∈ A , u 6 = 0. W e ma y write u in terms of the canonica l ba sis e i ( i = 1 , . . . , n ), u = P n i =1 λ i e i and without loss of generality w e may assume λ 1 6 = 0. Denote a representativ e of λ 1 by ( λ ε 1 ) ε . λ 1 6 = 0 in pa rticular ensur es the existence of a zero sequence ε k ց 0 in I and an m > 0 such that f or all k ≥ 1, | λ ε k 1 | ≥ ε m k . Deﬁne D := { ε k | k ≥ 1 } ⊂ I , le t χ D ∈ e R be the characteristic function on D . Clearly , χ D u ∈ A , furthermo re, if the submo dule generated by χ D u is not a strict s ubmo dule of A , o ne may replace D b y ¯ D := { ε 2 k | k ≥ 1 } to achiev e one in the sa me w ay , which how ev er is a strict s ubmo dule of A and we ar e done.  The preceding prop osition in conjunction with Theor em 2.3 1 gives rise to the following conclusio n: Corollary 2.33. e R n is not semisimple. 2.4. Ene rgy tens o rs and a domi nan t energy condition In this section we elabo rate a dominant ener gy condition in the spirit of Hawk- ing and E llis ([ 21 ]) for generalized energy tensor s. The latter will be constructed as tensor pro ducts of generalize d Riemann metrics derived from a (genera liz e d) Lorentzian metric and time-like vector ﬁelds. They shall be helpful for an ap- plication of the Stokes theorem to generalized energy in tegrals in the co urse o f establishing a (lo cal) existence a nd uniqueness theorem for the wav e equa tion on a generalized space-time (cf. [ 49 ], how ever ongoing r esearch tr e ats a wide range 26 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y of genera lized space-times, cf. chapter 3). Throughout this sectio n g denotes a symmetric bilinear form of Lore n tz signature on e R n , and for u, v ∈ e R n we write h u, v i := g ( u, v ). W e introduce the notion of a (gener alized) Lore ntz transformation: Deﬁnition 2.34. W e ca ll a linear map L : e R n → e R n a Lorentz tra nsformation, if it pres erves the metric, that is ∀ ξ ∈ e R n : h Lξ , Lη i = h ξ , η i or equiv alently , L µ λ L ν ρ g µν = g λρ . In the or iginal (classica l) setting the following lemma is an exerc ise in a c ourse on relativ ity [ 6 ]: Lemma 2.35. L et ξ , η ∈ e R n b e time-like unit ve ctors with the same t ime-orientation. Then L µ λ := δ µ λ − 2 η µ ξ λ + ( ξ µ + η µ )( ξ λ + η λ ) 1 − h ξ , η i is a L or entz tr ansformation with the pr op erty Lξ = η . The fo llowing prop osition is a crucia l ingredient in the s ubs e quent pro of o f the (generalized) dominant energ y co ndition for certain energ y tenso rs of this section: Prop ositi o n 2.36. L et u, v ∈ e R n b e time-like ve ctors such that h u, v i < 0 . Then h µν := u ( µ v ν ) − 1 2 h u, v i g µν is a p ositive deﬁnite symmetric biline ar form on e R n . Proof. Symmetry a nd bilinea rity of h ar e clear . What would b e left is to show tha t the co eﬃcie nt matr ix of h with resp ect to a n arbitrary bas is is inv ertible. How ever, determining the deter mina nt o f h is nontrivial. So w e pro cee d by showing that for a n y free w ∈ e R n , h ( w , w ) is strictly p os itive (thus also de r iving the cla ssic statement). W e ma y assume h u, u i = h v , v i = − 1; this can b e achiev ed b y scaling u, v (note that this is due t o the f act that f or a tim e-like (resp. space-like) vector u , h u, u i is s trictly non-zero, thus inv ertible in e R ). W e may assume we have chosen an orthogo nal basis B = { e 1 , . . . , e n } of e R n with resp ect to g , i. e., g ( e i , e j ) = ε ij λ i , where λ 1 ≤ · · · ≤ λ n are the eigenv a lues of ( g ( e i , e j )) ij . Due to Lemma 2.35 we can treat u, v b y means of generalized Lorentz transfor mations suc h that b oth vectors a pp ea r in the form u = ( 1 λ 1 , 0 , 0 , 0), v = γ ( v )( 1 λ 1 , V λ 2 , 0 , 0 ), where γ ( v ) = p − g ( v , v ) = √ 1 − V 2 > 0 (therefore | V | < 1). L e t w = ( w 1 , w 2 , w 3 , w 4 ) ∈ e R n be free (in particular w 6 = 0). Then (2.7) h ( w, w ) := h ab w a w b = h u, w ih v , w i − 1 2 h w, w ih u, v i . Obviously , h u, w i = − w 1 , h v , w i = γ ( v )( − w 1 + V w 2 ) , h u, v i = − γ ( v ). Thus h ( w, w ) = γ ( v )( − w 1 )( − w 1 + V w 2 ) + γ ( v ) 2 ( − ( w 1 ) 2 + ( w 2 ) 2 + ( w 3 ) 2 + ( w 4 ) 2 ) = = − γ ( v ) V w 1 w 2 + 1 2 γ ( v )(+( w 1 ) 2 + ( w 2 ) 2 + ( w 3 ) 2 + ( w 4 ) 2 ) . 2.4. ENERG Y TENSORS AND A DOMINANT ENER GY CONDITION 27 If V w 1 w 2 ≤ 0, w e ar e done. If not, replac e V by | V | ( − V ≥ −| V | ) a nd rewrite the last formula in the follo wing form : (2.8) h ( w, w ) ≥ γ ( v ) 2  ( | V | ( w 1 − w 2 ) 2 + (1 − | V | )( w 1 ) 2 + (1 − | V | )( w 2 ) 2 + ( w 3 ) 2 + ( w 4 ) 2  . Clearly for the ﬁr st term on the right side of (2 .8 ) we hav e | V | ( w 1 − w 2 ) 2 ≥ 0. F rom v is time- like we further de duce 1 − | V | = 1 − V 2 1+ | V | > 0. Since w is free we may apply Theorem 2.17, which yields (1 − | V | )( w 1 ) 2 + (1 − | V | )( w 2 ) 2 + ( w 3 ) 2 + ( w 4 ) 2 > 0 and thus h ( w, w ) > 0 due to eq ua tion (2.8 and we are done.  Finally w e are pr epared to sho w a do minant energ y condition in the style o f Hawking and Ellis ([ 21 ], pp. 91– 93) for a genera liz e d energy tenso r. In wha t follows, we use a bstract index notation. Theorem 2.37. F or θ ∈ e R n the ener gy tensor E ab ( θ ) := ( g ac g bd − 1 2 g ab g cd ) θ c θ d has the fol lowing pr op erties (i) If ξ , η ∈ e R n ar e time-like ve ctors with t he same orientation, t hen we have for any fr e e θ , E ab ( θ ) ξ a η b > 0 . (ii) Supp ose h θ, θ i is invertible in e R . If ξ ∈ e R n is time-like, then η b := E ab ( θ ) ξ a is t ime-like and η a ξ a > 0 , i. e ., η is p ast- oriente d with r esp e ct to ξ . Conversely, if h θ, θ i i s a zer o divisor, then η fa ils to b e time-like. Proof. (i): Deﬁne a symmetric bilinear for m h ab := ( g ( ac g b ) d − 1 2 g ab g cd ) ξ c η d . Due to our assumptions on ξ and η , P rop osition 2.36 yields that h ab is a p ositive deﬁnite symmetric bilinear form. By Theo r em 2.17 we conclude that for an y free θ ∈ e R n , h ab θ a θ b > 0 . It is not har d to c heck that E ab ( θ ) ξ a η b = h ab θ a θ b and therefore we hav e pr oved (i). (ii): T o start with, a ssume η is time-like. Then g ( ξ , η ) = g ab ξ a η b = g ab ξ a E ( θ ) ac ξ c = E ab ( θ ) ξ a ξ b . That this expression is strictly gr eater than zer o follows from (i), i. e., E ab ( θ ) ξ a is past-directed with respect t o ξ whenever h θ, θ i is in vertible, since the latter implies θ is fre e . It rema ins to pr ov e that h η , η i < 0. A straig h tforward calculation yields h η , η i = h E ( θ ) ξ , E ( θ ) ξ i = 1 4 h θ, θ i 2 h ξ , ξ i . Since h θ, θ i is inv er tible and ξ is time-like, we conclude that η is time-like a s w ell. Conv er sely , if h θ , θ i is a zero-diviso r, also h E ( θ ) ξ , E ( θ ) ξ i clearly is one. Therefor e, η = E ( θ ) ξ c a nnot b e time-lik e, and we a re done.  A remar k on this sta temen t is in o rder. A compar ison with ([ 21 ], pp. 91–93 ) shows, tha t our ”dominant energy condition” on T ab is stronger, s ince the vectors ξ , η in (i ) need no t coincide. F urthermore, if in (ii) the condition ” h θ , θ i is inv ertible” was dropp ed, then (as in the c lassical (”smo oth”) theory) we could co nclude that η w as not s pa ce-like, ho w ever, unlike in the smo o th theory , this do es not imply η to b e time- like or n ull (cf. the short note after Deﬁnition 2.1 6). 28 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y 2.5. Generali zed p oint v alue c haracterizations of gene rali zed pseudo-Ri e mannian metrics a nd of causalit y of generalized v ector ﬁelds Throughout this s ection X denotes a pa racompac t smo oth Hausdor ﬀ manifold of dimension n . O ur g o al is to g ive ﬁrst a p oint v a lue characterization of gene r alized pseudo-Riemannian metrics. Then we descr ib e c ausality of genera lized vector ﬁelds on X by means of caus a lity in e R n with resp ect to the bilinear for m induced b y a generalized Lorentzian metric through ev aluation on compactly suppo rted p oints (cf. [ 38 ]). F or a r eview on the basic deﬁnition of genera lized sections of v ector bundles in the sens e of M. Kunzinger and R. Steinbauer ([ 31 ]) we refer to the int ro duction. W e start b y establishing a p oint-v alue c haracterizatio n of generalize d pseudo-Riemannian metrics with res p ect to their index: Theorem 2.3 8 . L et g ∈ G 0 2 ( X ) satisfy one (henc e al l) of the e qu ivalent statemen t s of The or em 1.1, j ∈ N 0 . The fol lowing ar e e quivalent: (i) g has (c onstant) index j . (ii) F or e ach chart ( V α , ψ α ) and e ach e x ∈ ( ψ α ( V α )) ∼ c , g α ( e x ) is a symmetric biline ar form on e R n with index j . Proof. (i) ⇒ (ii): Let e x ∈ ψ α ( V α ) ∼ c be suppor ted in K ⊂⊂ ψ α ( V α ) and choose a representativ e ( g ε ) ε of g as in Theorem 1.1 (iii) and Deﬁnition 1.2. Accor ding to Theorem 1.1 (i), g α ( e x ) : e R n × e R n → e R is symmetric and non-degenera te. So it mer ely re ma ins to prov e that the index of g α ( e x ) coincides with the index of g . Since e x is compactly suppo rted, we ma y shrink V α to U α such that the latter is an op en relatively compact subset of X and e x ∈ ψ α ( U α ). By Deﬁnition 1.2 there exists a symmetric representativ e ( g ε ) ε of g on U α and a n ε 0 such that for a ll ε < ε 0 , g ε is a pseudo-Riemannian metr ic on U α with constan t index ν . L et ( e x ε ) ε be a repr esentativ e o f e x ly ing in U α for eac h ε < ε 0 . Let g ε α, ij be the coor dinate expression of g ε with r esp ect to the chart ( U α , ψ α ). Then for each ε < ε 0 , g ε α, ij ( e x ε ) has precisely ν negative and n − ν p ositive eigenv alues, therefor e due to Deﬁnition 2.8, the class g ij := [( g ε α, ij ( e x ε )) ε ] ∈ M n ( e R ) has index ν . By Deﬁnition 2.1 1 it follows that the respective bilinea r form g α ( e x ) induced b y ( g ij ) ij with resp ect to the cano nical basis o f e R has index ν and we a re done. T o show the conv erse direction, one may pr o ceed by an indir ect pro o f. Assume the contrary to (i), tha t is, g has non-co nstant index ν . In view o f Deﬁnition 1 .2 there exists a n op en, relatively compact chart ( V α , ψ α ), a symmetric repr e sentativ e ( g ε ) ε of g on V α and a zero sequence ε k in I suc h tha t the sequence ( ν k ) k of indices ν k of g ε k | V α has at least t wo a ccumulation p oints, say α 6 = β . Let ( x ε ) ε lie in ψ α ( V α ) for eac h ε . Ther efore the n umber of negative eigenv alues of ( g ij ) ij := ( g ε α,ij ( x ε )) ij is not constant for suﬃciently small ε , a nd therefore fo r e x := [( x ε ) ε ], the res pec tive bilinear for m g α ( e x ) induce d by ( g ij ) ij with respect to the ca nonical ba s is of e R has no index and we ar e done.  Before we go on to deﬁne the notion of causality o f vector ﬁelds with resp ect to a generalized metric of Lorentz signature, we intro duce the notion of strict p ositivity of functions (in analogy with strict p ositivity of genera lized num b ers, cf. section 2.6): Deﬁnition 2.39 . A function f ∈ G ( X ) is called strictly po sitive in G ( X ), if for any compact subset K ⊂ X there exists some representative ( f ε ) ε of f such that 2.5. POINT V ALUES AND GENE RALIZED CA US ALITY 29 for some ( m, ε 0 ) ∈ R × I we h av e ∀ ε ∈ (0 , ε 0 ] : inf x ∈ K | f ε ( x ) | > ε m . W e write f > 0. f ∈ G ( X ) is called strictly negative in G ( X ), if − f > 0 on X . If f > 0 on X , it follows that the conditio n fro m a b ove holds for any represen- tative. Also, f > 0 implies t hat f is in vertible (cf. Theorem 2.46 below). Before giving the main res ult of this section, we hav e to characterize strict p ositivity (or negativity) o f generalized functions by str ict po s itivit y (or negativity) in e R . De- note by X ∼ c the se t of compactly supp orted p oints on X . Suita ble mo diﬁca tions of po int -wise c ha racteriza tions of generalize d functions (as Theorem 2. 4 in [ 38 ], pp. 150) or o f p oint-wise characterizations of p os itiv ity (e. g ., P rop osition 3. 4 in ([ 35 ], p. 5) as well, yield: Prop ositi o n 2.40. F or any element f in G ( X ) we have: f > 0 ⇔ ∀ e x ∈ X ∼ c : f ( e x ) > 0 . Now we have the appropria te machinery at hand to characterize c a usality of generalized vector ﬁelds: Theorem 2.41. L et ξ ∈ G 1 0 ( X ) , g ∈ G 0 2 ( X ) b e a L or entzian metric. The fol lowing ar e e quivalent: (i) F or e ach chart ( V α , ψ α ) and e ach e x ∈ ( ψ α ( V α )) ∼ c , ξ α ( e x ) ∈ e R n is time-like (r esp. sp ac e-like, r esp. nul l) with r esp e ct to g α ( e x ) (a sy mmetric bili ne ar form on e R n of L or entz signatur e). (ii) g ( ξ , ξ ) < 0 (r esp. > 0 , r esp. = 0 ) in G ( X ) . Proof. (ii) ⇔ ∀ e x ∈ X ∼ c : g ( ξ , ξ )( e x ) < 0 (due to the preceding prop ositio n) ⇔ for eac h chart ( V α , ψ α ) and for all e x c ∈ ψ α ( V α ) ∼ c : g α ( e x )( ξ α ( e x ) , ξ α ( e x )) < 0 in e R ⇔ (i).  The pre ceding theore m giv es r ise to the following deﬁnition: Deﬁnition 2.42. A generaliz e d vector ﬁeld ξ ∈ G 1 0 ( X ) is called time-like (resp. space-like, resp. n ull) if it sa tisﬁes one of the respective equiv alen t statemen ts of Theorem 2 .4 1. Moreover, t w o time-like vector ﬁelds ξ , η are said to ha ve the same time or ient ation, if h ξ , η i < 0. Due to the above, this notion is consistent with the po int -wise one giv en in 2 .16. W e conclude this se c tio n by harvesting co nstructions of generalized pseudo- Riemannian metrics by means of po int -wise r esults of the preceding section in con- junction with the p oint-wise characterizations of the glo bal ob jects of this chapter: Theorem 2.43. L et g b e a gener alize d L or entzian metric and let ξ , η ∈ G 1 0 ( X ) b e time-like ve ctor ﬁelds with the same time orientation. Then h ab := ξ ( a η b ) − 1 2 h ξ , η i g ab is a gener alize d Riema nnian metric. Proof. Use Pr op osition 2.36 together with Theorem 2.4 1 and Theorem 2.38.  30 2. ALGEBRAIC FOUNDA TIONS OF COLOMBEA U LORENTZ GEO M ETR Y 2.6. App endix. Inv ertibili t y and strict p ositi vit y in generalized function a lgebras revis i ted This section is devoted to elab or ating a new characteriza tion of inv ertibility as well a s of strict p o s itivit y of gene r alized nu m ber s resp. functions. The ﬁr st inv es- tigation on whic h many w ork s in this ﬁeld are based was done by M. Kunzinger and R. Steinbauer in [ 31 ]; the authors of the latter work established the fa ct that inv ertible genera liz ed num b ers a re pr ecisely such for which the mo dulus o f a ny rep- resentativ e is b ounded from below b y a ﬁxed pow er of the smoo thing pa rameter (cf. the prop osition b elow). It is, how ever, remark able, that (as the following state- men t shows) comp onent-wise inv ertibility suﬃces: W e here show that a nu m ber is inv ertible if e a ch comp onent of any repr esentativ e is inv e r tible for suﬃciently small smo othing par ameter. Prop ositi o n 2.44. L et γ ∈ e R . The fol lowing ar e e quivalent: (i) γ is invertible. (ii) γ is strictly nonzer o, that i s: for some (henc e a ny) r epr esentative ( γ ε ) ε of γ t her e exists a m 0 and a ε 0 ∈ I su ch that for e ach ε < ε 0 we have | γ ε | > ε m 0 . (iii) F or e ach r epr esen t ative ( γ ε ) ε of γ ther e exist s so me ε 0 ∈ I such that for al l ε < ε 0 we have α ε 6 = 0 . (iv) | γ | is strictly p ositive. Proof. Since (i) ⇔ (ii) by ([ 31 ], Theor em 1.2.3 8) and (i) ⇔ (iv ) follo ws from the deﬁnition of strict p os itivity , w e only need to establish the equiv alence (ii) ⇔ (iii) in or der to co mplete pr o of. As the reader can eas ily v erify , the deﬁnit ion of strictly non-zer o is independent of the representativ e, tha t is for each representativ e ( γ ε ) ε of γ we hav e some m 0 and so me ε 0 such that for all ε < ε 0 we have | γ ε | > ε m 0 . By this cons ideration (iii) follows from (ii). In or der to show the co n verse dir e ction, we pro ceed by an indirect a rgument. Assume there exists some repr esentativ e ( γ ε ) ε of γ such that for s o me zero sequence ε k → 0 ( k → ∞ ) we hav e | γ ε k | < ε k k for each k > 0. Deﬁne a modera te net ( ˆ γ ε ) ε in the followin g wa y: ˆ γ ε := ( 0 if ε = ε k γ ε otherwise . It can th en easily be seen tha t ( ˆ γ ε ) ε − ( γ ε ) ε ∈ N ( R ) whic h means that (ˆ γ ε ) ε is a representative o f γ as well. Howev er the la tter violates (iii) and we are done.  Analogously w e can character ize the strict order relation on the generalized real num b ers: Prop ositi o n 2.45. L et γ ∈ e R . The fol lowing ar e e quivalent: (i) γ is strictly p ositive, that is: for some (henc e any) r epr esentative ( γ ε ) ε of γ ther e exist s an m 0 and an ε 0 ∈ I s u ch that for e ach ε < ε 0 we have γ ε > ε m 0 . (ii) γ is strictly nonzer o and h as a r epr esentative ( γ ε ) ε which is p ositive for e ach index ε > 0 . (iii) F or e ach r epr esen t ative ( γ ε ) ε of γ ther e exist s so me ε 0 ∈ I such that for al l ε < ε 0 we have α ε > 0 . 2.6. APPENDIX. INVER TIBILITY REVIS ITED 31 The sta temen t can b e shown in a similar manner as the the preceding one. Next, w e ma y no te that the ab ov e has a n immediate generaliza tion to ge ne r - alized functions. Here X deno tes a par a compact, s mo oth Hausdor ﬀ manifold of dimension n . Theorem 2. 46. L et u ∈ G ( X ) . The fol lowing ar e e quivalent: (i) u is invertible (r esp. strictly p ositive). (ii) F or e ach r epr esentative ( u ε ) ε of u and e ach c omp act set K in X ther e exists some ε 0 ∈ I a nd some m 0 such that for al l ε < ε 0 we have inf x ∈ K | u ε | > ε m 0 (r esp. inf x ∈ K u ε > ε m 0 ). (iii) F or e ach r epr esentative ( u ε ) ε of u and e ach c omp act set K in X ther e exists some ε 0 ∈ I such that ∀ x ∈ K ∀ ε < ε 0 : u ε 6 = 0 ( re sp. u ε > 0 ). Proof. W e only show that the characteriza tion of inv ertibility holds, the r est of the sta tement is then clear . Since (i) ⇔ (ii ) due to ([ 31 ], P rop osition 2.1) we only need to establis h the equiv alence of the third statement. Since (ii) ⇒ (iii ) is evident, we ﬁnish the proof by sho wing the conv er se dir ection. Assume (ii) do es not h old, then there exists a co mpa ctly suppor ted sequence ( x k ) k ∈ X N such that for some representative ( u ε ) ε of u we hav e | u ε k ( x k ) | < ε k k for eac h k . Similarly to the pro of of Pro po sition 2.44 we obser ve that ( ˆ u ε ) ε deﬁned as ˆ u ε := ( u ε − u ε ( x k ) if ε = ε k u ε otherwise yields a nother repr esentativ e o f u which, howev er , viola tes (iii) and we are done.  CHAPTER 3 The w a v e equation on singular space-times W e are interested in a lo cal existence and uniqueness re s ult for the sc a lar wa ve equation on a generalized four dimensional space- time ( M , g ), the Lorentzian metr ic g b eing mo deled a s a symmetric generalized tenso r ﬁeld g ∈ G 0 2 ( M ) with index ν = 1. As usua l the d’Alembertian  is deﬁned b y  := ∇ a ∇ a := g ab ∇ a ∇ b where ∇ denotes the cov ariant der iv ative induced by g . The appropriate initial v alue problem for the wa ve equa tion shall be f ormulated as so o n as w e ha v e in tr o duced the sp eciﬁc class of g eneralized metrics sub ject to our disc us sion. 3.1. Preli minaries T o star t with, we c ollect some basic material from (smo oth) Lore ntzian ge- ometry and ﬁx s ome notation. Throug hout this se c tion, ( M , g ) deno tes a smooth space-time. W e follow the conv ention that the sig nature of g is ( − , + , + , +). The (quite standar d) constructions revisited in the subsections 3.1.1, 3 .1.2 be low have suitable gener alizations in the Colo mbeau setting; these ar e established in chapter 2. 3.1.1. Constructions of R iemannian metrics from Loren tzian metrics. The ﬁnal results in the end o f this section inv olve p oint-wise a r guments. Therefore, we start b y reca lling elementary results from four-dimensiona l Minko wski spa ce- time ( M , η µν ) (where η = diag( − 1 , 1 , 1 , 1) and M = R 4 ). F ollowing the con vention concerning the signa ture of the Lor ent zian metric, we hav e the following co nv entions on causa lit y (using the no tation h ξ , η i := g ab ξ a η b ): A vector ξ ∈ M is called (i) time-like, if h ξ , ξ i < 0, (ii) space- like, if h ξ , ξ i > 0 and (iii) null, if h ξ , ξ i = 0. It should be noted that w e follo w the con v ent ion that ξ = 0 is deﬁned to be a null vector. T o begin with we show: Lemma 3.1. L et u, v b e time-like ve ctors in ( M , η µν ) su ch that h u, u i = h v , v i = − 1 and h u, v i < 0 (that is, u and v have the same t ime-orientation). Then the fol lowing statements hold : L µ ν := δ µ ν − 2 v µ u ν + ( u µ + v µ )( u ν + v ν ) 1 − h u, v i is a L or entz T ra nsformation, me aning L µ ν L λ ρ η µλ = η ν ρ , and has the pr op erty Lu = v . 33 34 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Proof. The ﬁr s t pa r t o f the statement is shown by means of simple algebr aic manipulations: L µ ν L λ ρ η µλ =  δ µ ν − 2 v µ u ν + ( u µ + v µ )( u ν + v ν ) 1 −h u,v i   δ λ ρ − 2 v λ u ρ + ( u λ + v λ )( u ρ + v ρ ) 1 −h u,v i  η µλ =  δ µ ν − 2 v µ u ν + ( u µ + v µ )( u ν + v ν ) 1 −h u,v i   η µρ − 2 v µ u ρ + ( u µ + v µ )( u ρ + v ρ ) 1 −h u,v i  = η ν ρ − 2 v ν u ρ + ( u ν + v ν )( u ρ + v ρ ) 1 −h u,v i − 2 v ρ u ν + 4 h v , v i u ν u ρ + ( − 2 h u,v i u ν − 2 h v, v i u ν )( u ρ + v ρ ) 1 −h u,v i + ( u ρ + v ρ )( u ν + v ν ) 1 −h u,v i + ( − 2 h u,v i u ρ − 2 h v, v i u ρ )( u ν + v ν ) 1 −h u,v i + ( h u,u i + h u,v i + h u,v i + h v ,v i )( u ν + v ν )( u ρ + v ρ ) (1 −h u,v i ) 2 = η ν ρ − 2 v ν u ρ − 2 v ρ u ν − 4 u ν u ρ + 2 u ν ( u ρ + v ρ ) + 2 u ρ ( u ν + v ν ) = η ν ρ . The other c la im is obtained by a further ca lculation: L µ ν u ν = u µ − 2 v µ h u, u i + ( u µ + v µ )( h u, u i + h u, v i ) 1 − h u, v i = (3.1) = u µ + 2 v µ − u µ − v µ = v µ , that is, Lu = v and we are done.  Constructions o f Riemannia n metrics by means of Lo rentzian metrics and time- like vector ﬁelds will b e used later on. Here is the result in full generality (w e will also use simpler constructions, wher e u = v , cf. the coro llary below): Lemma 3.2. L et u , v b e time-like ve ctors in ( M , η ) with the same time-orientation. Then h uv ab := u ( a v b ) − 1 2 h u, v i η ab is a symmetric p ositive deﬁnite biline ar form on M . Proof. Step 1. By scaling u, v appropriately it can be seen that w e may ass ume without loss of generality that u 2 = v 2 = − 1 and that u , v lie in the future ligh t cone. Step 2. By the pr eceding lemma, the Lorentz group acts transitively on the future light cone. Therefor e, there exists a Lorentz transfor mation L 1 such that ¯ u := L 1 u = (1 , 0 , 0 , 0) and we set ¯ v := L 1 v . By means of a rotatio n L 2 of the space co ordinates it can further be achieved that ˆ u := L 2 ¯ u = (1 , 0 , 0 , 0) and ˆ v := L 2 ¯ v = L 2 L 1 v = γ ( V )(1 , V , 0 , 0) with γ ( V ) = (1 − V 2 ) − 1 / 2 , | V | < 1. Step 3. W e denote by L := L 2 L 1 the comp osition of the tw o Lorentz transformations L 1 , L 2 . With this notation w e ha ve by the above, ˆ u = L u, ˆ v = Lv . Since h uv ab is evidently a symmetric bilinear form, w e only need to show that for each non-zer o v ec tor w , we hav e h uv ( w, w ) > 0. Since for ˆ w := Lw , h uv ( w, w ) = h ˆ u ˆ v ( ˆ w, ˆ w ), and since L is a linear isomorphism, it therefore suﬃces to show that for eac h no n-zero w , h ˆ u ˆ v ( w, w ) > 0. Let w = ( w 1 , w 2 , w 3 , w 4 ) ∈ M , w 6 = 0 and set h = h ˆ u ˆ v . Then w e hav e h ( w, w ) := h ab w a w b = h ˆ u, w ih ˆ v , w i − 1 2 h w, w ih ˆ u , ˆ v i . 3.1. PRELIMINARIES 35 Obviously , h ˆ u, w i = − w 1 , h ˆ v , w i = γ ( V )( − w 1 + V w 2 ) , h ˆ u, ˆ v i = − γ ( V ). Thus h ( w, w ) = γ ( V )( − w 1 )( − w 1 + V w 2 ) + 1 2 γ ( V )( − ( w 1 ) 2 + ( w 2 ) 2 + ( w 3 ) 2 + ( w 4 ) 2 ) = − γ ( V ) V w 1 w 2 + 1 2 γ ( V )(+( w 1 ) 2 + ( w 2 ) 2 + ( w 3 ) 2 + ( w 4 ) 2 ) If V w 1 w 2 ≤ 0, then we are done. Otherwise V w 1 w 2 = | V || w 1 || w 2 | < | w 1 w 2 | ≤ ( w 1 ) 2 +( w 2 ) 2 2 , b ecause of | V | < 1 and V w 1 w 2 6 = 0. Inserting this information in to the latter equation yields h ( w, w ) = − γ ( V ) V w 1 w 2 + 1 2 γ ( V )(+( w 1 ) 2 + ( w 2 ) 2 + ( w 3 ) 2 + ( w 4 ) 2 ) > > 1 2 γ ( V )(( w 3 ) 2 + ( w 4 ) 2 ) ≥ 0 , i. e. h ( w, w ) > 0 and w e are done.  An immediate corollary is: Corollary 3 . 3. L et ( M , g ) b e a smo oth sp ac e-time. L et ξ , η b e time-like ve ctor ﬁelds on ( M , g ) with the same time orientation. Then h ab := ξ ( a η b ) − 1 2 h ξ , η i g ab is a Riemannia n metric on M . As a c onse qu en c e we have: if θ is a time-like u nit ve ctor ﬁeld, then also k ab := g ab + 2 θ a θ b is a R iemannian metric. Proof. Let p ∈ M and choo se a lo cal chart ( U, ξ ) ∋ p such that the co or dinate expression of g is Minko wskian at p . Then we are in the setting o f Lemma 3.2, according to which h ab is a p ositive deﬁnite bilinear form a t p . F urthermore h ab is smo oth, since ξ , η and g are. T o prov e the second ass e rtion, we set ξ = η = θ . Due to the ﬁrst claim, k ab = 2( ξ ( a η b ) − 1 2 h ξ , η i g ab ) = g ab + 2 θ a θ b is a Riemannian metric, and we a re done.  A r emark on the Riemannian metric cons tr ucted a b ov e is in o r der. The ﬁrst observ atio n is, that in general h ab := 2( ξ ( a η b ) − 1 2 h ξ , η i g ab ) is not the in verse of h ab = 2( ξ ( a η b ) − 1 2 h ξ , η i g ab ) as deﬁned in the pre ceding corolla ry , but just the metric equiv alent cov ariant tensor . How ever, if ξ = η , then it is the case! F or the sake of simplicity , w e assume h θ , θ i = − 1. Then we have k ab = 2 h ab = g ab + 2 θ a θ b , and similarly , k bc = 2 h bc = g bc + 2 θ b θ c . Therefore we o btain (3.2) k ab k bc = ( g ab + 2 θ a θ b )( g bc + 2 θ b θ c ) = δ c a + 2 θ a θ c + 2 θ a θ c − 4 θ a θ c = δ c a , and we have shown the ass ertion. W e shall make use of such metric constructions in the deﬁnition of certain energy integrals (cf. section 3.6). How ever, in or der to en tirely understand their structure we inv estigate further in e ne r gy tens o rs and certain positivity s tatements, which in the physics liter ature ar e referred to as ”dominant energy condition(s)”: 3.1.2. Energy tens ors and domi nan t e nergy condi tion. Let ( M , g ) b e a smo oth spa c e-time. T he sta tement of this section are to b e understo o d p oint-wise. W e start to rev is it a notion of ([ 21 ], pp. 90). Deﬁnition 3.4. A symmetric tenso r T ab is said to satisfy the dominant energy condition if for every time like vector ξ a , η b := T ab ξ a is not space-like and if further T ab ξ a ξ b ≥ 0. 36 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES A rema rk on this is in order: The condition T ab ξ a ξ b ≥ 0 implies tha t the non- space lik e vector − η b = − T ab ξ a has the same time-o rientation as ξ a . This follows from − η b ξ b = − T ab ξ a ξ b ≤ 0 , that is g ( ξ , − η ) ≤ 0, which is eq uiv alent to saying tha t ξ , − η ha ve the s a me time- orientation. A conse quence of the dominant energy co ndition is the following Lemma 3 . 5. L et T ab b e a symmetric tens or satisfy ing the dominant ener gy c ondi- tion. Then for any t ime-like ve ctors ξ a , η b with the same time-orientation, we have T ab ξ a η b ≥ 0 . Proof. By the dominant energy condition, θ b := T ab ξ a is time-like or n ull, and − θ a has the same t ime-orientation as ξ a , that is g ab ξ a ( − θ b ) ≤ 0. T he r efore, by assumption, − θ b also has the same time-orientation as η c . As a co nsequence we hav e − T ab ξ a η b = g ab η a ( − θ b ) ≤ 0 , and we a re done.  F ollowing J. Vick ers and J. Wilso n ([ 49 ]) w e deﬁne a class of (symmetric) energy tensors T ab,k . Let e ab be a Riemannian metric with e ab its in verse, let W a 1 ...a k be an ar bitrary tenso r o f type (0 , k ), k ≥ 0 and let ξ a , η b be time-like vectors with the same time-or ientation. W e de ﬁne for k = 0 T ab, 0 ( W ) := − 1 2 g ab W 2 , and for k ≥ 1, we set T ab,k ( W ) := ( g ac g bd − 1 2 g ab g cd ) e p 1 q 1 . . . e p k − 1 q k − 1 W cp 1 ...p k − 1 W dq 1 ...q k − 1 . Then we hav e the follo wing: Prop ositi o n 3. 6. F or e ach k ≥ 0 , T ab,k ( W ) is a symmetric tensor whic h satisﬁes the dominant ener gy c ondition. Proof. The case k = 0 is trivial. Hence we start with k = 1. W e have η b := ( g ac g bd − 1 2 g ab g cd ) ξ a W c W d = ( ξ c g bd − 1 2 ξ b g cd ) W c W d = = ξ c W c W b − 1 2 ξ b W d W d = = W ( ξ ) W b − 1 2 ξ b h W , W i . F rom this we obta in g ( η , η ) = η b η b = ( W ( ξ ) W b − 1 2 ξ b h W , W i )( W ( ξ ) W b − 1 2 ξ b h W , W i ) = = 1 4 h ξ , ξ ih W, W i 2 ≤ 0 , where the last inequalit y ho lds because ξ a is t ime-like. W e hav e therefore sho wn that η b = T ab, 1 ξ a is time-like or null. It remains to sho w that the time-orie n tation of − η b is the same as the one of ξ a : T ab, 1 ( W ) ξ a ξ b = { ( g ac g bd − 1 2 g ab g cd ) ξ a ξ b } W c W d = ξ c W c ξ d W d − 1 2 ξ a ξ a W b W b . 3.1. PRELIMINARIES 37 Due to Co rollar y 3.3, { ( g ac g bd − 1 2 g ab g cd ) ξ a ξ b } = ξ c ξ d − 1 2 h ξ , ξ i g cd is a Riema nnia n metric, therefore, T ab, 1 ( W ) ξ a ξ b ≥ 0 and we a re done with the cas e k = 1. W e re duce the proo f for hig he r o rders k > 1 to the case k = 1 . T o this end, ﬁx p ∈ M and let B := { b 1 , . . . , b 4 } b e an o rthonormal basis of ( T p M ) ∗ with resp ect to e ab . With r e s pe ct to this basis T ab,k ( W ) r eads T ab,k ( W ) : = ( g ac g bd − 1 2 g ab g cd ) δ p 1 q 1 . . . δ p k − 1 q k − 1 W cp 1 ...p k − 1 W dq 1 ...q k − 1 = = X p 1 ...p k − 1 ( g ac g bd − 1 2 g ab g cd ) W cp 1 ...p k − 1 W dp 1 ...p k − 1 . Now for eac h tupel ( p 1 , . . . , p k − 1 ) we hav e a s in the case k = 1, ( g ac g bd − 1 2 g ab g cd ) W cp 1 ...p k − 1 W dp 1 ...p k − 1 ξ a ξ b ≥ 0 . Therefore, by summing over all these indices, we hav e T ab,k ( W ) ξ a ξ b ≥ 0 . It rema ins to show that T ab,k ( W ) ξ a is time-like o r nu ll, supp os ing that ξ a is time- like. T o show this, we use the following prop erty of the light cone: F o r each λ, µ ≥ 0 , λ + µ > 0 and each v a , w a in the future (res p. past) ligh t cone, also λv a + µw a lies in the future (resp. past) ligh t cone. Again, we ma y reduce to the case k = 1, and see that for each tuple ( p 1 , . . . , p k − 1 ), − θ b p 1 ,...,p k − 1 := − ( g ac g bd − 1 2 g ab g cd ) W cp 1 ...p k − 1 W dp 1 ...p k − 1 ξ a lies in the same light cone a s ξ a . Ther efore, by the conv exity prop erty of the light cone, also the sum over all such indices does, that is, − T ab,k ( W ) ξ a = X p 1 ...p k − 1 − θ b p 1 ,...,p k − 1 is time-like or null, and we ar e do ne.  As a c o nsequence of Lemma 3.5 and Prop osition 3.6, w e hav e for all time-lik e vectors with the same time-o rientation, T ab,k ( W ) ξ a η b ≥ 0 . This also may be concluded by directly applying Coro llary 3.3 by means of which we hav e the even stro nger result: Corollary 3 .7. F or e ach non- zer o tensor W a 1 ,...,a k , and for al l time-like ve ctors ξ a , η b with the same time-orientation, we have (3.3) T ab,k ( W ) ξ a η b > 0 38 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Proof. By coro llary 3.3, h cd := ( g a ( c g d ) b − 1 2 g ab g cd ) ξ a η b is a Riemannian metric. Ther efore, h cd e p 1 q 1 . . . e p k − 1 q k − 1 is a Riemannian metric on ⊗ k i =1 ( T M ) ∗ as well, and since W 6 = 0, w e hav e T ab,k ( W ) ξ a η b = h cd e p 1 q 1 . . . e p k − 1 q k − 1 W cp 1 ...p k − 1 W dq 1 ...q k − 1 > 0 and we have shown the claim.  Finally , we mention that the dominant energ y condition has recently been gen- eralized to a so-called sup er energy condition on sup er-ener gy tenso rs (cf. [ 45 ]). 3.1.3. The d’Alembe rtian in lo cal co ordi n ates . The aim o f this section is to justify the co o rdinate form of the d’Alem ber tian. Lemma 3.8. L et g b e a smo oth L or entzian metric. In l o c al c o or dinates ( x i ) ( i = 1 , . . . , 4) , the d’A lemb ertian takes the form (3.4)  u = | g | − 1 2 ∂ i ( | g | 1 2 g ij ∂ j u ) . Proof. Let U b e the do main of the co or dinate chart s ystem ξ = ( x 1 , . . . , x 4 ). By ([ 39 ], Lemma 19, p. 195), there exists a volume Element ω o n U such that (3.5) ω ( ∂ 1 , . . . , ∂ 4 ) = | g | 1 2 (the pro of essentially uses loca l orthogona l fra me ﬁelds). A further fa c t ([ 39 ], Lemma 21, p. 195) is that for any lo cal volume element ω on M w e hav e (3.6) ( L ξ ω ) bcde = ( ∇ a ξ a ) ω bcde W e claim that the div ergence of ξ can b e decompo sed in the follo wing wa y: (3.7) ∇ a ξ a = | g | − 1 2 ∂ a ( | g | 1 2 ξ a ) . Assuming that this identit y holds, we may set ξ a := ∇ a u and derive  u = ∇ a ( ∇ a u ) = | g | − 1 2 ∂ a ( | g | 1 2 ∇ a u ) = = | g | − 1 2 ∂ a ( | g | 1 2 g ab ∇ b u ) = = | g | − 1 2 ∂ a ( | g | 1 2 g ab ∂ b u ) and w e ha ve prov ed the lemma. In order to show the subclaim, we calcula te the left and right hand side of (3.7 ) separately . W e make use of (3.6) and the fa ct that, since we are dealing with a 4-form ω , it is suﬃcien t to ev aluate the for mu la at ( ∂ 1 , . . . , ∂ 4 ) only: the right side o f (3.6) yields by mea ns of (3.5) (3.8) ( ∇ a ξ a ) ω ( ∂ 1 , . . . , ∂ 4 ) = | g | 1 2 ∇ a ξ a . The left side of (3.6) yields: L ξ ω ( ∂ 1 , . . . , ∂ 4 ) = (3.9) L ξ ( ω ( ∂ 1 , . . . , ∂ 4 )) − X i ω ( ∂ 1 , . . . , L ξ ∂ i , . . . , ∂ 4 ) . Now we hav e (3.10) L ξ ∂ i = [ ξ , ∂ i ] = X j [ ξ j ∂ j , ∂ i ] = X j ( ξ i ∂ j ∂ i − ∂ i ( ξ j ∂ j )) = − X j ( ∂ i ξ j ) ∂ j . 3.1. PRELIMINARIES 39 By (3.5) a nd (3.1 0) we ther efore obtain L ξ ω ( ∂ 1 , . . . , ∂ 4 ) = L ξ ( | g | 1 2 ) + X i,j ∂ ξ j ∂ x i ω ( ∂ 1 , . . . , ∂ j , . . . , ∂ 4 ) = (3.11) = X i,j ξ i ∂ ( p | g | ) ∂ x i + X i ∂ ξ i ∂ x i ( δ ij p | g | ) = = X i ∂ ∂ x i ( p | g | ξ i ) . Since (3.11) ≡ (3.8) beca use of (3.6) we hav e succeeded to show (3.7) and w e are done with the sub claim.  3.1.4. General Loren tzian me trics in sui table co ordi nates . F or com- putational purpo ses it is advisable to ﬁnd coor dinates in whic h the metric has a sp ecial form, such that calculations ca n b e carrie d out more easily . In this section we ﬁrst recall wha t a metric lo oks like in Gaussian nor mal co o r dinates, and we ﬁn- ish by showing that in suitable co ordinates a static metric can be written without ( t, x µ )–cross terms. A t the end o f section (3.3 .3) w e shall return to this topic from a gener alized po int o f view. Theorem 3.9. L et Σ b e a t hr e e dimensional sp ac e-like manifold. Any p oint p ∈ Σ has a neighb orho o d such that in Gaussian normal c o or dinates, the L or entz ian metric g o n M lo c al ly takes the fo rm (3.12) ds 2 = − V 2 ( t, x γ ) dt 2 + g αβ ( t, x γ ) dx α dx β , that is, without ( t, x µ ) –cr oss t erms (her e the variables i n Gr e ek l etters ar e ra nging b etwe en 1 and 3 , ther efor e x α denote the sp ac e-variables, wher e as x 0 = t is the t ime variable). It c an further b e achieve d that V 2 ≡ 1 . Proof. F or the pro o f o f this statement we fo llow the lines of ([ 50 ], pp. 42 -43). A pro of for the r e sp ective statement in a more general context ca n b e found in ([ 39 ], pp. 199 -200, Lemma 25). Since Σ is space-like, the normal n a is time-like at each p o int of Σ. Fix p ∈ Σ and assume n a (initially o nly deﬁned on Σ) is extended to a geo desically conv ex neig hborho o d U of p . Through each po in t q ∈ U we constr uct the unique g eo desic γ q ( t ) with ˙ γ q ( t = 0) = n a ( q ). W e ma y now lab el each q ∈ U ∩ Σ by co ordinates x µ ( µ = 1 , 2 , 3), and cho ose t as the par a meter along the geo desic γ q ( t ). Then ( U, ( t ( q ) , x µ ( q )) is a lo cal chart at p , and ∂ t | t =0 = n a | Σ ∩ U . F rom n a ⊥ g Σ it follows that th e ( t, x µ ) cross-ter ms g 0 µ of the metric v anish at t = 0, since g 0 µ ( t = 0 , x µ ) = g ( ∂ t , ∂ µ ) | t =0 . Moreover, since parallel transpo rt is an isometry , w e hav e that g ( ∂ t , ∂ µ ) ≡ 0 on all of U . W e hav e thus pr ov ed (3.12). Since n a is time-like, w e can normalize it by the condition g ab n a n b = − 1, and therefore it ca n even be ac hieved tha t V 2 ≡ 1. This completes the pr o of of the theorem.  Next, we deﬁne cer tain space- time symmetries: Deﬁnition 3.10. A space-time ( M , g ) is called stationary , if ther e exists a time- like vector ﬁeld ξ a such that ∇ ( a ξ a ) = 0. This is equiv alent to L ξ g = 0. ξ a is called a time-like Killing vector. A stationar y space-time ( M , g ) with time-like Killing vector ξ a is called static, if ξ a is h ype r surface-o rthogona l, that is, through each p oint p there is a three dimensional spa ce-like hypersur face Σ such that ξ a is orthog onal to Σ. 40 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES In general, the c o eﬃcien ts − V 2 , g αβ in (3.12) which deter mine the metric v ia Theorem 3.9, are not independent of the time t . How ever, if g is a static space time, w e ha ve (for a pro of cf. the res pec tive statement in th e generalized setting, 3.16): Theorem 3. 11. A static sp ac e-time ( M , g ) c an lo c al ly b e written as (3.13) ds 2 = − V 2 ( x γ ) dt 2 + g αβ ( x γ ) dx α dx β . Such co or dinates w e call static co ordinates throughout. As a conseq uence of the pr eceding theorem, we see that the d’Alembertian takes a quite s imple for m in static co or dinates: Prop ositi o n 3.12 . L et ( M , g ) b e a static s p ac e-time. L et V , g αβ b e the c o eﬃcients of g in static c o or dinates as given in T he or em 3.11. Then the d’Alemb ertian takes the fol lowing form: (3.14)  u = − V − 2 ∂ 2 t u + | g | − 1 / 2 ∂ α  | g | 1 / 2 g αβ ∂ β  u. Proof. This follows basically from Lemma 3.8 and Theorem 3.11: in static co ordinates the time der iv atives ∂ t g ab v anish, and the ( t, x µ ) cross ter ms of the metric v anis h a s well. As a consequence, we ha ve ∂ t V − 2 ≡ 0 , ∂ t g αβ ≡ 0 , ∂ t | g | ≡ 0, and we a re done.  3.1.5. The wa v e equation on a smo oth space-time. W e b egin with re- calling causality notions. Let ( M , g ) b e a smoo th time-orient able space time. F or a point q in M , we call D + ( q ) the future dep endence do ma in o f q , that is the set of all po int s p which ca n be re a ched by future dir ected time-like geo des ics thro ug h p . F urthermo re, for a set S , D + ( S ) := S q ∈ S D + ( q ) is the future emission o f S . The closure o f the la tter is deno ted by J + ( S ) := D + ( S ). Reversing the time-or ientation, we may similarly deﬁne D − ( q ), D − ( S ) and J − ( S ). A set S is called pas t-compact if the in tersection S ∩ J − ( q ) is compact for each q ∈ S . Let S be a rela tively compact three dimensional spac e-like submanifold and let ξ be a time like vector ﬁeld. In the smo oth setting, loca l smo o th solutions for the initial v alue problem  u = f u | S = v (3.15) ∇ a ξ a u | S = w are guar a nteed to ex ist by the following theor em ([ 14 ], Theorem 5 .3.2): Theorem 3. 13. L et S b e a p ast- c omp act sp ac e-like hyp ersurfac e, such that ∂ J + ( S ) = S . Supp ose that f is C ∞ and t hat C ∞ Cauchy data v , w ar e given on S . Then the Cauchy pr oblem (3.15) has a unique solution in J + ( S ) such that u ∈ C ∞ ( J + ( S )) . 3.1.6. Lera y forms. This section is dedicated to rec a lling how to decomp os e volume integrals inside a foliated domain. Suppo se ( M , g ) is a smo oth spac e-time. Deno te by µ the volume form induced by g (as ment ioned ab ove in the pro of o f Lemma 3 .8); in co ordinates we may write µ a s µ = | g | 1 2 dt ∧ dx 1 · · · ∧ dx 3 , with | g | , the absolute v alue of the determinant o f g (that is | g | = − g ). 3.1. PRELIMINARIES 41 Let Ω b e an open domain in M , and let S be in C ∞ (Ω) with d S 6 = 0 o n Ω. Cho ose coo r dinates x i ( i = 0 , . . . , 3) suc h that S = t := x 0 . By ([ 14 ], Lemma 2.9.2 ), we may decomp ose µ as µ = dS ∧ µ S , with a 3 − form µ S and the restriction of µ S on S τ := { ( t, x µ ) | t = τ } is uniq ue. W e shall write µ S | S τ =: µ τ . More explicitly , w e have µ τ = | g | 1 2 dx 1 ∧ dx 2 ∧ dx 3 . A conse quence of F ubini’s theorem in this setting is ([ 14 ], Lemma 2.9.3): An y lo cally in tegrable function ψ with co mpact suppo rt in Ω may be integrated as follows (3.16) Z ψ µ = Z dτ Z S τ ψ µ τ . 3.1.7. F oliations and integration. In this section w e s how in which wa y we shall integrate energy integrals subsequently . In particula r w e discuss asp ects of integration and lo ca l foliations of co mpa ct subregions o f spa ce-time which will b e tailo red to our needs in s uch a wa y that Stokes’ theor em can be applied in a con venien t wa y . This w ill be needed later on when we deriv e estimates for an inﬁnite hier arch y of (gener alized) energy integrals. W e p oint out that the setting o f this se c tion is still the smo oth o ne; this, ho wever, is suﬃcient for displaying the co ncepts which will ﬁnally be used in the g eneralized setting . F rom now on we shall suppose tha t the given space-time ( M , g ) has the follow- ing feature: Each p oint p o n a a given initial space-like surface Σ admits a region Ω with p ∈ Ω space-like b oundary S and S 0 , with S 0 := Σ ∩ Ω (cf. ﬁgur e 1 . Note that Ω is not a neigh bo rho o d of p in the usual topolo gy). W e call such a region semi-neighborho od o f p . F urthermor e, we assume tha t Ω lies en tirely in a re g ion of space-time which can b e foliated b y thre e dimensional s pace-like hyper surfaces Σ τ meaning that there exists a coo r dinate system ( t, x µ ) such that Σ τ := { ( t, x µ ) | t = τ ) } . F urthermo re, Σ = Σ τ =0 and we deﬁne S τ := Σ τ ∩ Ω. Let γ > 0. W e shall in tegr ate ov er the compact reg ion Ω γ which is the part of Ω which lies b etw een Σ 0 and Σ γ . Therefore, the b oundar y of Ω γ is given by S 0 , S γ and S Ω ,γ := S ∩ Ω γ (cf. Figur e 3.1.7; note that the b oundar y is space- like throughout). A t the e nd of the pres ent sectio n we sha ll prov e that in static space - times ( M , g ) any po int p in Σ, (the lo cal space-like manifold through p orthogonal to the given symmetry ξ a ) a dmits such a semi-neighbor ho o d Ω, and in a subsequent se c tio n we establish an analo gous re s ult for generaliz e d static space-times. Finally , we sho w how to use this to integrate energies . Assume T ab , a sy mmetr ic tensor- ﬁeld of type (2,0) is g iven, whic h satisﬁes the dominant energy condition. Let ξ a be a time-lik e Killing vector ﬁeld, and let Σ t be or tho gonal to ξ a . W e denote b y n a the unit no rmal v e ctor ﬁeld to S Ω ,γ . Let µ be the volume ele ment induced by the metric . W e s eek to ca lculate the follo wing int egral on Ω γ : (3.17) Z Ω γ ξ b ∇ a T ab µ. 42 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Figure 1. Lo cal folia tio n of space-time First, we a pply Sto kes’s theorem in the following fashion (cf. W ald, pp. 432– 434): Theorem 3. 14. L et N b e an n –dimensional c omp act oriente d manifold with b ound- ary ∂ N , µ the n atu r al volume element induc e d by the metric g , and µ ∂ N the r e- sp e ctive surfac e form on ∂ N . Assume ∂ N is nowher e nul l. L et furt her v a ∈ X ( M ) and denote by n a the unit normal to ∂ N (that is g ab n a n b = ± 1 ). Then we have: Z N ∇ a v a µ = Z ∂ N n a v a µ ∂ N In the present setting, the bounda r ies o f Ω γ are S 0 , S γ with time-like nor mal ξ a , the Killing vector, and S Ω ,γ with normal n a . In general, ξ is not a unit vector ﬁeld. Denote therefore b y ˆ ξ := ξ √ − g ( ξ ,ξ ) the resp ective unit vector ﬁeld. Since ξ a is a K illing vector and T ab is sy mmetric, we have: ∇ b ( T ab ξ a ) = ξ a ( ∇ b T ab ) + T ab ∇ b ξ a = ξ b ( ∇ a T ab ) + T ( ab ) ∇ [ b ξ a ] = ξ b ( ∇ a T ab ) + 0 . The int egral (3.1 7) can therefor e be decompos ed in the following wa y b y Stok e s ’s Theorem: (3.18) Z Ω γ ∇ b ( T ab ξ a ) µ = Z Ω γ ξ b ∇ a T ab µ = Z S γ T ab ξ a ˆ ξ b µ γ − Z S 0 T ab ξ a ˆ ξ b µ 0 + Z S Ω ,γ T ab ξ a n b µ S Ω ,γ . How ever, since T ab satisﬁes the dominant energy condition, we have by Lemma 3.5: Z S Ω ,γ T ab ξ a n b µ S Ω ,γ ≥ 0 . Using this fact we co nclude by means of (3.18) that (3.19) Z S γ T ab ξ a ˆ ξ b µ γ ≤ Z S 0 T ab ξ a ˆ ξ b µ 0 + Z Ω γ ξ b ∇ a T ab µ. 3.2. Descripti on of the me tho d W e a re going to prove a n existence and uniquenes s theorem for the sca la r w av e equation in G ( M ) following the metho d of J. Vick ers and J. Wilson ([ 49 ]) develop ed 3.3. THE ASSUMPTIONS 43 in the con tex t of conical space times. Hence w e g eneralize the r esult in ([ 49 ]) from conical space times to g eneralized static space times. The pro gram is a s follows: (i) W e star t with sp ecifying the ingr edients of the theorem; these a re in particular the (a) assumptions on the ge ne r alized Lo rentzian metric in ter ms o f a ce r - tain asymptotic gr owth behavior o f the r epresentativ es. The metric is designed for admitting lo cal fo liations of space- time by space-like hypersurfaces . (b) Ener g y int egrals and So b o lev nor ms are in tro duced. (ii) Part A of the pro of establishes that energy integrals (on the three– dimensional submanifolds S τ ) and the three-dimensional So bo lev norms as deﬁned be low ar e equiv alent. This e na bles us to work with energ ie s of arbitrar y order instead o f Sob ole v norms. (iii) Part B is devoted to providing mo derate bounds on initial energies via mo derate b ounds on the initia l data . (iv) In pa rt C we plug in the information from the wa ve equation into the energy integrals in order to derive an energ y inequality . (v) Part D employs Gr onw all’s Lemma and shows that, if the initial energies of a ll orders are mo derate nets of real num b ers , then the same holds for all energ ies for all times 0 ≤ τ ≤ γ . (vi) Part E employs the Sobo lev embedding theo rem to show that the de s ired asymptotic g rowth pro per ties of the so lutions and their deriv atives follow from the respective growth o f energ ies of a ll order s. (vii) In P art F, a n existence and uniq ue ne s s result is achiev e d by putting the pieces A, B , C , D a nd E of the puzzle tog ether. (viii) In Part G we show that the solution is indep endent of the choice of (symmetric) r epresentativ es of the metric. It should b e ment ioned that Part A o f the metho d is the cr ucial part (the appro- priate sta temen t is lemma 1 in [ 49 ]); the rest o f the pr o of of the main theo rem basically follows the lines of [ 49 ], how ever, with a few modiﬁcatio ns. Instead of using a pseudo -foliation a s Vic kers a nd Wilso n (the three dimensiona l s ubmani- folds intersect in a t wo dimensional submanifold o f spa ce-time) we use the natural foliation Σ τ := { t = τ } stemming from the static co ordinates. F ur thermore, for the purp ose o f integration, we make use of the fact tha t the tensor-ﬁelds T k ab,ε ( u ) satisfy the domina nt energy c ondition. As a consequence of t he c ho sen f oliation, we do no t need to deal with improper in tegrals, as has been done in [ 49 ]. 3.3. The assumptions 3.3.1. In tro ductio n. Generalized static space-times. W e b egin with in- tro ducing a generalized static space-time. Deﬁnition 3.15. Let g ∈ G 0 2 ( M ) be a generalized Lorentz metric on M . W e say ( M , g ) is sta tic if the following tw o co nditions a re satisﬁed: (i) ( M , g ) is stationary , that is, there exists a smo oth time-lik e v ector ﬁeld ξ such that ∇ ( a ξ b ) = 0; this v ector ﬁeld w e call Killing as in the smoo th 44 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES setting and the one para meter group of isometrie s 1 generated by the ﬂow of ξ w e deno te by φ t . F ollowing the new concept of causa lit y in this generalized setting (Deﬁnition 2.16), ξ time-lik e mea ns t hat g ( ξ , ξ ) is a strictly nega tive gener alized function o n M (cf. Deﬁnition 2.42). (ii) There is a thr ee dimensio nal space-like h yper surface Σ thr ough each po int of M whic h is orthog onal to the or bits of the symmetry . An imp ortant observ a tio n is the follo wing: Theorem 3.16. L et ( M , g ) b e a gener alize d st atic sp ac e time. Then for e ach p oint p ∈ U ther e exist a r elatively c omp act op en lo c al c o or dinate chart ( U, ( t, x µ )) , p ∈ U , such that for e ach ε > 0 the gener alize d line element takes the form (3.20) ds 2 ε = − V 2 ε ( x 1 , x 2 , x 3 ) dt 2 + h ε µν ( x 1 , x 2 , x 3 ) dx µ dx ν = g ε ab dx a dx b wher e ( g ε ) ε is a suitable symmetric r epr esentative of g . Also in this setting we c al l the re sp e ctive c o or dinates static. F urther, V 2 ( x 1 , x 2 , x 3 ) is a strictly p ositive function, and h µν ( x 1 , x 2 , x 3 ) is a gener alize d Riema nnian metric on U . Proof. On a relatively compact op en neig hborho o d of p we pick a symmetric representative ( g ε ) ε of g such that on f or each ε > 0, g ε is Lor entz (cf. D eﬁnition 1.2 a nd Theore m 1.1 (iii)). F urther, denote by ∇ ε the cov ariant deriv ative induced by the metric g ε . T o show the claim w e pro ceed in t wo steps. Step 1. As in the standar d setting, an algebr aic manipulation s hows the equiv alence (3.21) L ξ g ab = 0 ⇔ ∇ ( a ξ b ) = 0 . Let p ∈ M lie in a r elatively compact neighborho o d Ω of Σ whic h c a n be rea ched by unique o rbits of ξ a through Σ. Cho os e a rbitrary co or dinates x µ lab eling Σ and let t b e the Killing par ameter. The n ( t, x µ ) ar e lo cal co or dinates near p 2 . In view of the ab ov e equiv alence (3.21) we ha ve a negligible symmetric tenso r ﬁeld ( n ε ab ) ε on Ω suc h that ( ∂ t g ε ab ( t, x µ )) ε = ( n ε ab ( t, x µ )) ε . Since Ω is re la tively compa c t, w e may replace g ε ab ( t, x µ )) ε by ˆ g ε ab ( t, x µ )) ε which is again a lo cal repr esentation of a suitable repr esentativ e of the metric, given for each ε by: ˆ g ε ab ( t, x µ ) := g ε ab ( t, x µ ) − Z t 0 n ε ab ( τ , x µ ) dτ . F or this r epresentativ e we hav e in static co o rdinates by deﬁnition: ( ∂ t ˆ g ε ab ( t, x µ )) ε = 0 . Step 2. Fina lly , we hav e to show that for a suitable re pr esentativ e ( e g ε ) ε , the ( t, x ) cross terms v anish. This is ea sily seen: B y the hypersurfa ce orthogo nality we know that h ∂ ∂ t , ∂ ∂ x µ i = g 0 µ = 0 in G ( ϕ (Ω)) for µ = 1 , 2 , 3 ((Ω , ϕ ) denoting the lo ca l ch art) Therefore we hav e neg ligible nets ( m ε 0 ,µ ) ε such that ˆ g ε µ, 0 = ˆ g ε 0 ,µ = m ε 0 ,µ . 1 T o see this, note that due to ident it y (3.21) we hav e L ξ g ≡ 0 in G . Therefore d dt (( F l ξ t ) ∗ g )( x ) = ( L ξ g )( F l ξ t ( x )) ≡ 0 in G . This implies that ( F l ξ t ) ∗ g = (( F l ξ 0 ) ∗ g )( x ) = g holds in G , and we hav e prov en that φ t is a generalized group of isometries of g . 2 T o see this, assume the con trary , that is ξ p = ξ | p ∈ T p Σ. Since Σ is space-like also ξ p is space-like , but this contradict s the assumption that ξ p is time-like. 3.3. THE ASSUMPTIONS 45 Since Ω was chosen to b e re la tively compact, we may even set the ( t, x µ ) c ross terms zero and s till have a lo cal repr esentation o f a suita ble r epresentativ e of g . W e hav e shown that the line element of the metric takes the form (3.20). A simple observ a tion is, that − V 2 = g ( ξ , ξ ), ther efore V 2 is a strictly positive function, and h µν is a generalized Riemannian metric.  This concludes the gener al dis cussion of generalize d space-times. F rom a the- oretical point of view, how ever, it is in teresting to further in vestigate characteri- zations o f genera lized space- times ( M , g ) via standar d space-times. W e ﬁnish this section with the following conjectur e Conjecture 3.17. On r elatively c omp act op en s ets, a gener alize d s t ationary s p ac e- time ( M , g ) admits a (symmet ric) r epr esentative ( g ε ) ε of t he metric g such that ( M , g ε ) is stationary (with Kil ling ve ct or ξ a ) for e ach ε > 0 . W e are now pr epared to pr e s ent the setting of this no te: 3.3.2. The setting. Assumptions on the metric. Throughout the rest of the chapter we suppose ( M , g ) is a g eneralized sta tic space- time. F urthermor e we shall w o r k on ( U , (( t, x µ )), ( p ∈ U ), a n open relatively compact c ha r t such that according to Theorem 3.16, ( t, x µ ) ar e static co o rdinates at p . ξ a shall denote the Killing vector ﬁeld on U and Σ is the three dimensiona l space-like hyper surface through p ∈ U , in static co or dinates given by t = 0 . Let m ab be a ba ckground Riemannian metric on U and denote by k k m the norm induced on the ﬁbres of the respec tive tensor bundle o n U . W e further imp ose the following assumptions on the metric g and the Killing vector ξ : (i) ∀ K ⊂⊂ U and for one (hence a n y) s ymmetric representativ e ( g ε ) ε we hav e: sup p ∈ K k g ε ab ( p ) k m = O (1) , sup p ∈ K k g ab ε ( p ) k m = O (1) ( ε → 0) . (ii) ∀ K ⊂⊂ U ∀ k ∈ N 0 ∀ ξ 1 , . . . , ξ k ∈ X ( U ) and for o ne (hence an y) symmetric repr esentativ e ( g ε ) ε we hav e: sup p ∈ K kL ξ 1 . . . L ξ k g ε ab k m = O ( ε − k ) ( ε → 0 ) . (iii) ∀ K ⊂⊂ U ∀ η ∈ X ( U ) : sup p ∈ K kL η ˆ ξ ε k m = O (1) , ( ε → 0) . where ( ˆ ξ ε ) ε := ξ √ − g ε ( ξ, ξ ) is a repr esentativ e o f the (g e neralized) o bserver ﬁeld ˆ ξ given b y ˆ ξ := ξ √ −h ξ ,ξ i . This is well deﬁned b y the fact that − g ( ξ , ξ ) = −h ξ , ξ i is a str ictly p ositive function on U , the squa re ro ot o f the latter is strictly p ositive as well, and this means p −h ξ , ξ i is inv er tible. Hence ˆ ξ in fact is a generalized unit v ector ﬁeld o n U , i. e., g ( ˆ ξ , ˆ ξ ) = − 1 in G ( U ). (iv) F or each symmetric representativ e ( g ε ) ε of the metric g on U , for suf- ﬁcient ly small ε , Σ is a past-compact s pace-like hypersurfa ce such that ∂ J + ε (Σ) = Σ. Here J + ε (Σ) denotes the top ologica l closure (with respect to the top olo gy inherited by U ) of the future emission D + ε (Σ) ⊂ U o f Σ 46 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES with resp ect to g ε . Mo reov er, there exists an o pen set A ⊆ M and an ε 0 such that A ⊆ \ ε<ε 0 J + ε (Σ) . Note, that (iv) is nece s sary to ensure exis tence of smo oth solutions on the level of r epresentativ es (cf. Theorem 3.1 3): F o r each suﬃcien tly sma ll ε there exists a unique smo o th function u ε on at lea st A ⊆ T ε<ε 0 J + ε (Σ). F urthermo r e the conditions (i)–(iii) are indep endent of the Riemannian metric m . Prop erty (iv) is an assumption o n e ach symmetr ic representative. A conjecture, how ever, is the following: Conjecture 3.18. If for one symmetric r epr esentative ( g ε ) ε of the metric g , for suﬃciently smal l ε , Σ is a p ast-c omp act sp ac e-like hyp ersurfac e such that ∂ J + ε ( S ) = S , so it is for every sy mmetric r epr esentative of the metric. In the remainder o f this section we interpret the setting of Deﬁnition 3.3.2 in ter ms of the static co ordina tes ( t, x µ ) o f Theo rem 3.1 6. With resp ect to these co ordinates, condition (i) means that a ll the co eﬃcients of g ε are b ounded b y a po sitive cons ta nt M 0 for suﬃciently small ε , and s o are the co eﬃcien ts of the inv erse of the metric. Finally , conditio n (ii) reads in static co ordinates ( t, x µ ): F or each k > 0 there exis ts a p o sitive constant M k such that for s uﬃcie ntly s ma ll ε w e hav e | ∂ ρ 1 . . . ∂ ρ k g ε ab | ≤ M k ε k , | ∂ ρ 1 . . . ∂ ρ k g ab ε | ≤ M k ε k , where ∂ ρ i ( i = 1 , 2 , 3) are partial deriv atives with resp ect to the space v ar iables x µ ( µ = 1 , 2 , 3); diﬀeren tiation with resp ect to time is no t in teresting, since in these co ordinates time dep endent contributions to t he metric coeﬃcients are negligible, anyw ay (cf. Theorem 3.16). Moreov er, condition (i) implies that there is a po sitive c onstant M suc h that for suﬃciently small ε w e hav e for the sca lar pr o duct o f the Killing vector ξ : (3.22) g ε ( ξ , ξ ) = g ε 00 = − V 2 ε ≤ − M < 0 . 3.3.3. The setting. F orm ulation of the i nitial v alue problem. Let v , w ∈ G (Σ). The initial v alue problem w e are interested in is the wa ve equation for u ∈ G ( M ) sub ject to the initial conditio ns :  u = 0 (3.23) u | Σ = v ξ a ∇ a u | Σ = w . An immediate consequence is that in static co ordinates ( t, x µ ) (cf. Theorem 3.16) which employ the Killing parameter t , on the lev el of representatives the initial v alue proble m (3.23) simply reads:  ε u ε = f ε (3.24) u ε ( t = 0 , x µ ) = v ε ( x µ ) ∂ t u ε ( t = 0 , x µ ) = w ε ( x µ ) , since Σ is lo ca lly parameterized as t = 0. Here ( f ε ) ε ∈ N ( ϕ (Ω)), and ( v ε ) ε , ( w ε ) ε ∈ E M ( ϕ (Ω ∩ Σ)) are lo ca l represe ntations o f arbitrary repres entativ es of v , w and  ε is the d’Alem b ertian with resp ect to an ar bitrary symmetric representativ e of g . 3.3. THE ASSUMPTIONS 47 How ever, from no w on we pic k a repr e sentativ e of the metr ic which in lo cal co ordinates tak es the for m of Theorem 3.16. Based on this choice w e establish an existence and uniqueness r e sult in the sense of Colo mbea u. Only in the last section we justify this choice in the sense that w e s how that choo sing any other s ymmetric representative would hav e lea d to the same generalized solution. Except f or Part A we also use the f act that ( u ε ) ε is a so lution of the initial v alue p roblem on th e level of r epresentativ es, i. e., u ε satisﬁes (3.2 4) for eac h ε . A remark on the setting is in order. W e hav e chosen the static setting bas ically for the reason that the initia l v alue pr oblem (3.23) can b e translated to (3.24) for each ε > 0. In particular, this mea ns that we can treat all equations in one and the sa me co ordina te patch ; in par ticular lo cal a symptotic estimates , which are required for a proof o f existence and uniqueness of the wav e equation, can be achiev ed nicely in co ordinates . How ever, in general, a con venien t coor dinate form of t he metric represen tative ( g ε ) ε cannot be achiev ed jointly for each ε > 0. F or instance, supp ose the mere a ssumption that we are giv en a generalized metric for which a three-dimensional submanifold Σ is space- like (in the se nse of chapter 2, Deﬁnition 2.1 6). Assume ( g ε ) ε is a sy mmetric r epresentativ e. Let p ∈ Σ. Then for each ε > 0 it is p ossible to in tro duce Gauss ian normal co or dina tes at p such that the metric can be written without ( t, x µ ) cro ss-terms (cf. Theore m 3.9). How ever, the metric g ε will in general dep end o n ε , the construction giv en in the mentioned theorem will therefore depend on the resulting geodesic s initially p e r p endicular to Σ; for diﬀerent ε they will not coincide in general. That means, for each ε > 0 there could emerge diﬀerent co or dinate charts, a nd the domain of these c harts might even shrink when ε → 0. 3.3.4. Lo cally foliated se mi-neig h b orho o d s . This section is dev oted to showing that in the chosen setting, for a ny p oint p ∈ Σ there is a compact semi- neighborho o d Ω γ which can b e foliated by space-like (in the gener a lized sense) hypersurfaces Σ t (cf. ﬁgur e 2). Throughout, we follo w the notation a s has b een set out in section 3 .1 .7. Ho wever, since the problem is a lo cal one, it suﬃces to c o nstruct the compac t r egion Ω γ (with s pa ce-like b oundar y througho ut) in a co ordina te c hart. F or the sa ke of simplicity w e will no t distinguish notationally b etw een the image of the foliated region inside the co o r dinate c hart a nd the foliated region o n the manifold. Let p ∈ M and let Σ be the initial surface through p , perp endicular to ξ a , the (smo oth) K illing vector. Due to Theorem 3.16 we have an op en rela tively co mpact co ordinate chart ( U, ( t, x µ )) a t p such tha t x µ ( p ) = 0, Σ is para meterized by t = 0 and U is fo liated by the spac e -like hypersur faces Σ τ : t = τ orthogonal to ξ = ∂ /∂ t . Due to Theorem 3.1 6 we may ﬁnd a r epresentativ e ( g ε ) ε such that the line element asso ciated to g ε reads in these co ordina tes for suﬃciently small ε ds 2 ε = − V 2 ε ( x α ) dt 2 + h ε µν ( x α ) dx µ dx ν . F urthermo re, w e hav e p ositive cons tants such that on all of U , M − 1 ≤ V − 2 ε ≤ M − 1 0 and | h µν ε | ≤ M − 1 0 for suﬃciently small ε . Let h > 0 , ρ > 0. W e take a para b oloid with boundary t = 0 and S ( t, x µ ) = 0 , t ≥ 0 , the zero level set o f the function S giv e n b y S := t − h 1 − P µ ( x µ ) 2 ρ 2 ! =: t − h (1 − k x k 2 ρ 2 ) , 48 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Figure 2. Lo cal folia tio n of space-time where height h and maximal radius ρ of the par ab oloid shall be determined in such a wa y that the b o unda ry S is spa c e-like with resp ect to the generalized metric (cf. below). Ω is the compact region with bo undaries S and S 0 , the subregio n of Σ, in co ordinates given b y t = 0, k x k ≤ ρ . Ω is therefor e foliated by the three dimensio nal hypersurfaces S ζ , the intersection of Σ ζ : t = ζ with Ω, all with nor mal vector ξ a . W e ﬁx for a ll that fo llows γ with 0 < γ < h , and call Ω γ the pa rt of Ω lying betw een t = 0 a nd t = γ . S Ω ,γ denotes the part of the b oundary S of Ω which lie s betw een t = 0 and t = γ . Ther efore, Ω γ has b oundar ie s S 0 , S γ and S Ω ,γ . Similarly , for 0 ≤ τ ≤ γ we use the notation S Ω ,τ , S 0 , S τ for the b oundarie s of Ω τ . Finally , we show that n a ε := g ab ε n b , the normal to S (hence to the subset S Ω ,γ ) given by the ( g ε -) metric equiv alent co vector dS , is time-lik e, if the ratio h/ρ ≤ 1 2 q M 0 6 M . In lo cal coor dinates we have dS = dt + 2 h ρ 2 δ ij x i dx j . Therefore (3.25) h n a ε , n a ε i ε = − V − 2 ε +  2 h ρ 2  2 h ij ε δ ik δ j l x k x l ≤ − V − 2 ε +  2 h ρ 2  2 (3 M − 1 0 k x k 2 ) . With P i ( x i ) 2 = k x k 2 ≤ ρ 2 we obtain b y means o f (3.25) the estimate h n a ε , n a ε i ε ≤ − 1 M + 12( h ρ ) 2 ≤ − 1 2 M for suﬃciently small ε . W e hav e shown that n a ε is time-like for each ε . In the generalized se nse of causa lit y which is established in chapter 2, this mea ns that n a := g ab n b is (ge neralized) space-like. 3.3.5. Energy integrals and Sob olev norms. Thro ug hout this and all s ub- sequent sections, we may ass ume that we hav e picked a p oint p ∈ Σ tog ether with a semi-neighborho od Ω γ which e ntirely lies in an op en rela tively compact c o ordinate patch ( U, ( t, x µ )), where ( t, x µ ) denote the static coo r dinates a t p , in which the metric g takes the for m (3.20) o n the level of represent atives. All the results will 3.3. THE ASSUMPTIONS 49 be prov ed on the level of r e pr esentativ es inside the c hosen co ordinate patch. Since the Killing vector ξ is a sta ndard vector ﬁeld, we may alwa ys take the c o nstant net ( ξ ) ε as a representativ e of ξ . W e hav e r evisited constructions of Riemannian metrics by means of Lor ent zian metrics in the preliminary section 3.1 and we ha v e further men tioned that there are analo gous constructions in the g eneralized setting (cf. chapter 2, section 2.4); these we a pply now in o rder to deﬁne Sob o lev norms and energy integrals. W e s hall deal with tw o diﬀerent speciﬁc co nstructions of Riemann metrics. F or the ﬁrst , we take g and ξ , the given K illing v e c tor, and deﬁne the Riemannian metric e ab := [( e ab ε ) ε ] on the level o f representativ es by (3.26) e ab ε := g ab ε − 2 g ε ( ξ , ξ ) ξ a ξ b = g ab ε + 2 V 2 ε ξ a ξ b . F or suﬃcient ly small ε > 0, e ab ε is a Riemann metric on U due to Corollary 3.3. F urthermo re e ab is even a gener a lized Riemannian metric on U : this follows, for instance, from the respective statemen t in the generalized s etting (cf. chapter 2, Theorem 2.4 3). How ever, since g ab has blo ck diagonal form in static co ordinates, and the metric constr uctio n (3.26) is quite simple, we can even directly co nﬁr m that e ab is a genera lized Riemannian metric. Indeed, due to the assumptions of the setting, the metric g has the line element ds 2 = − V 2 dt 2 + h µν dx µ dx ν , where h µν is a generalized Riemannian metric on Σ t ∩ U a nd g ( ξ , ξ ) = − V 2 is an inv ertible element of G ( U ) (which follows from the fa c t that g is assumed to b e non-degenera te). Therefor e, the line element of e takes the form ds 2 = + V 2 dt 2 + h µν dx µ dx ν . It follows that ds 2 is the line-element of a generalized Riemann metric on U . In the se c ond construction, g ab , ξ a and n a are in volved; ξ a and n a := g ab n b play the role of time-like vector ﬁelds in the construction (cf. Corolla ry 3 .3, how ever in the generalized setting: ξ a is the Killing vector (restric ted to S Ω ,γ = Ω γ ∩ S ) and n a is the normal to S Ω ,γ . W e deﬁne a Riemann metric on S Ω ,γ by (3.27) G cd := ( g a ( c g d ) b − 1 2 g ab g cd ) ξ a n b . Since both ξ a and n b are time-lik e with the same time-o rientation, G cd is a g e neral- ized Riemannia n metric on S Ω ,γ . This again follows from Theorem 2.43. W e have omitted to explicitly denote the restr ictions of g ab and ξ a to S Ω ,γ . O n the level of representatives G cd reads: (3.28) G cd ε := ( g a ( c ε g d ) b ε − 1 2 g ab ε g cd ε ) ξ ε a n b . W e proc e e d now to energy tensor s a nd ener g y int egrals. Let u now b e a smo o th function deﬁned on the coo rdinate patch U , and let ∇ ε denote the cov ariant deriv ativ e with resp ect to g ε for ea ch ε > 0. F or e ach non-negative integer k , we deﬁne energ y tensor s T ab,k ε ( u ) o n Ω γ as well as ener gies E k τ ,ε ( u ) on S τ (0 ≤ τ ≤ γ ) of order k as follows. F or k = 0 we set (3.29) T ab, 0 ε ( u ) := − 1 2 g ab ε u 2 . 50 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES F or k > 0 we deﬁne energy tensors T ab,k ε ( u ) := ( g ac ε g bd ε − 1 2 g ab ε g cd ε ) e p 1 q 1 ε . . . e p k − 1 q k − 1 ε (3.30) × ( ∇ ε c ∇ ε p 1 . . . ∇ ε p k − 1 u )( ∇ ε d ∇ ε q 1 . . . ∇ ε q k − 1 u ) . W e are now pr e pa red to deﬁne the e ne r gy in teg rals E k τ ,ε ( u ) via the energy tensors T ab,k ε ( u ) of any or der k . The energ y integral o f the k - th hierarchy is given by (3.31) E k τ ,ε ( u ) := k X j =0 Z S τ T ab,j ε ( u ) ξ ε a ˆ ξ ε b µ ε τ . Here µ ε τ is the unique three-for m induced on S τ by µ ε such that dτ ∧ µ ε τ = µ ε holds on S τ . (cf. [ 14 ], p. 6 6, Lemma 2.9.2) . F urthermore, ˆ ξ ε a := ξ a √ − g ε ( ξ, ξ ) . More over, it sho uld be noted tha t the tensors T ab,k ε ( u ) are symmetric tensors satisfying the dominant energ y condition. This ho lds due to P rop ositio n 3.6. Since ξ a is a Killing v ecto r ﬁe ld a nd T ab,k ε ( u ) is a symmetric vector satisfying the domina nt energy condition for each ε (cf. Prop os itio n 3.6), we hav e as a n application o f Stok e s ’s theorem (cf. (3.19)), (3.32) E k τ ,ε ( u ) ≤ E k τ =0 ,ε ( u ) + k X j =0 Z Ω τ ξ ε b ∇ ε a T ab,j ε ( u ) µ ε . The ineq uality is due to the fact that as a consequence o f the dominant ener gy condition, the integrand of the surface integral ov er S Ω ,γ is non negative, hence can be neglected. This inequa lity clear ly holds for eac h ε > 0 and each 0 ≤ τ ≤ γ . In the r emainder of the sectio n we intro duce Sob olev norms o n the co ordinate patch U . Let ε > 0 and 0 ≤ τ ≤ γ . The three dimensional Sob o lev-norms are int egrals of the cov a riant deriv ative ov e r S τ : (3.33) ∇ k u k k τ , ε :=   k X j =0 Z S τ |∇ ( j ) ε ( u ) | 2 µ ε τ   1 2 where, as usua l, the in tegrand is expres s ed by contraction of the cov a riant deriv ative of j th o rder of u with the Riemannian metric e ab : (3.34) |∇ ( j ) ε ( u ) | 2 := e p 1 q 1 ε . . . e p j q j ε ∇ ε p 1 . . . ∇ ε p j u ∇ ε q 1 . . . ∇ ε q j u. Similarly , the three dimens ional So bo lev norms inv olv ing par tial deriv a tives only , are: (3.35) ∂ k u k k τ , ε :=    X p 1 ...p j 0 ≤ j ≤ k Z S τ | ∂ p 1 . . . ∂ p j u | 2 µ ε τ    1 2 . On Ω τ we hav e the r esp ective (usual) Sobolev norms given b y (3.36) ∇ k u k k Ω τ , ε :=   k X j =0 Z Ω τ |∇ ( j ) ε ( u ) | 2 µ ε   1 2 3.4. EQUIV ALENCE: ENERGIES AND SOBOLEV NORMS 51 as well as (3.37) ∂ k u k k Ω τ , ε :=    X p 1 ...p j 0 ≤ j ≤ k Z Ω τ | ∂ p 1 . . . ∂ p j u | 2 µ ε    1 2 . 3.4. Equi v alence of ene rgy i n teg rals and Sobo lev norms (P art A) W e start by establishing that the three dimensional Sob olev norms and the energy in tegrals are equiv alent. In this section, inequalities a r e meant to hold for suﬃciently sma ll ε and for each 0 ≤ τ ≤ γ and for ea ch smoo th function u given inside the coo rdinate pa tch U . In the end of the s ection we shall give an int erpretation of these inequalities. The m ain statemen t of this section is the following (the re sp e ctive statemen t in conical space-times is ([ 49 ], Lemma 1)): Prop ositi o n 3.19. F or e ach k ≥ 0 , ther e exist p ositive c onstants A, A ′ such t hat for suﬃciently smal l ε we have E k τ ,ε ( u ) ≤ A ( ∇ k u k k τ , ε ) 2 (3.38) A ′ ( ∇ k u k k τ , ε ) 2 ≤ E k τ ,ε ( u ) (3.39) F or e ach k ≥ 1 , t her e ex ist p ositive c onst ant s B k , B ′ k such t hat for suﬃciently smal l ε we have ( ∇ k u k k τ , ε ) 2 ≤ B ′ k k X j =1 1 ε 2( k − j ) ( ∂ k u k j τ , ε ) 2 (3.40) ( ∂ k u k k τ , ε ) 2 ≤ B k k X j =1 1 ε 2( k − j ) ( ∇ k u k j τ , ε ) 2 (3.41) Mor e over, for k = 0 we cle arly h ave ( ∇ k u k 0 τ , ε ) 2 = ( ∂ k u k 0 τ , ε ) 2 . Before we present the pro of of the statement, we notice: (i) Note, that the term ” suﬃciently small ε ” in the statement in particular means that the index ε 0 from which on the inequalities a bove hold, de- pends on the or de r k : F or the la tter t wo inequalities this may happ en; the tw o ﬁrst inequalities pos sess a uniform ε 0 from w hich on they ho ld. (ii) The four inequalities ho ld for each 0 ≤ τ ≤ γ . (iii) Note that the s calar pro duct e p 1 q 1 ε . . . e p k q k ε η p 1 ...p k η q 1 ...q k and the euclidean scala r pro duct deﬁned on the coor dinate patch only , δ p 1 q 1 . . . δ p k q k η p 1 ...p k η q 1 ...q k are equiv ale nt on U for small ε in the sense that the resp ective norms ar e, that is: there ex ist p os itive constants C k, 1 , C k, 2 such that for suﬃcient ly small ε w e hav e C k, 1 δ p 1 q 1 . . . δ p k q k η p 1 ...p k η q 1 ...q k ≤ e p 1 q 1 ε . . . e p k q k ε η p 1 ...p k η q 1 ...q k (3.42) ≤ C k, 2 δ p 1 q 1 . . . δ p k q k η p 1 ...p k η q 1 ...q k . 52 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Similarly there exist p ositive constants D k, 1 , D k, 2 such tha t for suﬃ- ciently small ε we have D k, 1 δ p 1 q 1 . . . δ p k q k η p 1 ...p k η q 1 ...q k ≤ e ε p 1 q 1 . . . e ε p k q k η p 1 ...p k η q 1 ...q k (3.43) ≤ D k, 2 δ p 1 q 1 . . . δ p k q k η p 1 ...p k η q 1 ...q k . It is suﬃcient to show this for k = 1. Then (3.4 3) ma y b e refor- m ulated as follows: There exist positive constan ts C 1 , C 2 such that for suﬃciently small ε we have (3.44) C 1 δ ab η a η b ≤ e ε ab η a η b ≤ C 1 δ ab η a η b . Since there are p ositive constants M , M 0 such that for suﬃciently sma ll ε we hav e M ≤ V 2 ε ≤ M 0 , we may reduce the problem to the three spa ce dimensions (the g reek le tters therefore ranging b etw een 2 and 4): W e claim tha t on compact s ubregions of U we hav e for suﬃciently small ε : (3.45) C 1 δ µν η µ η ν ≤ h ε µν η µ η ν ≤ C 2 δ µν η µ η ν . Since we lo cally hav e | h ε µν | = O (1) ( ε → 0), the right hand inequality of (3.45) is tr ivial. The pro of o f the left ha nd inequality requires a little work: Let x range in a compa ct subset K of U . W e may assume that fo r small ε , h ε µν ( x ) h ν ρ ε ( x ) = δ ρ µ ( x ) + n ρ µ,ε ( x ) with neg ligible ( n ρ µ,ε ) ε , therefor e for a negligible ( n ε ( x )) ε (3.46) det( h ε ν ρ ( x )) = 1 + n ε ( x ) det( h ν ρ ε ( x )) , since det( h ν ρ ε ( x )) is inv ertible for suﬃciently small ε . Moreov er, since | h ν ρ ε ( x ) | = O (1), w e hav e | det( h ν ρ ε ( x )) | = O (1) holds on K . In view of this and the fact that ( n ε ( x )) ε is negligible (in particular we ma y assume that | n ε ( x ) | < 1 / 2 for all x in K and for small ε ), ther e exists a pos itive constant M ′ such that w e ha ve for all x and suﬃciently small ε (3.47) | det( h ε ν ρ ( x )) | ≥ 1 2 M ′ . F urthermo re we know that det( h ε ν ρ ( x )) = λ ε 2 ( x ) · · · · · λ ε 4 ( x ) , where λ ε i ( x ) ( i = 2 , 3 , 4 ) are the eig e n v alues o f h ε ν ρ ( x ) at x . Therefor e , b y using (3.47) and the fact that for ea ch i we have λ ε i = O (1), we see that there is a p ositive co nstant C 1 such that for eac h i = 2 , . . . , 4 w e hav e | λ ε i ( x ) | ≥ C 1 , whenever ε is small eno ugh. Since fo r suﬃciently small ε , h ε µν is sym- metric p o s itive deﬁnite, w e ha v e (3.48) inf η h ε µν ( x ) η µ η ν δ µν ( x ) η µ η ν = min i =1 ,...,n λ ε i ≥ C 1 , which proves the left inequality of (3.45) . Therefore we have shown that (3.44) holds. In a similar manner one can show the estimates (3.42), (3.43). W e are r eady to presen t a pro o f of P rop ositio n 3.1 9: 3.4. EQUIV ALENCE: ENERGIES AND SOBOLEV NORMS 53 Proof. Part 1: Ine qualities (3.38) and (3.39). T o establish these ineq ua lities, we c o nsider the cases k = 0 and k > 0 s e pa rately . F or k = 0, the situation is r elatively simple. W e hav e (3.49) T ab, 0 ε ( u ) ξ ε a ˆ ξ ε b = − 1 2 g ab ε ξ ε a ˆ ξ ε b u 2 = − 1 2 g ε ab ξ a ˆ ξ b u 2 = − 1 2 p − g ε ( ξ , ξ ) u 2 = V ε 2 u 2 . By the assumption on the metric, there exist po sitive co nstants M , M 0 such that for suﬃciently small ε (3.50) M ≤ V ε ≤ M 0 It follows that for A := M 0 / 2 and A ′ := M / 2 w e ha v e (3.51) A ′ u 2 ≤ T ab, 0 ε ( u ) ξ ε a ˆ ξ ε b ≤ Au 2 . Int egrating over S τ yields (3.52) A ′ ( ∇ k u k 0 τ , ε ) 2 ≤ E 0 τ ,ε ( u ) ≤ A ( ∇ k u k 0 τ , ε ) 2 and we a re done with k = 0. Next, we inv estiga te the case k > 0 . T o start with, note that ( g ac ε g bd ε − 1 2 g ab ε g cd ε ) ξ ε a ˆ ξ ε b =  ξ c ξ d − 1 2 h ξ , ξ i ε g cd ε  1 V ε (3.53) = 1 2 V ε  g cd ε + 2 V 2 ε ξ c ξ d  = = 1 2 V ε e cd ε . By the deﬁnition (3.30) o f T ab,k ε ( u ) and b y (3.53), we therefore hav e T ab,j ε ( u ) ξ ε a ˆ ξ ε b : =  ( g ac ε g bd ε − 1 2 g ab ε g cd ε ) ξ ε a ˆ ξ ε b  e p 1 q 1 ε . . . e p j − 1 q j − 1 ε × (3.54) × ( ∇ ε c ∇ ε p 1 . . . ∇ ε p j − 1 u )( ∇ ε d ∇ ε q 1 . . . ∇ ε q j − 1 u ) = = 1 2 V ε e cd ε e p 1 q 1 ε . . . e p j − 1 q j − 1 ε × × ( ∇ ε c ∇ ε p 1 . . . ∇ ε p j − 1 u )( ∇ ε d ∇ ε q 1 . . . ∇ ε q j − 1 u ) . By inequality (3.50) a nd (3.54), for all 1 ≤ j ≤ k we hav e for suﬃciently small ε (3.55) M 2 |∇ ( j ) ε ( u ) | 2 ≤ T ab,j ε ( u ) ξ ε a ˆ ξ ε b ≤ M 0 2 |∇ ( j ) ε ( u ) | 2 . Since A = M 0 2 , A ′ = M 2 , for a ll 1 ≤ j ≤ k w e hav e for suﬃciently small ε (3.56) A ′ |∇ ( j ) ε ( u ) | 2 ≤ T ab,j ε ( u ) ξ ε a ˆ ξ ε b ≤ A |∇ ( j ) ε ( u ) | 2 . Therefore we hav e by summing up (3.5 6) and (3.51) the follo wing es timate: (3.57) A ′ k X j =0  |∇ ( j ) ε ( u ) | 2  ≤ k X j =0 T ab,j ε ( u ) ξ ε a ˆ ξ ε b ≤ A k X j =0  |∇ ( j ) ε ( u ) | 2  . Int egration over S τ therefore yields (3.58) A ′ ( ∇ k u k k τ , ε ) 2 ≤ E k τ ,ε ( u ) ≤ A ( ∇ k u k k τ , ε ) 2 and we a re done with the pro of for k > 0. Part 2: Ine quality (3.40). 54 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES T o prov e this inequalit y , w e use the asymptotic gr owth be havior of par tial der iv a- tives of the metric as well a s the form ula whic h expresses the cov ar iant deriv ative of u in terms of partial deriv atives of u and Chris toﬀel s y m bo ls (see identit y (3.6 8) below). The case k = 0 is a trivia lity . Also , the ca se k = 1 is quite simple. Independently of ε we have ∇ ε a u = ∂ a u. There exists a constant M ′ 0 such that for s uﬃcient ly small ε w e ha ve | e ab ε | ≤ M ′ 0 . Therefore (3.40) holds for B ′′ 1 := 2 M ′ 0 , since (3.59) |∇ (1) ε u | 2 = e ab ε ∂ a u∂ b u ≤ 2 M ′ 0 X p ( ∂ p u ) 2 = B ′′ 1 X p ( ∂ p u ) 2 . With B ′ 1 := max(1 , B ′′ 1 ) we obtain (3.60) u 2 + |∇ (1) ε u | 2 ≤ B ′ 1 ( u 2 + X p ( ∂ p u ) 2 and integration ov er S τ yields ( ∇ k u k 1 τ , ε ) 2 ≤ B ′ 1 ( ∂ k u k 1 τ , ε ) 2 . This is the claim for k = 1. So let k = 2. Then (3.61) ∇ ε a ∇ ε b u = ∇ ε a ( ∂ b u ) = ∂ a ∂ b u − Γ c ab,ε ∂ c u. Since e ab ε = O (1) on Ω γ ⊇ Ω τ and Γ c ab,ε = O ( 1 ε ), there is a po sitive constant B ′′′ 2 such that (3.62) X p 1 p 2 |∇ ε p 1 ∇ ε p 2 u | 2 ≤ B ′′′ 2 1 ε 2 X p ( ∂ p u ) 2 + X p 1 p 2 ( ∂ p 1 ∂ p 2 u ) 2 ! . Using the rig h t hand side of (3.4 2) we co nc lude that for the po sitive cons tant B ′′ 2 := C 2 , 2 B ′′′ 2 and for s uﬃcient ly sma ll ε we hav e (3.63) |∇ (2) ε u | 2 ≤ B ′′ 2 1 ε 2 X p ( ∂ p u ) 2 + X p 1 p 2 ( ∂ p 1 ∂ p 2 u ) 2 ! . As a consequence of (3.60) and (3.63), there exists a p ositive co nstant B ′ 2 such that for suﬃciently small ε > 0 we have: (3.64) u 2 + |∇ (1) ε u | 2 + |∇ (2) ε u | 2 ≤ B ′ 2 1 ε 2 ( u 2 + X p ( ∂ p u ) 2 ) + ( u 2 + X p ( ∂ p u ) 2 + X p 1 p 2 ( ∂ p 1 p 2 u ) 2 ) ! and in tegra ting this inequality ov er S τ yields the claim for k = 2. T he pro o f for arbitrar y k is inductiv e . 3.4. EQUIV ALENCE: ENERGIES AND SOBOLEV NORMS 55 W e claim that for each 2 ≤ j < k we may write the j th cov ariant deriv ative of u as ∇ ε a 1 . . . ∇ ε a j u = ∂ a 1 . . . ∂ a j u + (3.65) + X b 1 ,...,b j − 1 B b 1 ...b j − 1 a 1 ...a j , ε ∂ b 1 . . . ∂ b j − 1 u + + X b 1 ,...,b j − 2 B b 1 ...b j − 2 a 1 ...a j , ε ∂ b 1 . . . ∂ b j − 2 u + + . . . + X b 1 B b 1 a 1 ...a j , ε ∂ b 1 u, with functions (deﬁned in the co ordinate patch ( U, ( t, x i ))): B b 1 ...b j − r a 1 ...a j , ε , 1 ≤ r ≤ j − 1 , where for ea ch non-neg ative integer m , w e hav e the following growth estimate on compact sets: (3.66) B b 1 ...b j − r a 1 ...a j , ε = O  1 ε r  . Of course , some o f the co eﬃcient functions B b 1 ...b j − r a 1 ...a j , ε might even v anish. W e use the induction principle for the pr o of o f this sub claim. The inductiv e basis k = 2 holds due to for m ula (3.61). F or the inductive step, ba sically tw o ingredients are needed: Firs t, the as ymptotic growth o f the Chr istoﬀel symbols on compact subsets of U when ε → 0, which for ev e ry non-neg ative integer m is (3.67) ∂ ρ 1 . . . ∂ ρ m Γ c ab,ε = O  1 ε m +1  . This formula follows by induction directly fr om the asymptotic gr owth of the metric co eﬃcients, the coe ﬃcie n ts of the inv erse o f the metric a nd their deriv atives. The second ing redient is the co o rdinate formula for the cov ariant deriv ative of a tensor of t ype (0 , n ), which is: (3.68) ∇ a ω b 1 ...b n = ∂ a ω b 1 ...b n − n X j =1 Γ c ab j ω b 1 ...b j − 1 cb j +1 ...b n . By using the tw o in gredients (3.67) and (3.68) the pr o of of claim (3 .65) is ea sily prov e n for j = k . Having s howed deco mpo sition (3.6 5) for ea ch non-neg ative integer k , the pro of of inequalit y (3.40) lies a t hand. One only nee ds the right hand side of e s timate (3.42). Then it follo ws by (3.65) that for so me p ositive co nstant A ′′ k |∇ ( k ) ε | 2 ≤ B ′′ k   X 0 ≤ j ≤ k 1 ε 2( k − j ) X p 1 ...p j | ∂ p 1 ...p j u | 2 |   (cf. inequa lit y (3 .63) in the ca s e k = 2 ) a nd integration of the res pec tive inequality for j = 0 , . . . , k over S τ yields inequa lit y (3.40) f or any k ≥ 2. W e are done with Part 2. Part 3. Ine quality (3.41). This problem is analo gous to inequalit y (3.40). O ne starts by expressing partial 56 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES deriv atives in terms o f co v ariant deriv atives using iden tity (3.6 8). Then one may ﬁnd estimates of squares of partial deriv atives via squares of cov ar iant deriv atives of the resp ective order s. Finally we may use the left ha nd inequality of (3.42) given in the preceding remark and integrate the achiev ed inequalities ov er S τ and we are done.  3.5. Bounds on in i tial ene rgies via bo un ds on i ni tial data (P art B) T o start with, we esta blish a symptotic estimates of deriv atives of arbitrar y order of the smoo th net ( u ε ) ε on (the co mpact set) S 0 . This may b e used la ter to establish the asymptotic growth behavior of the initial energies E k τ =0 ,ε ( u ε ). The following notation is useful: Deﬁnition 3.20. Let O b e an op en subset o f R n and let K ⊂⊂ O be a compact subset. A net ( g ε ) ε of smooth functions on O is said to satisfy moderate bounds on K , if there exis ts a num b er N such that sup x ∈ K | g ε ( x ) | = O ( ε N ) ( ε → 0 ) . In our ma in reference [ 49 ] (cf. pp. 1 341-1 344), the set o f suc h functions is denoted by E M ( K ), ho w ever, since this notation is mis le ading, we shall not use it. W e s hall esta blis h mo derate (r esp. negligible) bounds in all deriv atives o f the net of solutions ( u ε ) ε on a ﬁxed compact set only , namely Ω γ .The ﬁrst step is to establish mo der ate (re s p. neg ligible) bounds of ( u ε ) ε on S; this is the sub ject of this section. W e may go on now by reca lling that due to Pro po sition 3.1 2 the d’Alembertian takes the follo wing form in static co ordina tes : (3.69)  ε u ε = − V − 2 ε ∂ 2 t u ε + | g ε | − 1 / 2 ∂ α  | g ε | 1 / 2 g αβ ε ∂ β u ε  . W e may manipulate equation (3.69) by using  ε u ε = f ε and receive a formula for the seco nd deriv ative o f u ε : (3.70) ∂ 2 t u ε = − V 2 ε  f ε − | g ε | − 1 / 2 ∂ α  | g ε | 1 / 2 g αβ ε ∂ β u ε  . In order to der ive asymptotic b ounds on initial energies we shall need the following statement: Prop ositi o n 3.21. If ( v ε ) ε , ( w ε ) ε (as int ro duc e d in (3.24 ) ) satisfy mo der ate (r esp. ne gligible) b ounds on S 0 in al l derivatives, t hen for al l j, k ≥ 0 the deriv ative ( ∂ j t ∂ ρ 1 . . . ∂ ρ k u ε ) ε satisﬁes mo der ate (r esp. ne gligible b ounds) on S 0 . Proof. Part 1 Recall that ( f ε ) ε is negligible. W e star t by pro ving the estimates on S 0 for time deriv atives of ( u ε ) ε only . The inductive hypothesis is : If ( v ε ) ε , ( w ε ) ε satisfy mo d- erate (resp. negligible) b ounds on S 0 , so do es f or each j ≥ 0, the net ∂ j t u ε (0 , x α ). The inductiv e ba s is ma y b e j = 0 or j = 1: In these cases , the sta tement holds trivially: Due to the initial v alue formulation (3.24), w e ha ve ∂ 0 t u ε (0 , x α ) = u ε (0 , x α ) = v ε ( x α ) and ∂ t u ε (0 , x α ) = w ε ( x α ) 3.5. BOUNDS ON INITIAL ENERGIES 57 which a re b oth mo derate (r e sp. neg ligible) due to our assumption. E mploying the fact that ( u ε ) ε solves (3.2 4) as well a s the identit y (3.70), we have: (3.71) ∂ 2 t u ε (0 , x α ) = − V 2 ε  f ε (0 , x α ) − | g ε | − 1 / 2 ∂ α  | g ε | 1 / 2 g αβ ε ∂ β  v ε  . Here we have only explicitly written down the indep endent v ar iables, if the resp. functions a r e not functions of the s pa ce-v aria bles only . T o co nﬁrm that the claimed e s timates hold for the second deriv ative with re- sp ect to time, we only need to kno w that the pro duct of a net having moderate (resp. neglig ible) b ounds with a net having mo der ate b ounds, has mo derate (r esp. negligible) b ounds . There fo re, the hypothesis holds for or der j = 2 as w e ll, since ( v ε ) ε satisﬁes mo dera te (resp. neg ligible) b ounds (and of co urse, the representativ es of the metric coeﬃcients are modera te by deﬁnitio n, and so are the determinan t and its in verse). F or the inductive step, a s sume that for 2 ≤ j < m ( m ≥ 2) the des ired asymptotic g r owth is kno wn on S 0 . Diﬀerentiating equation (3.70) m − 2 times with res pec t to time yields: (3.72) ∂ m t u ε = − V 2 ε  ∂ m − 2 t f ε − | g ε | − 1 / 2 ∂ α  | g ε | 1 / 2 g αβ ε ∂ β ∂ m − 2 t u ε  . Here again we have used t he fact that V ε and the metric co e ﬃcie n ts are indep e n- dent of the time v a riable t . Due to the inductiv e hypothesis, ( ∂ m − 2 t u ε ) ε is mo dera te (resp. negligible), wherea s ( ∂ m − 2 t f ε ) ε is negligible by ass umption (since ( f ε ) ε is). By a similar rea soning as for the second deriv ative, w e ﬁnd that ( ∂ m t u ε ) ε satisﬁes mo derate (res p. negligible) b ounds on S 0 and we are done. Part 2 Estimates for the deriv a tives of ( u ε ) ε with resp e ct to space -v ariables a re easily achiev ed, since ( u ε (0 , x α )) ε = ( v ε ) ε is mo dera te (resp. neglig ible) due to our as- sumption, and deriv a tion with res p ect to s pace-v aria bles co mmu tes with ev aluatio n at t = 0. Part 3 It is left to be shown that mixed deriv a tives of ( u ε ) ε of an y order hav e mo der ate (resp. negligible) b ounds on S 0 . Her e again an inductiv e ar g ument as in the Part 1 applies. W e ﬁrst re wr ite (3.72) by using the Leibniz r ule: ∂ m t u ε = − V 2 ε  ∂ m − 2 t f ε − | g ε | − 1 / 2 ∂ α  | g ε | 1 / 2 g αβ ε  ∂ β ∂ m − 2 t u ε − g αβ ε ∂ α ∂ β ∂ m − 2 t u ε  . (3.73) W e ma y deﬁne the net G β ε ( x µ ) := | g ε | − 1 / 2 ∂ α  | g ε | 1 / 2 g αβ ε  . It is worth mentioning that ( G β ε ( x µ )) ε is a mo derate net in the coor dinate patch for each β = 1 , 2 , 3 . With this notation, (3.73) r e ads ∂ m t u ε = − V 2 ε  ∂ m − 2 t f ε − G β ε ( x µ ) ∂ β ∂ m − 2 t u ε − g αβ ε ∂ α ∂ β ∂ m − 2 t u ε  . (3.74) The inductive h yp othesis now sta tes that for each o rder m we have for each order k that ( ∂ ρ 1 . . . ∂ ρ k ∂ m t u ε )( t = 0 , x α ) is a mo derate (resp. negligible) function. The basis of induction is m = 0 holds according to Ca se 2. Assume therefore that the claim holds for 0 ≤ j ≤ m of order 58 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES of time-deriv atives of u ε . Diﬀeren tiating (3.7 4) with respect to time yie lds ∂ m +1 t u ε = − V 2 ε  ∂ m − 1 t f ε − G β ε ( x α ) ∂ β ∂ m − 1 t u ε − g αβ ε ∂ α ∂ β ∂ m − 1 t u ε  . (3.75) Now we ma y set t = 0 a nd diﬀerent iate k times with resp ect to the space- v ariables. By assumption, for each l ≥ 0, and for ea ch n ≤ m ( ∂ ρ 1 . . . ∂ ρ l ∂ n t u ε )( t = 0 , x α ) is a mo derate (resp. neg lig ible) function. Plugging this infor mation in to the right hand side of (3.75), we see tha t ( ∂ ρ 1 . . . ∂ ρ k ∂ m +1 t u ε )( t = 0 , x α ) has mo dera te (resp. neg ligible) b ounds for each k ≥ 0, and w e are done.  As a consequence of the preceding statement, we have Prop ositi o n 3.22. If ( v ε ) ε , ( w ε ) ε ar e mo der ate (r esp. ne gligible), then for e ach k the initial ener gies ( E k 0 ,ε ) ε ar e mo der ate (re sp. ne gligible) nets of r e al numb ers. Proof. This sta tement is a direc t consequence of the fo rm of the energ y inte- grals (rewr itten in ter ms of partial deriv atives using for mu la (3.68)).  3.6. Ene rgy i ne qualities (Pa rt C) F rom no w on, we will use the fact that ( u ε ) ε is a solution of (3.24) on Ω γ . W e start with the simplest cas e k = 1. Then we have the following inequa lity: Prop ositi o n 3.23. Ther e ex ist p ositive c onstants C ′ 1 and C ′′ 1 such that we have for e ach 0 ≤ τ ≤ γ and for suﬃciently smal l ε : (3.76) E 1 τ ,ε ( u ε ) ≤ E 1 0 ,ε ( u ε ) + C ′ 1 ( ∇ k f ε k 0 Ω τ , ε ) 2 + C ′′ 1 Z τ ζ =0 E 1 ζ ,ε ( u ε ) dζ . Proof. W e start with (3.32), whic h for k = 1 reads (3.77) E 1 τ ,ε ( u ε ) ≤ E 1 τ =0 ,ε ( u ε ) + Z Ω τ ξ ε b ∇ ε a T ab, 0 ε ( u ε ) µ ε + Z Ω τ ξ ε b ∇ ε a T ab, 1 ε ( u ε ) µ ε W e calculate the integrals on the right hand side of the inequalit y (3.77). F o r k = 0 the ener g y tensor is deﬁned by (3.78) T ab, 0 ε ( u ε ) = − 1 2 g ab ε u 2 ε . The cov ariant deriv ative is: ∇ ε a T ab, 0 ε ( u ε ) = − 1 2 ∇ ε a g ab ε u 2 ε − ( 1 2 g ab ε )(2 u ε ∇ ε a u ε ) = = 0 − u ε ∇ b ε u ε = − u ε ∇ b ε u ε (3.79) Moreov er, for k = 1 the ener gy tensor reads T ab, 1 ε ( u ε ) = ( g ac ε g bd ε − 1 2 g ab ε g cd ε ) ∇ ε c u ε ∇ ε d u ε . 3.6. ENERG Y INEQUALITIES 59 Therefore we obtain for the cov ariant deriv ative ∇ ε a T ab, 1 ε ( u ε ) = ( g ac ε g bd ε − 1 2 g ab ε g cd ε )( ∇ ε a ∇ ε c u ε ∇ ε d u ε + ∇ ε c u ε ∇ ε a ∇ ε d u ε ) = = ( g ac ε g bd ε − 1 2 g ab ε g cd ε )( ∇ ε a ∇ ε c u ε ∇ ε d u ε + ∇ ε c u ε ∇ ε d ∇ ε a u ε ) = = ∇ c ε ∇ ε c u ε ∇ b ε u ε + ( ∇ a ε u ε ∇ b ε ∇ ε a u ε − − 1 2 ∇ b ε ∇ ε c u ε ∇ c ε u ε − 1 2 ∇ d ε u ε ∇ b ε ∇ ε d u ε ) = = ∇ c ε ∇ ε c u ε ∇ b ε u ε = (  ε u ε ) ∇ b ε u ε = = f ε ∇ b ε u ε . (3.80) W e ma y now insert (3.79) and (3.8 0) into (3 .77). This yields E 1 τ ,ε ( u ε ) ≤ E 1 τ =0 ,ε ( u ε ) + Z Ω τ ξ ε b ∇ ε a  T ab, 0 ε ( u ε ) + T ab, 1 ε ( u ε )  µ ε = = E 1 0 ,ε ( u ε ) + Z Ω τ ξ ε b ∇ b ε u ε ( f ε − u ε ) µ ε = = E 1 0 ,ε ( u ε ) + Z Ω τ ξ a ∇ ε a u ε ( f ε − u ε ) µ ε . Using the Cauc h y Sch warz inequality we further obtain (3.81) E 1 τ ,ε ( u ε ) ≤ E 1 0 ,ε ( u ε ) +  Z Ω τ ( ξ a ∇ ε a u ε ) 2 µ ε  1 2  Z Ω τ ( f ε − u ε ) 2 µ ε  1 2 W e may no w estimate again by means o f the Cauch y Sch warz inequality for the scalar pro duct induced in eac h tangent s pace by e ε ab , (3.82) ξ a ∇ ε a u ε = g ε ab ξ a ∇ b ε u ε ≤ e ε ab ξ a ∇ b ε u ε ≤ p e ε ( ξ , ξ ) |∇ (1) ε u ε | . Note that the ﬁrs t inequalit y holds due to the fact tha t the diﬀerence be t ween the line elements o f g ε ab and e ε ab merely lies in the switc h of signs in the ﬁrst summand from − V 2 ε to + V 2 ε (therefore this inequality is trivial). F urthermo re, there exists a p ositive co ns tant C 1 such that p e ε ( ξ , ξ ) ≤ C 1 on Ω γ for suﬃciently small ε , b ecause ξ is smo o th, e ε ab is lo cally bo unded (b ecaus e g ε ab is in our setting) a nd Ω ⊂⊂ U . It follows that Z Ω τ ( ξ a ∇ ε a u ε ) 2 µ ε ≤ C 2 1 Z Ω τ |∇ (1) ε u ε | 2 µ ε . This information w e plug into (3 .8 1) and a chiev e E 1 τ ,ε ( u ε ) ≤ E 1 0 ,ε ( u ε ) + C 1  Z Ω τ |∇ (1) ε u ε | 2 µ ε  1 2 × ×  Z Ω τ f 2 ε µ ε  1 2 +  Z Ω τ u 2 ε µ ε  1 2 ! = = E 1 0 ,ε ( u ε ) + C 1  ∇ k u ε k 1 Ω τ , ε  2 + C 1 2  ∇ k f ε k 0 Ω τ , ε  2 , (3.83) where for the second integrand of the right hand side o f (3 .81) w e ha ve used the triangle inequality for the Sobo lev nor m (a nd further that a ( b + c ) ≤ ( a 2 + c 2 ) + b 2 2 ). 60 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Next we employ inequa lit y (3.39) of P rop osition 3.1 9: W e have (3.84)  ∇ k u ε k 1 Ω τ , ε  2 = Z τ ζ =0 ( ∇ k u ε k 1 ζ , ε ) 2 dζ ≤ 1 A ′ Z τ ζ =0 E 1 ζ ,ε ( u ε ) dζ . W e may set C ′ 1 := C 1 2 , C ′′ 1 := C 1 / A ′ . P lugging (3.8 4) into (3.83) yields the claim (3.76).  F or energies hierarchies larger than one, we simila rly have: Prop ositi o n 3.24. F or e ach k > 1 ther e exist p ositive c onstants C ′ k , C ′′ k , C ′′′ k such that for e ach 0 ≤ τ ≤ γ and suﬃciently smal l ε we have, E k τ ,ε ( u ε ) ≤ E k 0 ,ε ( u ε ) + C ′ k ( ∇ k f ε k k − 1 Ω τ , ε ) 2 + C ′′ k Z τ ζ =0 E k ζ ,ε ( u ε ) dζ + (3.85) + C ′′′ k k − 1 X j =1 1 ε 2(1+ k − j ) Z τ ζ =0 E j ζ ,ε ( u ε ) dζ . Before we prov e this pr op osition, we establish a couple of technical lemmas. The ﬁrst one gives a for m ula for the cov ariant deriv a tive o f the energy tensor T ab,k ε ( u ). F or the sa ke of simplicity , we o mit the smo othing parameter in the techn ical lemmas. Moreover, we write ∇ I u := ∇ p 1 . . . ∇ p k − 1 u and for t he tensor pro duct e I J := e p 1 q 1 . . . e p k − 1 q k − 1 . Lemma 3. 25. F or e ach k ≥ 2 , the diver genc e of T ab,k ( u ) = ( g ac g bd − 1 2 g ab g cd ) e I J ∇ c ∇ I u ∇ d ∇ J u c an b e written in the fol lowing form: ∇ a T ab,k ( u ) = ( ∇ a e I J )( g ac g bd − 1 2 g ab g cd ) ∇ c ∇ I u ∇ d ∇ J u (3.86) + e I J ( g bd ∇ d ∇ J u )( g ac ∇ a ∇ c ∇ I u ) (3.87) − 2 e I J ( ∇ d ∇ J u )( g ab g cd ∇ [ a ∇ c ] ∇ I u ) . (3.88) 3.6. ENERG Y INEQUALITIES 61 Proof. W e ha ve ∇ a T ab,k ( u ) = ( ∇ a e I J )( g ac g bd − 1 2 g ab g cd ) ∇ c ∇ I u ∇ d ∇ J u + g ac g bd e I J ( ∇ a ∇ c ∇ I u ∇ d ∇ J u + ∇ c ∇ I u ∇ a ∇ d ∇ J u ) (3.89) − 1 2 g ab g cd e I J ( ∇ a ∇ c ∇ I u ∇ d ∇ J u + ∇ c ∇ I u ∇ a ∇ d ∇ J u ) = = ( ∇ a e I J )( g ac g bd − 1 2 g ab g cd ) ∇ c ∇ I u ∇ d ∇ J u + g bd e I J ( g ac ∇ a ∇ c ∇ I u )( ∇ d ∇ J u ) + ∇ c ∇ I u ∇ c ∇ b ∇ I u − 1 2 ( ∇ d ∇ I u )( ∇ b ∇ d ∇ I u ) − 1 2 ( ∇ c ∇ I u )( ∇ b ∇ c ∇ I u ) = = ( ∇ a e I J )( g ac g bd − 1 2 g ab g cd ) ∇ c ∇ I u ∇ d ∇ J u + g bd e I J ( g ac ∇ a ∇ c ∇ I u )( ∇ d ∇ J u ) + ∇ c ∇ I u ∇ c ∇ b ∇ I u − ∇ c ∇ I u ∇ b ∇ c ∇ I u = ( ∇ a e I J )( g ac g bd − 1 2 g ab g cd ) ∇ c ∇ I u ∇ d ∇ J u + g bd e I J ( g ac ∇ a ∇ c ∇ I u )( ∇ d ∇ J u ) − 2 ∇ c ∇ I u ∇ [ b ∇ c ] ∇ I u = ( ∇ a e I J )( g ac g bd − 1 2 g ab g cd ) ∇ c ∇ I u ∇ d ∇ J u + g bd e I J ( g ac ∇ a ∇ c ∇ I u )( ∇ d ∇ J u ) − 2 ∇ d ∇ I u ∇ [ b ∇ d ] ∇ I u = ( ∇ a e I J )( g ac g bd − 1 2 g ab g cd ) ∇ c ∇ I u ∇ d ∇ J u + e I J ( g bd ∇ d ∇ J u )( g ac ∇ a ∇ c ∇ I u ) − 2 e I J ( ∇ d ∇ J u )( g ab g cd ∇ [ a ∇ c ] ∇ I u ) .  W e shall conside r (3.86), (3.87),(3.88) separ ately in the following lemmas. W e start with (3.86): Lemma 3. 26. On U we have k∇ ε a e I J ε k m = O (1) , ( ε → 0) . Proof. This follows directly from the assumptions on the Killing v e c tor ﬁeld ξ (cf. (3.22), and the ass umption (iii) on the metric in section 3.3.2) and the Leibniz rule: ∇ ε a e bc ε = ∇ ε a ( g ab ε − 2 h ξ , ξ i ε ξ b ξ c ) = − 2 ∇ ε a ξ b p −h ξ , ξ i ε ξ c p −h ξ , ξ i ε ! = − 2 ∇ ε a ξ b p −h ξ , ξ i ε ! ξ c p −h ξ , ξ i ε − 2 ∇ ε a ξ c p −h ξ , ξ i ε ! ξ b p −h ξ , ξ i ε = O (1) (3.90)  Next we inv estiga te (3 .8 8): 62 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Lemma 3. 27. − 2 ∇ [ a ∇ c ] ∇ p 1 . . . ∇ p k − 1 u = R d p 1 ac ∇ d ∇ p 2 . . . ∇ p k − 1 u + R d p 2 ac ∇ p 1 ∇ d . . . ∇ p k − 1 u + · · · + (3.91) + R d p k − 1 ac ∇ p 1 . . . ∇ p 2 . . . ∇ d u Proof. The pro of is a direct consequence of the Ricci Identities (3.92) − 2 ∇ [ a ∇ c ] X p 1 ...p k = R d p 1 ac X dp 2 ...p k + R d p 2 ac X p 1 d...p k + · · · + R d p k ac X p 1 p 2 ...d .  This concludes the algebra ic treatment of (3.88). What is left is to give an asymptotic estimate on compact sets. Firs t w e ne e d infor mation on the asymptotic growth of the Riemann tensor : Lemma 3.28. F or e ach c omp act set K in U and for e ach k ≥ 0 , ther e exist p ositive c onstants F k > 0 s u ch that fo r suﬃciently smal l ε the fol lowing holds on K : (3.93) | ∂ ρ 1 . . . ∂ ρ k R d,ε abc | ≤ F k ε 2+ k and (3.94) |∇ ε a 1 . . . ∇ ε a k R d,ε abc | ≤ F k ε 2+ k . Proof. (3.93) is an immedia te consequence of the formula for the co eﬃcients of the Riemann tensor in terms of Christoﬀel sym b ols (hence in terms of partial deriv atives of the metric co eﬃcients) and their asymptotic growth. F or (3.94) one needs in addition the formu la expressing the cov ariant deriv a tive in terms of pa rtial deriv atives and Christoﬀel symbols .  Lemma 3.27 and Lemma 3.2 8 in conjunction y ie ld: Lemma 3.29. F or e ach c omp act set K in U and for e ach k ≥ 2 , ther e exist p ositive c onstants G k > 0 s u ch that fo r suﬃciently smal l ε the fol lowing holds on K : | 2 ∇ ε [ a ∇ ε c ] ∇ ε p 1 . . . ∇ ε p k − 1 u | 2 ≤ G k ε 4 X p 1 ,...,p k − 1 |∇ ε p 1 ∇ ε p 2 . . . ∇ ε p k − 1 u | 2 . This is an immediate conclusion and therefore we o mit the pro of. Finally , w e inv estiga te term (3.87). The next calculation is a purely algebraic manipulation. Again, we omit to write down the smo othing par ameter ε explicitly . Lemma 3. 30. F or e ach k ≥ 2 , we have (3.95) g ac ∇ a ∇ c ∇ p 1 . . . ∇ p k − 1 u = ∇ p 1 . . . ∇ p k − 1  u + k − 1 X j =1 R ( k − 1 ,j ) u, wher e R ( k,j ) u r epr esents a line ar c ombination of c ontr actions of the ( k − j ) th c o- variant derivative of the Riemann tensor with t he j th c ovaria nt derivative of u , 0 ≤ j ≤ k . Proof. Before we start, we note tha t w e shall write R ( k,j ) u + R ( k,j ) u = R ( k,j ) u, 3.6. ENERG Y INEQUALITIES 63 to indicate that the sum of such linear combinations is a linear combination of the same t yp e (containing the same order of co v ariant de r iv atives o f the Riemann tensor and the function u ). In this sense , by the Leibniz rule we hav e: (3.96) ∇ p k k − 1 X j =1 R ( k − 1 ,j ) u = k X j =1 R ( k,j ) u. W e start by calcula ting the basis of induction, namely k = 2. Since the c o nnection is torsio n free, we hav e: (3.97) g ac ∇ a ∇ c ∇ p 1 u = g ac ∇ a ( ∇ p 1 ∇ c u − 2 ∇ [ p 1 ∇ c ] u ) = g ac ∇ p 1 ∇ a ∇ c u = ∇ p 1  u. So the cla im ho lds in the ca se k = 2 (s inc e the linea r co mb ination R (1 ,j ) is allow ed to v anish). F or the inductiv e step, assume (3.95) holds. T o manage the step k − 1 → k , we hav e to repeatedly use the Ricci identities (3.92) in order to sh uﬄe the co v ariant deriv ative indices of u . First, w e sh uﬄe the indices c, p 1 : g ac ∇ a ∇ c ∇ p 1 . . . ∇ p k u = = g ac ∇ a ∇ p 1 ∇ c ∇ p 2 . . . ∇ p k u − − 2 g ac ∇ a ∇ [ p 1 ∇ c ] ∇ p 2 . . . ∇ p k u = = g ac ∇ a ∇ p 1 ∇ c ∇ p 2 . . . ∇ p k u + + g ac ∇ a k X i =2 R d p i p 1 c ∇ p 2 . . . ∇ p i − 1 ∇ d ∇ p i +1 . . . ∇ p k u ! = g ac ∇ a ∇ p 1 ∇ c ∇ p 2 . . . ∇ p k u + k X j =1 R ( k,j ) u. Repe a ting the same pro cedure a second time b y sh uﬄing p 1 and a , w e receive (3.98) g ac ∇ a ∇ c ∇ p 1 . . . ∇ p k u = ∇ p 1 ( g ac ∇ a ∇ c ∇ p 2 . . . ∇ p k u ) + k X j =1 R ( k,j ) u. W e may now us e the induction h ypo thesis (3.95). Inserting in to (3.98) yields by means of (3 .9 6), g ac ∇ a ∇ c ∇ p 1 . . . ∇ p k u = ∇ p 1 . . . ∇ p k  u + ∇ p 1   k − 1 X j =1 R ( k − 1 ,j ) u   + k X j =1 R ( k,j ) u = = ∇ p 1 . . . ∇ p k  u + k X j =1 R ( k,j ) u. and we a re done.  The last helpful estimate we establish b efor e proving P rop osition 3.85 is the following: Lemma 3.31. F or e ach c omp act set K in U and for e ach k ≥ 2 , ther e exist p ositive c onstants G k > 0 s u ch that fo r suﬃciently smal l ε the fol lowing holds on K : (3.99) |R ( k − 1 ,j ) ε u | 2 ≤ G k ε 2( k − j +1) X q 1 ...q j 1 ≤ j ≤ k − 1 |∇ ε q 1 . . . ∇ ε q j u | 2 . 64 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Proof. The pro o f follows dir ectly fr o m Lemma 3.28 and the deﬁnition of R ( k − 1 ,j ) ε (a linear combination of cov ar iant deriv ativ es of the Riemann tensor of k − 1 − j order and cov aria nt der iv atives of u of o rder j ).  Finally , w e are prepared to prov e the main statement, Prop os ition 3 .24: Proof. W e sta rt with (3.32) , where w e insert the solution ( u ε ) ε of the wav e equation (3.2 4): (3.100) E k τ ,ε ( u ε ) ≤ E k τ =0 ,ε ( u ε ) + k X j =0 Z Ω τ ξ ε b ∇ ε a T ab,j ε ( u ε ) µ ε . Hence, fo r each ener g y hierar ch y m , w e hav e to es timate the div ergence of T ab,k ε ( u ε ) for each 2 ≤ k ≤ m (the case k = 1 has been pr oved in Prop osition 3 .23 and the case k = 0 can b e easily be derived fr om the information given in the proo f of Prop ositio n 3 .23). So let k ≥ 2. By Lemma 3.2 5, we have ∇ ε a T ab,k ε ( u ε ) = ( ∇ ε a e I J ε )( g ac ε g bd ε − 1 2 g ab ε g cd ε ) ∇ ε c ∇ ε I u ε ∇ ε d ∇ ε J u ε (3.101) + e I J ε ( g bd ε ∇ ε d ∇ ε J u ε )( g ac ε ∇ ε a ∇ ε c ∇ ε I u ε ) (3.102) − 2 e I J ε ( ∇ ε d ∇ ε J u ε )( g ab ε g cd ε ∇ ε [ a ∇ ε c ] ∇ ε I u ε ) . (3.103) W e estimate the asymptotic gr owth of a ll the three terms (3.101), (3.102), (3.103) by mea ns of the preceding lemmas. The ﬁr s t term (3.101) can b e es timate by means of Lemma 3.26 a s follows. F or eac h k there e x ists a constant T k such that for suﬃciently small ε w e hav e (3.104) | ( ∇ ε a e I J ε )( g ac ε g bd ε − 1 2 g ab ε g cd ε ) ∇ ε c ∇ ε I u ε ∇ ε d ∇ ε J u ε | 2 ≤ T k X p 1 ,...,p k |∇ ε p 1 . . . ∇ ε p k u ε | 2 . So we a re done with the ﬁrs t term. By Lemma 3 .30, we hav e (3.105) g ac ε ∇ ε a ∇ ε c ∇ ε p 1 . . . ∇ ε p k − 1 u ε = ∇ ε p 1 . . . ∇ ε p k − 1  ε u ε + k − 1 X j =1 R ( k − 1 ,j ) ε u ε , and Lemma 3 .31 provides the a symptotic gr owth behavior of the quantities P k − 1 j =1 R ( k − 1 ,j ) ε u ε . W e further may use that ( u ε ) ε solves the initial v alue pr oblem (3.24) on the level of representativ es; taking the co v a riant deriv ative k times this implies ∇ ε p 1 . . . ∇ ε p k − 1  ε u ε = ∇ ε p 1 . . . ∇ ε p k − 1 f ε . Hence, ther e exists a po sitive cons tant T ′ k such tha t the left s ide of (3.105) is bo unded for small ε by | g ac ε ∇ ε a ∇ ε c ∇ ε p 1 . . . ∇ ε p k − 1 u ε | 2 ≤ T ′ k |∇ ε p 1 . . . ∇ ε p k − 1 f ε | 2 + + T ′ k X q 1 ...q j 1 ≤ j ≤ k − 1 1 ε 2(1+ k − j ) |∇ ε q 1 . . . ∇ ε q j u ε | 2 . (3.106) 3.7. BOUNDS ON ENERGIES 65 F or ter m (3.1 03) we obta in b y Lemma 3.29 that lo cally ther e exists a cons ta nt G k > 0 such that for suﬃciently sma ll ε we have: (3.107) | 2 ∇ ε [ a ∇ ε c ] ∇ ε p 1 . . . ∇ ε p k − 1 u ε | 2 ≤ G k ε 4 X p 1 ,...,p k − 1 |∇ ε p 1 ∇ ε p 2 . . . ∇ ε p k − 1 u ε | 2 . W e ﬁnally may use the estimates (3.10 4), (3.10 6) and (3 .107) to estimate the ener- gies ( T a,b,k ε ( u ε ) ε ). This yields |∇ ε a T ab,k ε ( u ε ) | ≤ S k X p 1 ...p k |∇ ε p 1 . . . ∇ ε p k u ε | 2 + + S k X p 1 ...p k − 1 |∇ ε p 1 . . . ∇ ε p k − 1 f ε | 2 + S k 1 ε 2(1+ k − j ) X q 1 ,...,q j 1 ≤ j ≤ k − 1 |∇ ε q 1 ∇ ε q 2 . . . ∇ ε q j u ε | 2 . Summation ov er k = 1 . . . m and in tegration yields for po sitive constants C ′ m E m τ ,ε ( u ε ) ≤ E m 0 ,ε ( u ε ) + C ′ m   ( ∇ k u ε k m Ω τ , ε ) 2 + ( ∇ k f ε k m − 1 Ω τ , ε ) 2 + m − 1 X j =1 1 ε 2(1+ m − j ) ( ∇ k u ε k j Ω τ , ε ) 2   . (3.108) This may b e turned in to an energy inequality by Prop osition 3 .19 (3.39) and the information from section 3.1.6. Indeed, for each j , we hav e a p os itive co nstant A ′ j such that for small ε (3.109) ( ∇ k u ε k j Ω τ , ε ) 2 = Z τ ζ =0 ( ∇ k u ε k j τ , ε ) 2 dζ ≤ A ′ j Z τ ζ =0 E j ζ ,ε ( u ε ) dζ . Inserting (3.1 09) into (3.1 08) therefor e yields: E m τ ,ε ( u ε ) ≤ E m 0 ,ε ( u ε ) + C ′ m ( ∇ k f ε k m − 1 Ω τ , ε ) 2 + C ′′ m Z τ ζ =0 E m ζ ,ε ( u ε ) dζ + C ′′′ m m − 1 X j =1 1 ε 2(1+ m − j ) Z τ ζ =0 E j ζ ,ε ( u ε ) dζ . and the proo f is ﬁnished.  3.7. Bounds on energi es via bounds on ini tial energie s (P art D) If we apply Gr onw all’s Lemma to (3.85) we obtain: Prop ositi o n 3.32. F or e ach k ≥ 1 ther e exist p ositive c onstants C ′ k , C ′′ k , C ′′′ k such that we ha ve for e ach ε > 0 and fo r e ach 0 ≤ τ ≤ γ , (3.110) E k τ ,ε ( u ε ) ≤   E k 0 ,ε ( u ε ) + C ′ k ( ∇ k f ε k k − 1 Ω τ , ε ) 2 + C ′′′ k k − 1 X j =1 1 ε 2(1+ k − j ) Z τ ζ =0 E j ζ ,ε ( u ε ) dζ   e C ′′ k τ Note that C ′′′ 1 = 0 (this r efers to the empty s um when k = 1 ) A direct consequence of the preceding prop os ition is the following statement: 66 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Prop ositi o n 3.33. L et 0 ≤ τ ≤ γ . If for e ach k , the initial ener gy ( E k 0 , ε ( u ε )) ε determines a mo der ate (r esp. ne gligible) net of r e al numb ers, then also ( sup 0 ≤ ζ ≤ τ E k ζ , ε ( u ε )) ε is mo der ate (r esp. ne gligible ) for e ach k . 3 Proof. The proo f is inductive. The basis of induction is k = 1: In this case, the s um in the bra ck ets of inequality (3.1 10) is empt y , and since ( f ε ) ε is a negligible function (since it is the representativ e of zer o), the net of real num b ers ( ∇ k f ε k 0 Ω τ , ε ) ε is negligible. By the assumption, also ( E j =0 0 , ε ( u ε )) ε is moderate (resp. negligible). As a co nsequence o f inequality (3.110), (sup 0 ≤ ζ ≤ τ E 1 ζ , ε ( u ε )) ε is mo der ate (res p. negligible). The inductive step is similar: Assume for 0 ≤ j < k we know that (sup 0 ≤ ζ ≤ τ E j ζ , ε ( u ε )) ε is mo derate (res p. negli- gible). B y a s sumption, ( E k 0 , ε ( u ε )) ε is mo der ate (resp. negligible) as well. F ur ther- more, ( f ε ) ε is a negligible function (since it is the representativ e of zer o), hence the net of rea l num ber s ( ∇ k f ε k 0 Ω τ , ε ) ε is negligible. By applying inequality (3.1 10) we achiev e that (sup 0 ≤ ζ ≤ τ E k ζ , ε ( u ε )) ε is mo derate (resp. negligible) and we a re done.  3.8. Es timates via a Sob olev embedding theorem (P art E ) In or der to translate the bounds on the energies ( E j ζ ,ε ( u ε )) ba ck to bo unds on the nets ( u ε ) ε and its deriv a tives, we shall need the follo wing ”g eneralized” So b o lev lemma express ed in terms of the ener gies ( E j ζ ,ε ( u ε )). Lemma 3.34 . F or m > 3 / 2 , ther e exists a c onstant K , a nu mb er N and an ε 0 such that for al l φ ∈ C ∞ (Ω τ ) and for al l ζ ∈ [0 , τ ] and for al l ε < ε 0 we have (3.111) sup x ∈ S ζ | φ ( x ) | ≤ K ε − N sup 0 ≤ ζ ≤ τ E m ζ ,ε ( φ ) . Before we prov e the statement, we note that since the right hand side o f (3.111) is indep endent of ζ , the statement is eq uiv alent to (3.112) sup x ∈ Ω τ | φ ( x ) | ≤ K ε − N sup 0 ≤ ζ ≤ τ E m ζ ,ε ( φ ) . Proof. By ([ 1 ], Lemma 5.17), there exists 4 a co ns tant K such that for ea ch 0 ≤ ζ ≤ τ we hav e fo r m > 3 / 2 (3.113) sup x ∈ S ζ | φ ( x ) | ≤ K k φ k m,S ζ . with k φ k m,S ζ , the thr e e dimensional Sobo lev nor m on S ζ with the V olume form o f R 3 , that is , k φ k m,S ζ = Z S ζ X ρ 1 ,...,ρ j 0 ≤ j ≤ m | ∂ ρ 1 . . . ∂ ρ j φ | 2 dx 1 dx 2 dx 3 , 3 In the statemen t of [ 49 ], a typing er ror occurs, and instead of k , k − 1 is wr itten. F ur- thermore, f or to pro v e Proposi tion 3.35, it is no t suﬃcient to hav e mo derate resp. negligible nets ( E k τ , ε ( u ε )) ε , but the suprem um of the energies o ver all 0 ≤ ζ ≤ τ must be moderate r esp. negligible. 4 this follows f rom the fact that b oundary of the paraboloid Ω is Lipschitz 3.8. SOBOLEV ESTIMA TES 67 where partial deriv atives ar e only taken with r esp ect to s pace-v ar iables, that is tangential to S ζ for each 0 ≤ ζ ≤ τ . Note tha t the expression is not inv ariant for t wo reasons. The ﬁrst is tha t partial deriv a tives are inv olved and not cov ar iant deriv atives. Secondly , the v olume elemen t of R 3 is taken. W e shall, howev er , derive an estimate by inv ariant express ions, namely , the energ ies. Next, w e int ro duce the determinant of the metr ic into the Sobole v norms. Note that o n Ω γ , whic h is a compact set, the absolute v alue of the determinant of the metric | g ε | for suﬃcien tly small ε is bo unded from below by a ﬁxed power of ε . This follows fr om inv ertibilit y of the metric. In our ca se, how ever, where the metric a nd its inverse lo ca lly are O (1 ), there exists a p ositive constan t C and a ε 0 ∈ I such that for all ε < ε 0 we hav e (3.114) | g ε | 1 2 ≥ C holds on Ω γ . Therefore, for s mall ε and for all ζ , 0 ≤ ζ ≤ τ , we hav e the estimate (3.115) k φ k m,S ζ ≤ C − 1 Z S ζ X ρ 1 ,...,ρ j 0 ≤ j ≤ m | ∂ ρ 1 . . . ∂ ρ j φ | 2 | g ε | 1 2 dx 1 dx 2 dx 3 . Clearly , this can further b e estima ted by the cruder three dimensional Sob olev norm ∂ k φ k m ζ , ε , which resp ects also time-deriv atives. Therefore, w e may es tima te (3.115) by (3.116) ∀ ζ ∈ [0 , τ ] ∀ ε < ε 0 : k φ k m,S ζ ≤ C − 1 ( ∂ k φ k m ζ , ε ) . Inserting (3.1 16) into (3.1 13) yields the estimate (3.117) ∀ ζ ∈ [0 , τ ] ∀ ε < ε 0 : sup x ∈ S ζ | φ ( x ) | ≤ K C − 1 ( ∂ k φ k m ζ , ε ) . Finally w e apply Pr op osition 3.19 t wice, namely the estimates (3.41) and (3 .39). This yields a n um b er N ′ such that for suﬃciently sma ll ε and for a ll 0 ≤ ζ ≤ τ we hav e (3.118) sup x ∈ S ζ | φ ( x ) | ≤ ε − N ′ E m ζ ,ε ( φ ) . On the righ t side of (3.118) w e may no w take the suprem um o ver ζ ∈ [0 , τ ] and achiev e (3.119) sup x ∈ S ζ | φ ( x ) | ≤ ε − N ′ ( sup 0 ≤ ζ ≤ τ E m ζ ,ε ( φ )) .  The main statemen t of this s ection is the following: Prop ositi o n 3 .35. L et 0 ≤ τ ≤ γ . If for e ach k , (sup 0 ≤ ζ ≤ τ E k ζ ,ε ( u ε )) ε is mo der ate (r esp. ne gligible), then ( u ε ) ε satisﬁes mo der ate b ounds (ne gligible b ounds) on Ω τ . Proof. Inserting ( u ε ) ε int o (3.11 1) yie lds (3.120) sup x ∈ S τ | u ε ( x ) | ≤ K ε − N sup 0 ≤ ζ ≤ τ E m ζ ,ε ( u ε ) . Similarly , for higher deriv a tives of ( u ε ) ε , one ac hieves b ounds via higher ener g ies: (3.121) sup x ∈ Ω τ | ∂ ρ 1 . . . ∂ ρ k ∂ l t u ε ( x ) | ≤ K ε − N sup 0 ≤ ζ ≤ τ E m + k + l ζ ,ε ( u ε ) .  68 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES Figure 3. Choice of the op en set V for the existence result 3.9. Existence and uniqueness (Par t F) In this section we collect all the preceding material a nd pr ov e a lo c a l existence and uniqueness re sult for the wav e e q uation; this, how e ver, is based on the sp eciﬁc choice of representativ e of the metr ic g ab . In the next section w e show t hat the generalized solution do es indeed not dep end on the (symmetric) choice of the metric representative. T o b egin with, we note that the wav e equation fo r the static representativ e written down in co o rdinates is time reversible, meaning: the diﬀeren tial equatio n (3.24) is inv a r iant under a transformation of the form t 7→ − t . In other words: If ( u ε ( t, x i )) ε is a so lution of (3.24) for t ≤ 0, also ( u ε ( − t, x i )) ε solves (3.24), how ever for t ≥ 0 . Therefore, similarly as in the above w e ma y achiev e es tima tes for ( u ε ( t, x i )) for t ≤ 0. T he compact r egion o n which the es timates a re esta blished we call Ω − τ , 0 ≤ τ ≤ γ which is the (time–)reﬂected Ω τ (cf. ﬁgure 3 ). It is, howev er , also p ossible to deﬁne the Ω τ as in se c tion 3.1 .7 and apply Stokes’ theorem, thus rep eating the whole pr o cedure on estimating o f Part A to P art E, just with Ω τ replaced by Ω − τ , 0 ≤ τ ≤ γ . W e are now prepar ed to presen t the existence and uniqueness theo rem for the Cauch y proble m of the wav e equatio n in o ur setting: Theorem 3. 36. F or e ach p oint p in Σ ther e exists an op en neighb orho o d V ⊂ U on which a unique gener alize d solution u ∈ G ( V ) of t he initial value pr oblem (3.23) exists. 3.9. EXISTENCE AND UNIQUE NESS 69 Even thoug h there will be redundancies, w e shall present a detailed pr o of of the theor em. Proof. Let ( U, ( t, x µ )) be an open relatively compact static co o rdinate chart at p . By Theorem 3.16, we c ho ose a representativ e ( g ε ab ) ε of the metric whic h is static for each ε and whic h (for small ε ) satisﬁes the res p ective b ounds according to the s e tting. F ur thermore, a r epresentativ e ( e ε ab ) ε of e ab may b e directly constructed from the representativ e ( g ε ab ) ε of the metric. Under these conditions Prop o sition 3.19 may be applied. On the lev el of representativ es the initial v alue problem (3.23) tak es the form (3.24) with ( f ε ) ε negligible, and ( v ε ) ε , ( w ε ) ε mo derate. Part 1. Existenc e of a lo c al mo der ate net of solutions The smo oth theory then pr ovides smo o th solutions ( u ε ) ε on U . W e ﬁrst show that the net ( u ε ) ε satisﬁes m o derate bounds on Ω γ : Mo derate data ( v ε ) ε , ( w ε ) ε translate by means of Pro po sition 3.22 to mo der ate initial energies ( E k 0 ,ε ( u ε )) ε for ea ch hierarchy k . Moreover, by means of Prop osition 3.32, mo derate initial energies ( E k 0 ,ε ( u ε )) ε ( k ≥ 1) tra nslate to moderate energies ( E k τ ,ε ( u ε )) ε ( k ≥ 1), where 0 ≤ τ ≤ γ , this is the s ta tement of Pr op osition 3.33. Fina lly Pro po sition 3.35 states that mo derate energies ( E k τ ,ε ( u ε )) ε ( k ≥ 1 , 0 ≤ τ ≤ γ ) translate to mo derate bo unds of ( u ε ) ε and of its deriv a tives of all orders on Ω γ . Due to the preceding in tr o ductory remark, estimates of the sa me kind hold on Ω − γ . W e pic k an op en subset V of Ω − γ , γ := Ω − γ ∪ Ω γ (see ﬁgure 3 .9). Due to our considerations in the be ginning o f Part B (section 3 .5), we hav e ther efore established that ( u ε ) ε is a mo dera te net on V . Part 2. Uniqueness of solutions W e ma y now deﬁne a lo cal genera lized solution u on V by u := [( u ε ) ε ] , the cla ss of ( u ε ) ε from Part 1. What is le ft to b e shown is that the so lutio n u do e s not dep end o n the c hoice of re presentativ e s of ( f ε ) ε , ( v ε ) ε , ( w ε ) ε of f ≡ 0 , v , w . Let ther efore ( ˆ f ε ) ε , ( ˆ v ε ) ε , ( ˆ w ε ) ε be further represen tatives of f ≡ 0 , v , w , and let ( ˆ u ε ) ε be the re s pe c tive net of smo oth so lutio ns. Setting e u ε := u ε − ˆ u ε , e f ε := f ε − ˆ f ε , e v ε := v ε − ˆ v ε , e w ε := w ε − ˆ w ε , we see that for each ε > 0 e u ε is a solution of the initial v alue problem  ε e u ε = e f ε e u ε ( t = 0 , x µ ) = e v ε ( x µ ) ∂ t e u ε ( t = 0 , x µ ) = e w ε ( x µ ) , Note, that here a ll the nets ( e f ε ) ε , ( e v ε ) ε , ( e w ε ) ε are negligible. Wha t is left to s how is that the net ( e u ε ) ε is neg ligible, as well; uniqueness of the ab ov e deﬁned solution u is then obvious, s inc e [( e u ε ) ε ] = [( u ε ) ε ] = u . Negligible da ta ( v ε ) ε , ( w ε ) ε translate by mea ns of Prop osition 3.22 to negligible initial energies ( E k 0 ,ε ( u ε )) ε for each hier a rch y k . Mo r eov e r , by means of Prop osition 3.32, negligible initial ener gies ( E k 0 ,ε ( u ε )) ε ( k ≥ 1) translate to negligible energies ( E k τ ,ε ( u ε )) ε ( k ≥ 1), where 0 ≤ τ ≤ γ , this is the statemen t of P rop osition 3.33. Finally Prop ositio n 3.35 states that neglig ible energies ( E k τ ,ε ( u ε )) ε ( k ≥ 1 , 0 ≤ τ ≤ 70 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES γ ) tr anslate t o a negligible bounds of ( u ε ) ε and of its deriv ativ es of all or ders on Ω γ . Due to the preceding introducto ry r emark, estimates of the sa me kind hold o n Ω − γ . Due to our consider ations in the beginning of Part B (section 3.5), w e hav e therefore established that ( u ε ) ε is a negligible net on V . This prov es uniqueness of the so lutio n u on V .  3.10. Dep endence on the representa tiv e of the metric (P art G) So far , we hav e prov ed that on V ⊂ Ω γ ∪ Ω − γ , a unique so lution to the initial v alue proble m exists . W e had, howev er, pick ed a speciﬁc symmetric r epresentativ e ( g ε ab ) ε of the metric g ab (to b e mor e precise , these are coo rdinate expres s ions o f the metric compo nents) and w orked with one and the same all the time. It is, therefore, advisable, to sho w that the gener alized solution u of the wav e equation is indep endent of the choice of the representative of the metric. This is the aim of this section. There is only one further a ssumption we imp ose o n the r epresentativ es ( g ε ab ) ε of the metr ic: they sha ll be symmetric (cf. the note in the end o f the se c tion). The initial v alue problem with resp ect to ( g ε ab ) ε is the follo wing:  ε u ε = f ε (3.122) u ε ( t = 0 , x α ) = v ε ( x α ) ∂ t u ε ( t = 0 , x α ) = w ε ( x α ) Now, let (ˆ g ε ab ) ε be another s y mmetric repre s ent ative of g ab . W e call ˆ  ε the d’Alem ber tian oper ator induced by (ˆ g ε ab ) ε . The initial v alue problem with resp ect to the latter reads quite similarly ˆ  ε ˆ u ε = f ε (3.123) ˆ u ε ( t = 0 , x α ) = v ε ( x α ) ∂ t ˆ u ε ( t = 0 , x α ) = w ε ( x α ) . W e may pause he r e for a momen t a nd co nsider why Pr op osition 3.19 (and therefore all subsequent statemen ts based o n the la tter) a lso holds true for the alternative choice ( ˆ g ε ab ) ε of metric re presentativ e : First, the diﬀerence betw een ( ˆ g ε ab ) ε and the static representative ( g ε ab ) ε (according to Theorem 3.16) is negligible by deﬁnition. As a co nsequence the diﬀerence betw e e n es tima tes established on compact sets and with res pe ct to these diﬀerent r epresentativ e is neglig ible . Since we only w or k on the compact regio n Ω γ , the estimates acco r ding to Prop osition 3.19 hold as well for other (symmetric) representativ es of the metr ic a nd for small ε ; how ev er, presumably with modiﬁed positive cons ta nt s A, A ′ , B k , B ′ k . The pro o f of Theorem 3.36 (Part 1) provides mo derate s olutions ( u ε ) ε and ( ˆ u ε ) ε of (3.122) and (3.123). It is o nly left to show that the diﬀerence ( e u ε ) ε := ( u ε ) ε − ( ˆ u ε ) ε is negligible on Ω τ . F or this diﬀerence we hav e ˆ  ε e u ε = f ε − ˆ  ε u ε (3.124) e u ε ( t = 0 , x α ) = 0 ∂ t e u ε ( t = 0 , x α ) = 0 . In view of the pro of of Theo rem 3.36 (Part 2) we only need to show that f ε − ˆ  ε u ε is negligible. T o this end we ﬁrst manipulate the right hand side of line 1 of (3.124) 3.11. POSSIBLE GE NERALIZA TIONS 71 as follows: (3.125) f ε − ˆ  ε u ε = ( f ε −  ε u ε ) + (  ε u ε − ˆ  ε u ε ) =  ε u ε − ˆ  ε u ε , bec ause ( u ε ) ε solves (3.122). Ther efore the pro blem is reduced to showing that (  ε u ε − ˆ  ε u ε ) ε is negligible. W e calculate the diﬀerence in lo c a l coo rdinates. W e use | det g ε ij | := | g ε | = − g ε and for the sake of simplicit y w e further omit the index ε . The diﬀerence then reads:  u − ˆ  u = ( − g ) − 1 2 ∂ a (( − g ) 1 2 g ab ∂ b u ) − ( − ˆ g ) − 1 2 ∂ a (( − ˆ g ) 1 2 ˆ g ab ∂ b u ) = (3.126)  ( − g ) − 1 2 ∂ a (( − g ) 1 2 g ab ∂ b u ) − ( − ˆ g ) − 1 2 ∂ a (( − g ) 1 2 g ab ∂ b u )  + +  ( − ˆ g ) − 1 2 ∂ a (( − g ) 1 2 g ab ∂ b u ) − ( − ˆ g ) − 1 2 ∂ a (( − ˆ g ) 1 2 ˆ g ab ∂ b u )  = (( − g ) − 1 2 − ( − ˆ g ) − 1 2 ) ∂ a (( − g ) 1 2 g ab ∂ b u ) + ( − ˆ g ) − 1 2 ∂ a  ( − g ) 1 2 g ab − ( − ˆ g ) 1 2 ˆ g ab  ∂ b u The diﬀerences within the brack ets of the la st line of (3.12 6) can easily b e s hown to b e negligible. Indeed, since ( g ε ab − ˆ g ε ab ) ε is neglig ible, also g ε − ˆ g ε is neglig ible, therefore, as can b e seen b y the following e le mentary alge br aic manipulation, the diﬀerence (3.127) ( − g ε ) − 1 2 − ( − ˆ g ε ) − 1 2 = g ε − ˆ g ε √ g ε ˆ g ε ( √ − ˆ g ε + √ − g ε ) is negligible. Also (3.128) √ − g ε g ab − p − ˆ g ε ˆ g ab ε = √ − g ε ( g ab ε − ˆ g ab ε ) + ˆ g ab ε ˆ g ε − g ε √ − g ε + √ − ˆ g ε is neglig ible . Plugging (3.127) and (3.128) into (3.12 6), we deriv e that (  ε u ε − ˆ  ε u ε ) ε is negligible, and by iden tit y (3.12 5), ( f ε − ˆ  ε u ε ) ε is a negligible net of smo oth funct ions as w e ll. This is the r ight hand side of the diﬀerential eq uation (3.124). Therefor e, Part 2 of the pro o f of Theor em 3.36) ensures that ( e u ε ) ε = ( u ε − ˆ u ε ) ε is negligible and we a re done. It g o es without s aying that no n-symmetric p er turbations of the metric are no t relev ant. Another fo rmulation o f the latter would be the following: The prese n t metho d for s olving the initial v alue pro blem (3 .23) basically lies in showing the existence result on the level o f r epresentativ es given an arbitrary choice of repre- sentativ es of the initial data as a w ell as a symmetric representative of the metric. The resulting gener alized solution do es not dep end on the choice of symmetric rep- resentativ es of the metric and neither do es it dep end on the choice of representativ es of the initia l data . 3.11. Possible g eneralizations W e ﬁnish this c ha pter by p ointing out po ssible improvemen ts of Theo rem 3.36 concerning generality of the statement as w ell a s reducing the lis t of necessary assumption on the metric as given in s ection 3.3.2. First w e co njecture th at condition (iv) in section 3.3.2, whic h guarantees ex- istence of smo oth solutions on the level of r epresentativ es (that is with r esp ect to each suﬃcien tly small ε –comp onent of the r e presentativ e of the metric ), pr esumably follows from co ndition (i). Moreov er, w e believe that the Cauchy problem (3.23) a lso admits u nique so- lutions in the sp ecial a lgebra of gener a lized functions even if t he condition (i) ar e 72 3. THE W A VE EQUA TION ON SINGULAR SP ACE-T IMES weak ened to logarithmic growth pr op erties of the metr ic coeﬃcients. In this ca se, the constants A, A ′ , B k , B ′ k of Pro po sition 3.19 might dep e nd on ε , say A ( ε ) = A log( ε ) with a po sitive constant A etc. . Therefore, a later application of Grown- wall’s Lemma w o uld yield modera te g rowth o f energies of a rbitrary o rder, since ( e A log ε ) ε = ( ε A ) ε is mo derate. CHAPTER 4 P oin t v alues and uniqueness qu estions in algebras of generalized functions 4.1. P oi n t v alue c haracterizations of ultrametric Egorov algebras As already men tioned in the in tro duction, a distinguishing feature (compared to spaces of distributions in the sense o f Sch wartz) of Colombeau- and Egorov type algebras is the av ailability of a generalized point v alue c haracter iz a tion for elements of such spaces (se e [ 38 ], resp. [ 30 ] fo r the manifold setting). Suc h a c haracteriza tio n may b e view ed as a nonstandard asp ect of the theo ry: for uniquely determining an ele men t of a Colombea u- or Egorov algebra , its v alues on classical (’standard’) po int s do no t suﬃce: there exist elements whic h v anish on each classical p oint yet are nonzer o in the quotient a lgebra underlying the resp ective construction. A unique determination can only be attained by taking into acc ount v a lues on generalized po in ts, themselves g iven as equiv alence classes of standard po int s. T his characteristic feature is re-encountered in practically all known v ariants o f such algebras of generalized functions. It therefore c ame as a surprise when in a ser ie s of pap ers ([ 2, 3 ]) it was claimed that, contrary to the ab ov e g eneral situation, in p -adic Colombea u-Egor ov algebra s a general p oint v alue characterizatio n using only standard po ints w a s av a ilable. This chapter is dedicated to a thorough study of (generalized) p oint v alue charac- terizations of p -adic Co lombeau-Eg orov algebra s and to showing that in fact also in the p -adic setting cla s sical p oint v alues do not suﬃce to uniquely determine elements of suc h. In the remainder of this section we recall some m aterial from ([ 2, 3 ]), using notation from [ 18 ]. Let N b e the natura l n um ber s starting with n = 1. F or a ﬁxed prime p , let Q p denote the ﬁeld of ratio nal p -adic num ber s. Let D ( Q n p ) denote the linear s pace of lo c ally constant complex v alued functions on Q n p ( n ≥ 1) with compact suppor t. Let further P ( Q n p ) := D ( Q n p ) N . P ( Q n p ) is endow ed with an algebra- structure by deﬁning addition a nd m ultiplication of s equences comp onent- wise. Let N ( Q n p ) b e the subalgebra of elements { ( f k ) k } ∈ P ( Q n p ) such that for a ny compact set K ⊆ Q n p there e x ists an N ∈ N suc h that ∀ x ∈ K ∀ k ≥ N : f k ( x ) = 0. This is an ideal in P ( Q n p ). The quotient algebra G ( Q n p ) := P ( Q n p ) / N ( Q n p ) is called the p -adic Colombea u- Egor ov algebr a. Finally , so called Colo mbea u-Egor ov generalized num b ers e C are intro duce d in the following w ay: Let ¯ C b e the one-p oint compactiﬁcation of C ∪ { ∞} . F actor izing A = ¯ C N by the ideal I := { u = ( u k ) k ∈ A | ∃ N ∈ N ∀ k ≥ N : u k = 0 } yields then the ring e C o f Colombeau-Eg orov generalized n um ber s. W e repla ce ¯ C by C and construct similarly C , the ring of gene r alized n umber s : Clea rly , ¯ C is not needed in this context, since representativ es of element s f ∈ G ( Q n p ) mer ely take on 73 74 4. POINT V ALUES & UNIQUENES S QUE STIONS v alues in C N . Le t f = [( f k ) k ] ∈ G ( Q n p ). It is clear that for a ﬁx e d x ∈ Q n p , the p oint value of f at x , [( f k ( x )) k ] is a w ell deﬁned element of C , i.e., we may consider f as a map (4.1) f : Q n p → C : x 7→ f ( x ) := ( f k ( x )) k + I . Note that the abov e co nstitutes a slight abuse of notation: The letter f denotes bo th a generalized function (an element of G ( Q n p )) a nd a mapping on Q n p . Finally , let A b e a set and let R b e a r ing. F or B ⊂ A, θ ∈ R we call the characteristic function of B the map χ B ,θ : A → R which is iden tica lly θ o n B and which v anishes on A \ B . F urthermore, if θ = 1 ∈ R we simply write χ B = χ B , 1 . 4.1.1. Uniqueness via p oint v alues and a coun terexample. The follow- ing statement is proved in Theo rem 4.4 o f [ 3 ]: L et f ∈ G ( Q n p ) , then: f = 0 in G ( Q n p ) ⇔ ∀ x ∈ Q n p : f ( x ) = 0 in C . How ever, inspired by ([ 43 ], p. 218) we construct the following coun ter example to this claim, whic h shows that po int v a lues cannot uniquely determine elemen ts in G ( Q n p ) uniquely . F or the sake of simplicit y w e assume tha t n = 1 . Example 4 . 1. F or any l ∈ N , set B l := { x ∈ Z p : | x − p l | < | p 2 l |} ⊂ { x ∈ Z p : | x | = | p l |} . F or any i ∈ N , w e set f i := χ B i . Clear ly B i ∩ B j = ∅ whenever i 6 = j and since f i ∈ D ( Q p ) for all natural n umber s i , ( f i ) i is a representativ e of some f ∈ G ( Q p ). Now, for any α ∈ Q p , f ( α ) = 0 in C , since either α ∈ B i for some i ∈ N (whic h implies that f j ( α ) = 0 ∀ j > i ) o r α ∈ Q p \ S B i , where ea ch f i ( i ∈ N ) is identically zero. Cons ide r now the sequence ( β i ) i ≥ 1 ∈ N N ⊆ Z N p , where β i = p i ∀ i ∈ N . It follows that f i ( β i ) = 1 ∀ i ∈ N . In particular , for K = Z p or a ny dressed ball containing 0 , there is no representativ e ( g j ) j of f such that for so me N > 0, g j = 0 ∀ j ≥ N . Hence f 6 = 0 in G ( Q p ) although all standar d point v alues of f v anish. Remark 4.2. By means of the ab ov e example w e may analyze the pro of of Theorem 4.4 in [ 3 ]. Let f b e the generalized function from 4.1. As a compact set c ho o se K := B ≤ p − 2 (0) = p 2 Z p . F or the repres ent ative ( f k ) k constructed in 4.1 and x = 0 we hav e N (0) = 1, whic h in the notation of [ 3 ] means that for any k ≥ 1 = N (0), f k (0) = 0. Also, r ecall that B γ ( a ) is the dressed ball B ≤ p γ ( a ). The “parameter of constancy” ([ 3 ], p. 6) of f 1 at x = 0, which is th e maximal γ suc h that f 1 is ident ically zero on B γ (0), is l 0 (0) = − 2. Now, there exists a c overing of K consisting of a sing le set, namely B l 0 (0) (0). Thus we ma y replac e the application of the Heine- Borel Lemma in [ 3 ] by our singleton- cov er ing. But th en the claim that (4.1) and (4.2) imply that for a ll k ≥ N (0) = 1 we hav e f k (0 + x ′ ) = f k (0) = 0 ∀ x ′ ∈ K do es not hold. This indeed follo ws f rom the deﬁnition of the sequence ( f k ) k of locally constant functions from ab ov e, s ince for a ny k ∈ N we have f k ( p k ) = 1. 4.1.2. Egoro v algebras on lo cally compact ultrametric spaces. In this section we inv estig ate the problem of point v alue characterization in Egorov alg e bras in full generality: to this end we consider a g e ne r al lo cally compact ultr ametric space ( M , d ) instead of Q n p , where M need not hav e a ﬁeld structure. Our aim is to show tha t even in such a general setting, the resp e ctive alg ebra canno t have a p o int v alue c har acterizatio n, unless M car ries the discrete top ology . Denote by E d ( M ) 4.1. POINT V ALUES IN EGORO V ALGEBRAS 75 the a lgebra of se q uences of lo ca lly co ns tant functions w ith co mpact suppor t, taking v alues in a commutativ e ring R 6 = { 0 } . Let N d ( M ) b e the set of neglig ible functions { ( f k ) k } ∈ E d ( M ) such that for any compact set K ⊂ M ther e exists an N ∈ N such that ∀ x ∈ K ∀ k ≥ N : f k ( x ) = 0. The s ubset N d ( M ) is an ideal in E d ( M ) and the quotient a lgebra G ( M , R ) := E d ( M ) / N d ( M ) is called the ultrametric Ego rov algebra asso ciated with ( M , d ). F urther more, the ring of generalized num b ers is deﬁned by R := R N / ∼ , where ∼ is the equiv alence rela tion on R N given by u ∼ v in R N ⇔ ∃ N ∈ N ∀ k ≥ N : u k − v k = 0 . W e call I ( R ) := { w ∈ R N : w ∼ 0 } the idea l of neglig ible sequence s in R . Analogous to (4.1), for f ∈ G ( M , R ) ev aluatio n o n sta nda rd point v alues is intro duce d b y means of the mapping: (4.2) f : M → R : x 7→ f ( x ) := ( f k ( x )) k + I ( R ) . Deﬁnition 4.3. An ultrametric Eg o rov algebra G ( M , R ) is said to admit a sta nda rd po int v alue c haracteriza tion if for each u ∈ G ( M , R ) w e ha ve u = 0 ⇔ ∀ x ∈ M : u ( x ) = 0 in R . Using this termino lo gy , Example 4.1 s hows that G ( Q n p ) do es not admit a stan- dard po int v alue characteriz ation. The main result of this sectio n is the f ollowing generaliza tion: Theorem 4. 4. L et ( M , d ) b e a lo c al ly c omp act ult r ametric sp ac e and let R 6 = { 0 } . Then G ( M , R ) do es not admit a standar d p oint value char acterization unless ( M , d ) is discr ete. Proof. The r e sult follows by g e ne r alizing the constructio n o f Ex ample 4.1. Assume ( M , d ) is not discrete, then there exis ts a po int x ∈ M and a sequence ( x n ) n of distinct p oints in M conv erg ing to x . W e may ass ume that d ( x, x i ) > d ( x, x j ) whenever i < j . Deﬁne stripp ed balls ( B n ) n ≥ 1 with centers ( x n ) n ≥ 1 by B n := { y ∈ M | d ( x n , y ) < d ( x n ,x ) 2 } . Due to the ultrametric prop erty “the strongest one wins” w e hav e B n ⊂ { z | d ( x, z ) = d ( x n , x ) } , whic h further implies that for all i 6 = j ( i, j ) ∈ N , the ba lls B i , B j are disj oint sets in M . Since R is a non-tr ivial ring, we may c hoo se s ome θ ∈ R \ { 0 } . Now we deﬁne a s equence ( f k ) k of lo cally constant functions in the following w ay: F or any i ≥ 1 set f i = χ B i ,θ . Clearly , f := [( f i ) i ] ∈ G ( M , R ), a nd similarly to Example 4.1 , for an y α ∈ M , f ( α ) = 0 in R . Nevertheless f or the sequence ( x n ) n , which without lo ss of genera lity ma y b e assumed to lie in a compact neig hborho o d of x , one ha s f i ( x i ) = θ ∀ i ≥ 1 which implies that f 6 = 0 in G ( M , R ).  Recall that a discrete top olo gical space X has the following prop er ties: (i) X is loca lly compact. (ii) Any compact se t in X contains ﬁnitely many p oints only . Therefore w e know that for a set D endo wed with the dis crete metric a nd for any commutativ e ring R , t he respective ultrametric Eg orov algebra G ( D , R ) admits a po int wise c haracteriza tion. W e therefore conclude: Corollary 4 .5. F or a lo c al ly c omp act ultr ametric sp ac e ( M , d ) and a non-trivial ring R , the fol lowing s t atements ar e e quivalent: (i) G ( M , R ) admits a standar d p oint value char acterization. (ii) The top olo gy of ( M , d ) is discr ete. 76 4. POINT V ALUES & UNIQUENES S QUE STIONS 4.1.3. Generalized p oin t v alues. In this section w e giv e a n appr opriate generalized point v a lue characterization in the s tyle o f ([ 38 ], pp. Theorem 2. 4) of G ( M , R ), where M is endow e d with a no n- discrete ultrametric d for which M is lo cally compact, and R 6 = { 0 } . First, we have to introduce a set f M c of compactly suppo rted generalized points o ver M . Let E = M N , the ring of sequences in M , and identify tw o sequence s , if for so me index N ∈ N o ne has d ( x n , y n ) = 0 for each po sitive integer n , that is, x n = y n ∀ n ≥ N ; w e write x ∼ y . W e ca ll f M = E / ∼ the ring of generalized n umbers . Fina lly , f M c is the subset of such elements x ∈ f M for which there exists a c o mpact subset K a nd some r epresentativ e ( x n ) n of x such that for so me N > 0 we ha ve x n ∈ K fo r all n ≥ N . It follows that ev alua ting a function u ∈ G ( M , R ) at a compactly suppor ted g eneralized point x is p o ssible, i.e., for repres e n tatives ( x k ) k , ( u k ) k of x re sp. u , [( u k ( x k )) k ] is a well deﬁned element o f R . Prop ositi o n 4.6. In G ( M , R ) , ther e is a gener alize d p oint value char acterization, i.e., u = 0 in G ( M , R ) ⇔ ∀ x ∈ f M c : u ( x ) = 0 in R . Proof. The condition on the right side obviously is necessa ry . Co nv ers e ly , let u ∈ G ( M , R ), u 6 = 0. This means that there is a repr esentativ e ( u k ) k of u and a compact set K ⊂ ⊂ M such that u k do es not v anis h on K for inﬁnitely man y k ∈ N . In particula r this means we hav e a sequence ( x k ) k in K s uch that for inﬁnitely many k ∈ N , u k ( x k ) 6 = 0 . Clea rly this means that u ( x ) 6 = 0 in R , where we hav e s et x := [( x k ) k ].  4.1.4. The δ -distributi o n. In [ 3 ], Theo rem 4.4 is illustr ated b y some exa m- ples, to highlight the adv antage of a p oint v alue concept in G ( Q n p ). In this section we discuss the δ -distribution (Example 4.5 on p. 1 2 in [ 3 ]) and construct a gener- alized function f ∈ G ( Q p ) diﬀere nt fro m δ which how ev er coincides with δ on all standard points in Q p . W e ﬁrst embed the δ -distribution in G ( Q p ) as in [ 3 ] (p. 9, Theorem 3.3) which yields ι ( δ ) = ( δ k ) k + N p ( Q p ), where δ k ( x ) := p k Ω( p k | x | p ) for each k , and Ω is the bump function on R + 0 given by Ω( t ) := ( 1 , 0 ≤ t ≤ 1 0 , t > 1 . Ev aluation of ι ( δ ) on standard p oints is shown in Example 4.5 of [ 3 ]. With ˜ c := ( p k ) k + I ∈ C one has: ι ( δ )( x ) = ( e c, x = 0 0 , x 6 = 0 ( x ∈ Q p ) . Let ϕ : N → Z b e a monotonous function suc h that lim k →∞ ϕ ( k ) = ∞ , and such that the ca rdinality of U ϕ := { k : ϕ ( k ) > k } is inﬁnite. Consider an elemen t f ∈ G ( Q p ) giv en by f := ( f k ) k + N ( Q p ) where for any k ≥ 1 , f k ( x ) := p k Ω( p ϕ ( k ) | x | p ). Then the standar d p oint v alues of ι ( δ ) and f coincide. F urthermo re, they coincide on compactly supp orted genera liz ed p o ints x ∈ e Q p,c with the pro pe r ty that for any representative ( x k ) k of x there exists an N ∈ N such that ∀ k ≥ N : | x k | p > p − min { k,ϕ ( k ) } , since in this ca se w e have δ k ( x k ) = f k ( x k ) = 0. How ever, there are co mpactly supp orted genera liz e d points vio lating this condition whic h yield diﬀerent gener a lized po int v alues of ι ( δ ) resp. f : for ins ta nce, take the generalized 4.2. SPHERICAL COMPLETENESS 77 po int x 0 := [( p k ) k ] ∈ e Q p,c . Then f ( x 0 ) 6 = e c , since θ k := f k ( x k ) = 0 for a n y k ∈ U ϕ and thus θ k = 0 for inﬁnitely man y k ∈ N . But ι ( δ )( x 0 ) = e c . 4.2. Spheri cal completeness o f the ring of generalized n umbers Let ( M , d ) b e an ultrametric spa ce. F or given x ∈ M , r ∈ R + , we call B ≤ r ( x ) := { y ∈ M | d ( x, y ) ≤ r } the dre ssed ball with cen ter x and radius r . Throughout N := { 1 , 2 , . . . } denote the p ositive integers. Let ( x i ) i ∈ M N and ( r i ) i be a seq uenc e of p ositive r eals. W e call ( B i ) i , B i := B ≤ r i ( x i ) ( i ≥ 1) a nested sequence of dress e d balls, if r 1 ≥ r 2 ≥ r 3 . . . and B 1 ⊇ B 2 ⊇ . . . . F ollowing standard ultrametric literature (cf. [ 43 ]), nested s equences of dre s sed balls might hav e empt y intersection. The conv erse prop erty is deﬁned as follo ws: Deﬁnition 4.7. ( M , d ) is called spherically complete, if ev er y nested sequence of dressed balls has a non- empt y intersection. It is evident that any s pherically complete ultrametric space is complete with resp ect to the top ology induced by its metric (using the well known fact that top o- logical completeness of ( M , d ) is equiv alen t to the proper t y of Deﬁnition 4.7 with radii r i ց 0) . How ever, there are p opular non-trivial examples in the literature, for which the co nverse is not true. As an example w e mention the ﬁeld o f complex p -adic num b ers together with its p -a dic v aluation considered as the co mpletion of the algebraic closure o f the ﬁeld C p of ra tional p -adic n um ber s. Due to Kras ner, this ﬁeld has nice algebraic prop erties (as it is algebraically closed, and ev en isomorphic to the co mplex num b er s cf. [ 43 ], pp. 13 4–14 5), but it als o has b een shown, that C p is not s pher ically complete. This is mainly due to the fact that the complex p -adic nu m ber s are a separable, co mplete ultrametric space with dense v a luation (cf. [ 43 ], pp. 143 –144). How ever, for an u ltrametric ﬁeld K , spher ical completeness is nec- essary in or der to ensur e K has the Hahn Banach extension prop er t y (to which we refer as HBEP), that is, any no rmed K -vector spa ce E admits contin uous linear functionals previously deﬁned on a strict subspace V of E to be extended to the whole space under conserv ation of their norm (cf. W. Ingleton’s proo f [ 24 ]). Since spherical completeness fails, it is natural to as k if the p -adic num b ers could at lea st be spherically completed, i.e., if there existed a spherically complete ultr ametric ﬁeld Ω in to whic h C p can b e em b edded. This question ha s a positive answer (cf. [ 43 ]). The nec e ssity of spherical completeness for the HBEP of K = C p is evident: even the iden tity map ϕ : C p → C p , ϕ ( x ) := x cannot b e extended to a functional ψ : Ω → C p under conserv ation of its norm k ϕ k = 1 (her e we co nsider Ω as a C p - vector space). 1 The pres ent work is motiv ated by the question if some version of Hahn-Banach’s Theorem holds on diﬀeren tial algebras in the sense of Colombeau considered as ultra pseudo normed modules ov er the ring o f g eneralized n umbers e R (r esp. e C ). 1 T o chec k this, let B i := B ≤ r i ( x i ) b e a nested sequence of dressed balls in C p with empty int ersection. Then ˆ B i := B ≤ r i ( x i ) ⊆ Ω ha ve n onempt y intersection, sa y Ω ∋ α ∈ T ∞ i =1 ˆ B i . Assume f urther, the iden tity ϕ on C p can be extende d to some linear map ψ : Ω → C p under conserv ation of its norm. Then | ψ ( α ) − x i | Ω = | ψ ( α ) − φ ( x i ) | Ω ≤ k ψ k| α − x i | C p = | α − x i | C p , therefore ψ ( α ) ∈ T ∞ i =1 B i which is a contradiction and w e are done. 78 4. POINT V ALUES & UNIQUENES S QUE STIONS Even though topo logical questions on top olo gical e C mo dules have been rece n tly inv estiga ted to a wide extent (cf. C. Gare tto’s recent pap ers [ 15, 16 ] as w e ll as [ 11 ]), a HBE P has not yet b een established in the literature. The analogy with the p - adic case lie s at hand, since the ring of genera lized nu m ber s ca n naturally b e endow ed with an ultra metric pseudo -norm. How ever, the presence of zero -divisor in e R as well a s the failing m ultiplicativit y of the pseudo- norm turns the question into a non-trivial one and Ingleton’s ultrametric version of the Ha hn Bana ch Theorem ca nnot b e c arried ov er to o ur setting unr estrictedly . On our ﬁrst step tackling this question we dis cuss spherical co mpleteness o f the ring of generalized n um ber s endow ed with the giv en ultrametric (induced by t he resp ective pseudo-no rm, cf. the preliminary sectio n). e R ﬁrst was introduce d a s the set of v alues of ge neralized functions at standard po int s; howev er , a subring co nsisting of compactly s uppo rted g eneralized num ber s turned o ut to be the set o f points for which ev aluatio n deter mines uniquenes s, whereas standard p oints do not suﬃce do determine g eneralized functions uniquely (cf. [ 34, 38 ]). A hint, that e R (or e C as well), the ring of ge ner alized real (or co mplex ) nu m ber s is spherically complete, is, that con trary to the above o utlined s itua tion on C p , the generalized n um ber s endow ed with the top olo gy induced by the s harp ultra-pseudo norm ar e not separa ble. This, fo r instance, follows fro m the fact that the res triction of the sharp v aluation to the r eal (or complex) n um ber s is discrete. Having motiv ated our w ork by now, w e may formulate the aim of this section, which is to prov e the following: Theorem 4. 8. The ring of gener alize d nu m b ers is sph eric al ly c omplete. W e therefore ha ve an independent pro o f of the fac t (cf. [ 16 ], Prop osition 1 . 30): Corollary 4.9. The ring of gener alize d numb ers is top olo gic al ly c omplete. In the last section o f this note we present a mo diﬁed version of Hahn-Bana ch’s Theorem which ba ses on spherically co mpleteness of e R (resp. e C ). Finally , a rema rk on the a pplicability of the ultra metric version of Banach ﬁxed po in t theorem can be found in the Appendix. 4.2.1. Preliminaries. In what follows w e re p ea t the deﬁnitions of the ring of (real o r complex) gener alized num b ers a lo ng with its non-archimedean v a luation function. The mater ial is taken from diﬀerent sourc es; as references we may recom- mend the r ecent works due to C. Garetto ([ 15, 16 ]) and A. D elcroix et al. ([ 11 ]) as well as one of the o riginal sour ces of this topic due to D. Scarpale z o s (cf. [ 12 ]). Let I := (0 , 1 ] ⊆ R , and let K de no te R resp. C . The r ing of generalized n um bers ov er K is co nstructed in the follo wing way: Given the ring of mo derate (nets of ) nu m ber s E := { ( x ε ) ε ∈ K I | ∃ m : | x ε | = O ( ε m ) ( ε → 0) } and, similarly , the idea l of negligible nets in E ( K ) which ar e of the form N := { ( x ε ) ε ∈ K I | ∀ m : | x ε | = O ( ε m ) ( ε → 0) } , we may deﬁne the genera lized num ber s as the factor ring e K := E M / N . W e deﬁne a (real v alued) v aluation function ν : on E M ( K ) in the following way: ν (( u ε ) ε ) := sup { b ∈ R | | u ε | = O ( ε b ) ( ε → 0) } . 4.2. SPHERICAL COMPLETENESS 79 This v aluation can b e carrie d ov er to the ring o f genera lized num b ers in a w ell deﬁned wa y , since for t wo representatives of a g eneralized num b er , the v a luations ab ov e coincide (cf. [ 16 ], section 1). W e then ma y endo w e K with an ultra -pseudo- norm (’pseudo’ refers to non-multiplicativit y) | | e in the following way: | 0 | e := 0, and whenever x 6 = 0 , | x | e := e − ν ( x ) . With the metric d e induced by the above norm, e K turns out to be a non-discr ete ultrametric space, with the follo wing topo lo gical prop erties: (i) ( e K , d e ) is to po logically co mplete (cf. [ 16 ]), (ii) ( e K , d e ) is no t sepa rable, since the restriction o f d e onto K is discrete. The latter prop erty holds, since on metric spaces second countabilit y and sepa r abil- it y ar e equiv alent and the well k nown fact that the prop erty of seco nd countabilit y is inherited b y subspaces (wherea s s e pa rability is not in ge ne r al). In o rder to av oid confusio n we henceforth denote close d balls in K b y B ≤ r ( x ) in distinction with dr essed balls in e K which we denote by e B ≤ r ( x ). Simila rly stripp ed balls a nd the sphere in the ring o f genera lized num b er s ar e denoted b y e B 0 , let ( B ε ) ε b e an euclide an mo del for e B ≤ r ( x ) a nd set ρ = − lo g r . Then we have: 80 4. POINT V ALUES & UNIQUENES S QUE STIONS (i) Any y ∈ e B ρ, and this implies that there exists some ρ ′ > ρ such that for any r epresentativ e ( y ε ) ε of y a nd any representativ e ( x ε ) ε of x w e hav e | y ε − x ε | = o ( ε ρ ′ ) , ε → 0 . This further implies that for an y choice of representatives of x resp. of y , there exists some η ∈ I with (4.3) | y ε − x ε | ≤ ε ρ ′ for each ε < η . Since C ε > 0 for eac h ε ∈ I and C ε is monotonously increasing with ε → 0, w e ha v e ε ρ ′ ≤ C ε ε ρ for suﬃciently small ε , therefore, a suitable c ho ice of y ε , for ε ≥ η , yields the ﬁrst claim (for instance, o ne may set y ε := x ε whenever ε ≥ η ). W e go o n by proving (ii): F or the ﬁrst part, set y ε := 2 C ε ε ρ + x ε Let y denote the class of ( y ε ) ε . It is ev ide nt, that y ∈ e B ≤ r ( x ). How ever, ( y ε ) / ∈ B ε for each ε ∈ I . Indeed, ∀ ε ∈ I : | y ε − x ε | = 2 C ε ε ρ > C ε ε ρ , since C ε > 0 for each ε . W e further show, that the same holds for an y representative ( ¯ y ε ) ε of y for suﬃcien tly small index ε . Indeed, the diﬀerence of tw o re presentativ e s being negligible implies that for a ny N > 0 w e hav e y ε − ˆ y ε = o ( ε N ) ( ε → 0) . Therefore, for N > ρ and suﬃcien tly small ε , w e hav e: | ˆ y ε − y ε | ≥ || ˆ y ε − y ε | − | y ε − x ε || ≥ 2 C ε ε ρ − ε N ≥ 3 2 C ε ε ρ > C ε ε ρ . Therefore we hav e shown the ﬁrst par t of (ii). Let y ∈ e S r ( x ). W e demonstr ate how to blow up ( B ε ) ε to catch some ﬁxed r epresentativ e ( y ε ) ε of y . Since | y − x | = e − ρ = r , there is a net C ′ ε ≥ 0 ( | ( C ′ ε ) ε | e = 1) such that ∀ ε ∈ I : | y ε − x ε | = C ′ ε ε ρ Set C ′′ ε = max η ≥ ε { 1 , C ′ η } . This ensur es that ( C ′′ ε ) is a monotonously increasing with ε → 0, above 1 for each ε ∈ I , and | ( C ′′ ε ) | e = 1 is preserved. Deﬁne B ′ ε := 4.2. SPHERICAL COMPLETENESS 81 B ≤ C ε C ′′ ε ε ρ ( x ε ). Then ( B ′ ε ) ε is a new model for e B ≤ r ( x ) containing t he old mo del and ( y ε ) ε as well, since the pro duct C ε C ′′ ε has the required prop erties, a nd | y ε − x ε | ≤ C ′′ ε ε ρ ≤ C ′′ ε C ε ε ρ and we a re done with (ii). Pro of of (iii): So far, we hav e shown that for eac h y ∈ e B ≤ r ( x ), there exists an euclidean mo del ( B ≤ C ε ε ρ ( x ε )) of B ≤ r ( x ) s uch that for some representative ( y ε ) ε of y ∈ e B ≤ r ( x ) we hav e ∀ ε ∈ I : y ε ∈ B ε . Therefore, by replacing C ε by 2 C ε ab ov e, again a model for e B ≤ r ( x ) is achieved, how ever with the further prop erty that | y ε − x ε | ≤ C ε / 2 ε ρ for each ε ∈ I which prov e s our claim.  Before going on by establishing the crucia l statement whic h will allow us to translate decr easing sequences of closed balls in the giv e n ultrametric space e K to decreasing sequences of their (appropr iately chosen) euclidean mo dels , we introduce a useful term: Deﬁnition 4.12. Suppos e , we have a nested se q uence ( e B i ) ∞ i =1 of c losed ba lls with centers x i and ra dius r i in e K and for each i ∈ N we have an euclidean mo de l ( B ( i ) ε ) ε . W e sa y , this a sso ciated sequence of euclidean mo dels is proper, if  ( B ( i ) ε ) ε  ∞ i =1 is nested as well, that is, if we hav e: ( B (1) ε ) ε ⊇ ( B (2) ε ) ε ⊇ ( B (3) ε ) ε ⊇ . . . . 4.2.3. Pro of of the main theorem. In order to prov e the main statement, we pro ceed by es ta blishing tw o imp or tant preliminar y statements. First, a remark on the notation in the sequel: If ( x i ) i , a sequence of points in the ring of generalized nu m ber s, is considered, then ( x ( i ) ε ) ε denote (certain) repr esentativ es of the x i ’s. F urthermo re, for subsequent choices o f nets of real num b ers ( C ( i ) ε ) ε , and positive radii r i , w e denote by ρ i the nega tive log arithms of the r i ’s ( i = 1 , 2 , . . . , ) a nd the euclidean models of the balls e B ≤ r i ( x i ) with radii r i ε := C ( i ) ε ε ρ i to be constructed are denoted b y B ( i ) ε := B ≤ r ( i ) ε ( x ( i ) ε ) . W e start with the fundamen tal prop osition: Prop ositi o n 4.13. L et x 1 , x 2 ∈ e K , and r 1 , r 2 b e p ositive numb ers such that e B ≤ r 1 ( x 1 ) ⊇ e B ≤ r 2 ( x 2 ) . L et ( x (1) ε ) ε b e a r epr esentative of x 1 . Then the fol lowing holds: (i) Ther e exists a net ( C (1) ε ) ε satisfying c ondition (E) su ch that for e ach ε ∈ I (4.4) x (2) ε ∈ B ≤ C (1) ε ε ρ 1 2 ( x (1) ε ) . (ii) F urthermor e, for e ach n et ( C (2) ε ) ε satisfying c ondition ( E) ther e exists an ε (1) 0 ∈ I such that for e ach ε < ε (1) 0 ∈ I we have B (2) ε ⊆ B (1) ε . Proof. Pro of of (i): Let ( x (2) ε ) ε be a repr esentativ e of x 1 . W e distinguish the following tw o cases: 82 4. POINT V ALUES & UNIQUENES S QUE STIONS (i) x 2 ∈ e S r 1 ( x 1 ), that is, | x 2 − x 1 | e = r 1 . Let ( x (2) ε ) ε be a r e pr esentativ e of x 2 . Deﬁne ˆ C (1) ε := | x (1) ε − x (2) ε | . No w, s et C (1) ε := 2 max( { ˆ C (1) η | η > ε } , 1). Then not o nly C (1) ε > 0 for each para meter ε , but also the net C (1) ε > 0 is mo no tonically incr easing with ε → 0, furthermore (4.4) holds, a nd we are done with this ca se. (ii) x 2 / ∈ e S r 1 ( x 1 ), that is, | x 2 − x 1 | e < r 1 . Set, for instance, C (1) ε = 1. F or each r e presentativ e ( x (2) ε ) ε of x 2 it follows that | x (2) ε − x (1) ε | = o ( ε ρ 1 ) and a representativ e satisfying the desired pr o p erties is easily found. Pro of of (ii): T o sho w this we consider the asymptotic growth o f ( C (1) ε ) ε , ( C (2) ε ) ε , ε ρ 1 , ε ρ 2 as well as the monotonicity of C (1) ε : let y ∈ B ≤ C (2) ε ε ρ 2 ( x (2) ε ). Then we hav e by the triang le inequality for each ε ∈ I : (4.5) | y − x (1) ε | ≤ | y − x (2) ε | + | x (2) ε − x (1) ε | ≤ C (2) ε ε ρ 2 + C (1) ε ε ρ 1 2 . W e kno w further that by the mo no tonicity ∀ ε ∈ I : C (1) ε ≥ C (1) 0 := C 0 so that (4.6) C (2) ε C (1) ε ε ρ 2 − ρ 1 ≤ C 0 C (2) ε ε ρ 2 − ρ 1 . Moreov er, since the sharp no rm of C (2) ε equals 1, for any α > 0 we hav e C (2) ε = o ( ε − α ) , ( ε → 0) . which in conjunction with the fact that ρ 2 > ρ 1 allows us to further estimate the right hand side of (4.6): Obta ining C (2) ε C (1) ε ε ρ 2 − ρ 1 = o (1) , ( ε → 0) , we plug this information into (4.5). This yields for suﬃciently small ε , say ε < ε (1) 0 : (4.7) | y − x (1) ε | ≤ C (1) ε ε ρ 1 2 + C (1) ε ε ρ 1 2 = C (1) ε ε ρ 1 ; the pro o f is ﬁnished.  Prop ositi o n 4.14. Any n este d se quenc e of close d b al ls in e K a dmits a pr op er se- quenc e of asso ciate d euclide an mo dels. Proof. W e proceed step by step s o that we may easily read oﬀ the inductive argument of the pro of in the end. W e may ass ume that for each i ≥ 1, r i > r i +1 . Deﬁne ρ i := − log( r i ) (so tha t ρ i < ρ i +1 for each i ≥ 1). Step 1. Cho ose a representativ e ( x (1) ε ) ε of x 1 . Step 2. 4.2. SPHERICAL COMPLETENESS 83 Due to Pro p osition 4 .13 (i) we may ch o ose a repres ent ative ( x (2) ε ) ε of x 2 and a net ( C (1) ε ) ε of real n umbers satisfying condition (E) such that suc h that for each ε ∈ I x (2) ε ∈ B ≤ C (1) ε ε ρ 1 2 ( x (1) ε ) . Denote by ε (1) 0 ∈ I be the max imal ε such that the inclusion r elation B (2) ε ⊆ B (1) ε as in (cf. (ii) of Prop osition 4.1 3) holds. Step 3. Similarly , tak e a r epresentativ e ( ˆ x (3) ε ) ε of x 3 and a ne t ( ˆ C (2) ε ) ε of real n um b er s satisfying co ndition (E) suc h that such that for each ε ∈ I (4.8) ˆ x (3) ε ∈ B ≤ ˆ C (2) ε ε ρ 2 2 ( x (2) ε ) . W e show now, how to adjust our choice of ˆ x (3) ε , ˆ C (2) ε such that condition (E) as well as the inclusion rela tion (4.8) is preserved, howev er , we do this in a w ay such that we more over achiev e the inclusion relatio n (4.9) B (2) ε ⊆ B (1) ε for e ach ε (for suﬃcien tly sma ll para meter this is g uaranteed b y the pro ceeding prop osition). F or ε ≥ ε (1) 0 we leave the choice unchanged, that is, w e set x (3) ε := ˆ x (3) ε , C (2) ε := ˆ C (2) ε ; for ε < ε (1) 0 , howev er, we set (4.10) x (3) ε := x (2) ε , C (2) ε := min ( C (1) ε 2 ε ρ 1 − ρ 2 , ˆ C (2) ε ) . Therefore, ( C (2) ε ) ε still satisﬁes condition (E), since it is still p ositive and mono ton- ically incre a sing with ε → 0, furthermore we have only modiﬁed for big parameter ε , the asymptotic growth with ε → 0 therefor e remains unc hanged (and so does the sharp norm o f ( C (2) ε ) ε , which it is identically 1). Next, it is e vident that x (3) ε ∈ B ≤ C (2) ε ε ρ 2 2 ( x (2) ε ) . still holds for each ε ∈ I . Finally , b y (4.10) it f ollows that the inclusion re lation (4.9) holds now for each ε ∈ I . F or the inductive pro o f of the statement one for ma lly pro ceeds as in Step 3. Let k > 1. Assume we hav e representativ es ( x (1) ε ) ε , . . . , ( x ( k +1) ε ) ε and nets of p o sitive n um ber s ( C ( j ) ε ) ε , (1 ≤ j ≤ k ) , satisfying co ndition (E), suc h that for ea ch ε ∈ I we hav e : B ≤ C (1) ε ε ρ 1 ( x (1) ε ) ⊇ B ≤ C (2) ε ε ρ 2 ( x (2) ε ) ⊇ · · · ⊇ B ≤ C ( k − 1) ε ε ρ k − 1 ( x ( k − 1) ε ) . and for s ome ε ( k − 1) 0 we hav e for ea ch ε < ε ( k − 1) 0 B ≤ C ( k − 1) ε ε ρ k − 1 ( x ( k − 1) ε ) ⊇ B ≤ C ( k ) ε ε ρ k ( x ( k ) ε ) . 84 4. POINT V ALUES & UNIQUENES S QUE STIONS F urthermo re w e supp ose the follo wing additional pr op erty is satisﬁed: F or eac h ε ∈ I we hav e: x ( k +1) ε ∈ B ≤ C ( k ) ε 2 ε ρ k ( x ( k ) ε ) , where ρ k := − log r k . In the very same manner a s ab ov e, we may now ﬁnd a representative ( x ( k +2) ε ) ε of x k +2 and a net of num b ers ( C ( k +1) ε ) ε satisfying condition (E) such that the ab ov e seque ntial construction can b e enla rged by one ( k → k + 1 ).  The prec e ding prop osition is a key ingredient in the pro of of our main statement Theorem 4.8: Proof. Let ( e B i ) ∞ i =1 , B i := e B ≤ r i ( x i ) ( i ≥ 1) b e t he g iven nes ted sequence of dressed balls; due to P rop ositio n 4.14, there exists a prop er sequence of a sso ciated euclidean mo dels ( B ( i ) ε ) ε such that for representatives ( x ( i ) ε ) ε of x i ( i ≥ 1) the ab ov e nets are given by B ( i ) ε := B ≤ C ( i ) ε ε ρ i ( x ( i ) ε ) , ρ i := − log r i , C ( i ) ε ∈ R + for each ( ε, i ) ∈ I × N . Since K is loc a lly compact, f or each ε ∈ I we may c ho o se some x ε ∈ R such that x ε ∈ ∞ \ i =1 B ( i ) ε since for each ε ∈ I w e h av e B (1) ε ⊇ B (2) ε ⊇ . . . . Since the sequence of euclidean mo dels of the e B i ’s is prope r , for e a ch ε ∈ I further holds: | x ε − x ( i ) ε | ≤ C ( i ) ε ε ρ i . This sho ws that not o nly t he net ( x ε ) ε is moderate (use the t riangle inequalit y), but also g ives r ise to a generalized num b er x := ( x ε ) ε + N ( K ) with the pro p e r ty | x − x i | e ≤ r i for each i . This shows that x ∈ ∞ \ i =1 e B i 6 = ∅ which yields the claim: e K is spherically complete.  4.2.4. A Hahn-Banac h Theorem. Let L b e a subﬁeld of e K such that ν e restricted to L is additiv e. Let E be an ultra pseudo-normed L -linea r space. W e call ϕ an L - linear functional on E , if ϕ is an L - linear mapping on E with v alues in e K . ϕ is con tinuous if k ϕ k := sup 0 6 = x ∈ E | ϕ ( x ) | k x k < ∞ and the space of all con tin uous L -linear functionals on E w e denote b y E ′ L . Remark 4.1 5. Note that nontrivial subﬁelds L of e K exist. F or instance, one may choose K ( α ) with α = [( ε ) ε ] ∈ e K or its co mpletio n with r esp ect to | | e -the Laurent series ov er e K . 4.2. SPHERICAL COMPLETENESS 85 Having in tro duced these notions we show that following version o f the Hahn- Banach Theor em holds: Theorem 4.16. L et V b e an L -line ar subsp ac e of E and ϕ ∈ V ′ L . Then ϕ c an b e extende d to some ψ ∈ E ′ L such that k ψ k = k ϕ k . Proof. W e follow the lines of the pro o f of Ingleton’s theor em (cf. [ 2 4 ]) in the fashion of ([ 43 ], pp. 194–1 95). T o star t with, le t V b e a str ict L -linear subspace of E and let a ∈ E \ V . W e ﬁrst show tha t ϕ ∈ V ′ L can b e extended to ψ ∈ ( V + La ) ′ L under conserv ation of its norm. T o do this it is suﬃcien t to prove that such ψ satisﬁes for ea ch x ∈ V : k ψ ( x − a ) k ≤ k ψ k · k x − a k (4.11) k ϕ ( x ) − ψ ( a ) k ≤ k ϕ k · k x − a k =: r x . T o this end deﬁne for each x in V the dressed ball B x := B ≤ r x ( ϕ ( x )) . Next we cla im that the family { B x | x ∈ V } of dressed balls is nes ted. T o see this, let x, y ∈ V . By the linearity of ϕ and the ultrametric (stro ng) triang le inequa lit y we hav e | ϕ ( x ) − ϕ ( y ) | ≤ k ϕ k · k x − y k ≤ k ϕ k max( k x − a k , k y − a k ) = max( r x , r y ) . Therefore w e hav e B x ⊆ B y or B y ⊆ B x or vice versa. According to Theorem 4.8, e K is spher ically co mplete, therefore w e ma y c hoo se α ∈ \ x ∈ V B x and further deﬁne ψ ( a ) := α . Due to (4.11) and the homogeneity of the shar p norm with res pec t to the ﬁeld L w e therefore hav e for each z ∈ V and for each λ ∈ L , | ψ ( z − λa ) | = | λ | · | ψ ( z /λ − a ) | ≤ | λ | r z /λ = | λ |k ϕ k · k z / λ − a k = k ϕ k · k z − λa k which shows that ψ is an extensio n of ϕ o nt o V + La and k ψ k = k ϕ k . The rest o f the pro of is the s tandard o ne- an application of Zo rn’s Lemma.  Let E be a ultra pseudo-no rmed e K mo dule and denote by E ′ all con tin uous linear functionals on E . W e e nd this section by p osing the following conjecture: Conjecture 4.17. L et V b e a submo dule of E and let ϕ ∈ V ′ . Then ϕ c an b e extende d to some element ψ ∈ E ′ such that k ψ k = k ϕ k . App endix. Finally , it is w orth ment ioning that apar t from the standard Fixe d Poin t Theorem due to Bana ch, a non-archimedean version is av ailable in spheri- cally complete ultrametr ic spaces ( therefor e, also o n e K , cf. [ 41 ], a nd for a recent generaliza tion cf. [ 42 ]): Theorem 4 .18. L et ( M , d ) b e a spheric al ly c omplete ultr ametric sp ac e and f : M → M b e a mappi ng having the pr op erty ∀ x, y ∈ M : d ( f ( x ) , f ( y )) < d ( x, y ) . Then f has a unique ﬁxe d p oint in M . 86 4. POINT V ALUES & UNIQUENES S QUE STIONS 4.3. Scaling in v ariance in algebras of generalized functions Recent research in the ﬁeld of ge neralized functions increasing ly fo cuses on int rinsic pro blems in algebras of generalized functions. This is emphasized b y a nu m ber of scie ntiﬁc pape r s on algebraic (cf. [ 5 ]) and top olo gical topics (cf. [ 11, 12, 15, 16 ]). In this c ha pter w e inv estig ate scaling in v ar iance of g eneralized functions. W e prov e tha t a generalized function on the real line which is inv ar iant under p osi- tive standard sca ling has to b e a constant. Also, we add a couple of further new characterizations of lo cally constant g eneralized functions to the well known ones. Our pro of is partia lly based on the solution o f the so-called ”Lobster problem”. It was a t the International Confer enc e on Gener alize d f unctions 2000 (April, 17–21) that Professo r Mic hael O ber guggenber ger oﬀered a lobster for the answer to the question: ” Are generalized f unctions whic h are inv ariant under standard transla- tions, merely the constant s?” A (positive) answ er to the latter was ﬁrst given b y S. P ilipovic, D. Sca rpalezos and V. V almorin in [ 40 ] and an independent pro of has recently b een established by H. V er naeve [ 48 ]. Note that there is also an ev iden t link b etw een the present w ork and that o f S. Konjik and M. K unzing er dealing with g roup in v a riants in algebras of generalized functions ([ 25, 26 ]) which a re also partially based on the so lutio n of the Lobster problem. 4.3.1. Preliminaries. The setting of this c ha pter is the sp e cial algebr a G ( R d ) of genera lized functions (cf. the intro ductio n). T o start with we shor tly review the sp eciﬁc concepts r esp. metho ds we are going to employ in the sequel: asso ciation and in tegration of generaliz e d functions, generalized p oints and sharp top olo gy as well as contin uity issues with re s pe ct to the latter . F or the sake of simplicit y w e set d = 1 . F o r the generalized p oint v a lue concept in algebras of ge ne r alized functions intro duced b y M. Kunzinger and M. Ob ergugg enberger in [ 38 ], we r efer to the in tro duction. Next, let us r ecall the so-called shar p topo logy o n the r ing of generalized num b ers: 4.3.1.1. The sharp top olo gy on e R . The - maybe mos t natural - top olo gy on the ring of g eneralized num b ers is the one whic h resp ects the a symptotic growth b y means of whic h they are deﬁned. Deﬁne a (real v alued) v alua tion function ν on E M ( R ) in the following wa y: ν (( u ε ) ε ) := sup { b ∈ R | | u ε | = O ( ε b ) ( ε → 0) } . This v aluation can b e carrie d ov er to the ring o f genera lized num b ers in a w ell deﬁned way , since fo r t wo representativ es o f a genera lized num b er, their v aluations coincide (cf. [ 16 ], chapter 1). W e then may endow e R w ith an ultra-pseudo-no rm (’pseudo’ refers to no n- mu ltiplicativity) | | e in the follo wing wa y: | 0 | e := 0, and whenever x 6 = 0, | x | e := e − ν ( x ) . With the metric d e induced b y the ab ove norm, e R turns out to b e a non-dis crete ultrametric space, with the following topo logical prop erties: (i) ( e R , d e ) is topo logically complete (cf. [ 16 ]), (ii) ( e R , d e ) is not separa ble , since the res triction of d e onto R is discrete. 4.3. SCALING INV ARIANCE 87 The latter prop erty holds, since on metric spaces second countabilit y and sepa r abil- it y ar e equiv alent and the well k nown fact that the prop erty of seco nd countabilit y is inherited b y subspaces (wherea s s e pa rability is not in ge ne r al). 4.3.1.2. Continuity issues. In ([ 4 ]) Aragona et al. develop a new concept of diﬀerentiabilit y of generalized functions f viewed as maps e f : e R c → e R , a concept which is co mpatible with par tial diﬀerentiation in G ( R d ) a nd ev alua tion of functions at ge ne r alized po int s. W e need not r e call this in detail; we only mention one no table consequence which we will make use of subsequently: F act 4 .19. If e f : e R c → e R is induc e d by a gener alize d fun ction, then e f is c ontinu ous with r esp e ct to the sharp top olo gy on e R c . 4.3.1.3. Int e gr ation of gener alize d functions. Generalized functions ma y be in- tegrated ov er relatively compa ct Leb esgue measur able sets. W e recall an elementary statement (this is Pro po sition 1.2.56 in [ 1 8 ]): F act 4.2 0. L et M b e a L eb esgue-me asur able set s u ch that ¯ M ⊂⊂ R and take u ∈ G ( R ) . L et ( u ε ) ε b e a r epr esentative of u . Then Z M u ( x ) dx :=  Z M u ε ( x ) dx  ε + N is a wel l-deﬁne d element of e R c al le d the inte gr al of u over M . Also, we are go ing to need the ’antideriv ative’ F o f a genera lized function. Let f ∈ G ( R ). This we introduce by F ( x ) := Z x 0 f ( s ) ds :=  Z x 0 f ε ( s ) ds  ε + N ( R ) ∈ G ( R ) where ( f ε ) ε is an a rbitrary repr e sentativ e of f . Note that F is the primitiv e of f with p oint v alue F (0) = 0 in e R (cf. Prop osition 1.2.5 8 in [ 18 ]). 4.3.1.4. The c onc ept of asso ciation. Finally we recall the concept of as so ciation in e R and in G ( R d ). First, let α ∈ e R . W e write α ≈ 0 and we say ” α is asso cia ted to zero” , if for some (hence any) represe ntative ( α ε ) ε we hav e α ε → 0 whenever ε → 0 . Similarly , we say u ∈ G ( R n ) is asso ciated with zero, if for each test function φ we hav e Z u ε ( x ) φ ( x ) dx n → 0 whenever ε → 0 . The relation ≈ is a n equiv alence r elation on e R resp. G ( R d ). By s lightly abusing the ab ov e terminolog y w e write u ≈ w , w ∈ D ′ ( R d ) and say ” u is asso cia ted with w ” (or , ” w is the distributional shadow o f u ” ), if we hav e Z u ε ( x ) φ ( x ) dx n → h w , φ i whenever ε → 0 . It is a well known fact that a generalize d function u has at most one distributional shadow (cf. [ 18 ], Pro po sition 1.2.67 ). 88 4. POINT V ALUES & UNIQUENES S QUE STIONS 4.3.2. Generalized functions supp orted at the origin. T o start with we establish a basic lemma: Lemma 4.21 . L et f ∈ G ( R ) b e a non-n e gative function with supp( f ) ⊆ { 0 } . If for some a > 0 we have I ( f ) = Z [ − a,a ] f ( x ) dx = 0 , then f = 0 . Proof. W e presen t t wo v ar iants of the proo f: First Pr o of. It ha s been sho wn r ecently (cf. [ 37 ]) that if for a generalized funct ion f we hav e for all ϕ ∈ G c ( R ) (the space of co mpactly supp orted generalized functions) Z f ( x ) ϕ ( x ) dx = 0 , then f = 0 in G ( R ). This is the so- called fundamen tal lemma o f the calculus o f v ariations in the genera lized context. Now we hav e by the no n-negativity of f ,     Z f ( x ) ϕ ( x ) dx     ≤ k ϕ k ∞ Z f ( x ) dx = 0 , therefore by the a bove we have f = 0 in G ( R ) and w e are do ne. Alternative Pr o of. This pro o f employs contin uity arguments of gene r alized functions wit h r esp ect to the sharp top olog y . In view of the ﬁr st pro of this may also yield a link b etw een the fundamen tal lemma of v ariatio na l calculus (in the g eneralized setting) and (sharp) top ological iss ue s . F or our (indirect) pro of we pro ceed in three steps. Step 1. Since f is non-negative a nd K := [ − a, a ] ⊂⊂ R is a compact set, we may choose a representative ( f ε ) ε of f which is non-neg ative on K , that is, ( f ε ) ε satisﬁes: ∀ x ∈ K ∀ ε > 0 : f ε ≥ 0 . Assume f 6 = 0 in G ( R ). Due to (cf. s ubsection 1.2.0.1), there exists a compactly suppo rted ge neralized p oint x c ∈ e R such that f ( x c ) = c 6 = 0 . F r om our assumption on the s uppo rt of f (supp( f ) ⊆ { 0 } ) it is further evident that x c ≈ 0; this informa- tion, how e ver, is not crucial for what fo llows). Step 2. Let ( x ε ) ε be a repr esentativ e of x c . W e shall prov e the following: (4.12) ∃ ε k → 0 ∃ m 0 ∃ ρ 0 ∀ k ∀ y k ∈ [ x ε k − ε ρ 0 k , x ε k + ε ρ 0 k ] : f ε k ( y k ) ≥ ε m 0 k . T o se e this, we ﬁrst o bserve by means of Step 1 that there exists a zero s equence ε k and a real n umber m 0 such that f or ea ch k ≥ 0 we hav e f ε k ( x ε k ) ≥ 2 ε m 0 k (w e shall take this zer o seq uenc e as the one o f our cla im). Next, we employ a co n tin uit y argument to prov e (4.12). Recall that f viewed as a ma p e f : e R c → e R is co nt in uous with resp ect to the shar p topo logy (cf. subsection 4.3.1 .2). Ass ume that (4.12) is not true. Then for each m and for each ρ there exists a sequence ( y k ) k with y k ∈ [ x ε k − ε ρ k , x ε k + ε ρ k ] for each k such that (4.13) 0 ≤ f ε k ( y k ) < ε m k 4.3. SCALING INV ARIANCE 89 (the ﬁr s t inequality holds because w e ma y assume without lo ss o f genera lity that everything takes place ins ide [ − a, a ], where we have found a non-neg ative r epresen- tative of f ). Deﬁne a (compactly supp o rted) generalized num b er y := ( y ε ) ε + N via y ε := ( y k , if ε = ε k x ε , otherwise Then we hav e for suﬃcien tly small m | f ε k ( x ε k ) − f ε k ( y ε k ) | > 2 ε m 0 k − ε m k > ε m 0 k , whereas for ε 6 = ε k we ha v e by the a bove co nstruction tha t f ε ( x ε ) − f ε ( y ε ) = 0 . In terms of the sharp nor m | | e we therefor e hav e: | f ( x c ) − f ( y ) | e ≥ e − m 0 ; by our assumption, how ever, it follows that | x c − y | e ≤ e − ρ 0 . The c hoice of ρ w as arbitrary , and ρ → 0 violates the contin uit y of f at x c . Therefore we hav e es tablished (4 .12). This we apply in the third and ﬁnal step: Step 3. F or suﬃcien tly large k we obtain (4.14) Z [ − a,a ] f ε k ( y ) dy > ε m 0 k (2 ε ρ k ) = 2 ε ρ + m 0 k . Since  R [ − a,a ] f ε ( y ) dy  ε is a representative of I ( f ), ineq uality (4.14) contradicts our assumption I ( f ) = 0 (the re presentativ e no t b eing a neg ligible net) a nd we are done.  A further ingredient in the subseq uen t pro of of our main result is the ele- men tary observ ation that gener alized scaling inv a r iant functions f ∈ G ( R ) with suppo rt contained in the or igin hav e to b e identically z e ro. T o motiv ate o ur pro of, we ﬁrst analyze the-mayb e- s implest non- trivial exa mple: a g e neralized function ˆ ρ asso ciated with a distribution supp or ted at the origin, say δ . In this situation in- v ariance under standard s caling is absurd: Assume we are given a standard molliﬁer ρ ∈ C ∞ c ( R ), that is R R ρ ( x ) dx = 1 . T he n ρ ε : ( 1 ε ρ ( x ε )) ε gives r ise to a generalized function ˆ ρ := [( ρ ε ) ε ] ∈ G ( R ) and, as it is well known, we hav e : ˆ ρ ≈ δ, that is, ∀ ϕ ∈ C ∞ c ( R ) : lim ε → 0 h ρ ε , ϕ i → h δ, ϕ i = ϕ (0) . Consider no w, h 6 = 0 , 1 and assume the identit y ˆ ρ ( hx ) = ˆ ρ ( x ) holds in G ( R ). This in particular means that ˆ ρ = [( ρ ε ( hx )) ε ] in G ( R ) holds as w ell. But fo r each ϕ ∈ C ∞ c ( R ) , ϕ (0) 6 = 0 w e ha v e lim ε → 0 h ρ ε ( hx ) , ϕ ( x ) i → 1 h ϕ (0) 6 = h δ, ϕ i therefore ˆ ρ h as more than one distributional shadow, namely δ, hδ, for arbitrary h 6 = 0 which is imp os s ible! 2 W e ma y now pre sent the statement in full generality: 2 Of course, δ i s scaling inv ar iant , ho w ev er, in the s ense that h δ ( ∗ h ) , ϕ i := h δ, ϕ ( ∗ h ) i . 90 4. POINT V ALUES & UNIQUENES S QUE STIONS Prop ositi o n 4.22. Assu me f ∈ G ( R ) has the fol lowing pr op ert ies: (i) f is invari ant under p ositive standar d sc aling. (ii) supp( f ) ⊆ { 0 } . Then f = 0 in G ( R ) . Proof. Assume f ∈ G ( R ) satisﬁes the ass umption of the pro po sition and without loss o f gener ality we further a s sume f ≥ 0 (otherwis e, take f 2 instead of f ). Let ( f ε ) ε be a repr esentativ e of f and a > 0. Since f is supp orted at the or igin, the integral I ( f ) := Z [ − a,a ] f ( x ) dx := Z [ − a,a ] f ε ( x ) dx ! ε + N ∈ e R is w ell deﬁned, that is , the v alue I ( f ) is indep endent o f the choice of a > 0 re sp. o f the repr esentativ e of f . F urther, for eac h h 6 = 0 , 1 and ea ch ε > 0 we hav e: Z [ − a,a ] f ε ( xh ) dx = 1 h Z [ − ah,ah ] f ε ( s ) ds. The scaling in v a riance of f , therefore, w hich in ter ms of representativ es reads ( f ε ( x )) ε − ( f ε ( hx )) ε = ( n ε ( x )) ε ∈ N ( R ) , combined with the fact tha t f is supp orted in the origin, yields I ( f ) = 1 h I ( f ) in e R . Since h 6 = 0 , 1 this implies I ( f ) = 0 . Now we may a pply L e mma 4.21 to the non-negative function f , and we obta in f = 0.  4.3.3. The main theorem. W e are now r eady to state the main theor em: Theorem 4. 23. L et f ∈ G ( R ) . The fol lowing ar e e quivalent: (i) f is c onstant , that is, ther e exists an a ∈ e R such that f = a holds in G ( R ) . (ii) e f is c onstant. (iii) e f is lo c al ly c onstant. (iv) f is tr anslation invariant, that is ∀ h ∈ R : f ( x + h ) = f ( x ) holds in G ( R ) . (v) f is invariant under p ositive standar d sc aling, that is, ∀ h ∈ R + : f ( hx ) = f ( x ) . (vi) F is addi tive, that is, ∀ h ∈ R : F ( x + h ) = F ( x ) + F ( h ) holds i n G ( R ) . (vii) e F is additive, that is, ∀ x c , h c ∈ e R : e F ( x c + h c ) = e F ( x c ) + e F ( h c ) holds in e R . In terms of the mo del delta net ab ov e this r efers to the fol lowing ’scaling’: ρ ε ( x ) 7→ hρ ε ( hx ) , h 6 = 0 and for each h 6 = 0 the ’scaled’ ob ject i s asso ciated to δ as well; furthermore even the identit y ˆ ρ ( x ) = h ˆ ρ ( hx ) holds i n G ( R ) (cf. the proof of Proposi tion 4.22). 4.3. SCALING INV ARIANCE 91 (viii) F has t he fol lowing pr op erty: Ther e exists γ ∈ (0 , 1) su ch tha t the i den- tity: (4.15) ∀ h ∈ R : F ( γ x + (1 − γ ) h ) = γ F ( x ) + (1 − γ ) F ( h ) holds in G ( R ) . Proof. W e establish the implications (iii) ⇒ (i) ⇒ (ii) ⇒ (iii) a s well a s (viii) ⇒ (iv) ⇒ (i) ⇒ (viii) and the equiv alences (i) ⇔ (vi), (i) ⇔ (v), T o begin with, as- sume (iii ), that is f is lo ca lly co nstant. W e show the implication by applying the generalized diﬀerential calc ulus fo r Co lombeau genera lized functions ev alu- ated on genera lized points as has b een developed by Arago na et al. in ([ 4 ]). Let κ : G ( R ) → e R e R c be the linear embedding o f ge ne r alized functions into mappings on compactly suppor ted po int s due to [ 38 ]. Due to ([ 4 ], Theorem 4.1) diﬀerentiation in G ( R ) resp. in e R e R c commute with κ . Clea rly κ ( f ) is diﬀerentiable with deriv a tive κ ( f ) ′ ≡ 0, and as just mentioned, κ ( f ′ ) = κ ( f ) ′ = 0, therefor e, due to the gener al- ized point c haracteriza tion in G ( R ) ([ 38 ]) we have f ′ = 0 in G ( R ) and integrating yields (i) that is, f is constant as a generalized function. The latter immediately implies (ii) by ev aluating f on compactly supp or ted generalized num b ers a nd the implication (ii) ⇒ (iii) is trivia l. Next, let γ ∈ (0 , 1) and assume (4.15) holds for F . Diﬀerentiating yields ∀ h ∈ R : f ( γ x + (1 − γ ) h ) = f ( x ) ho lds in G ( R ) . This is equiv alent to (4.16) ∀ h ∈ R : f ( x + h ) = f ( γ − 1 x ) ho lds in G ( R ) . Setting h = 0 shows tha t f ( x ) = f ( γ − 1 x ) in G ( R ) whic h further implies (4.17) ∀ h ∈ R : f ( x + h ) = f ( x ) holds in G ( R ) , i. e ., f is tra nslation inv ariant. This prov es (iv). The implication (iv) ⇒ (i) is prov en in ([ 40 ], Theo r em 6); for an alternative pro of cf. the a ppe ndix to [ 48 ]. Since f = a implies F = ax in G ( R ), the implica tion (i) ⇒ (viii ) holds. F urther w e e s tablish the equiv ale nce (i) ⇔ (vi). Aga in (i) implies that F is of the form F = ax with some generalized num b er a , therefore (vi) ho lds. Con v ersely , assume that f satisﬁes F ( x + h ) = F ( x ) + F ( h ) ho lds in G ( R ) for each h ∈ R . Diﬀerent iation yields ∀ h ∈ R : f ( x + h ) = f ( x ) holds in G ( R ) and by the a bove this implies (i). Finally we establish the equiv alence (i ) ⇔ (v). Since (i) ⇒ (v) is trivial, w e only need to sho w (i) ⇐ (v): Note that without loss of g e ne r ality w e may assume that f is symmetric (or equiv- alently , f is inv ariant under any non-zero standard scaling). Indeed, if f is not, we may introduce the tw o functions g ± resp. f ± given via g + ( x ) := f 2 + ( x ) := ( f ( x ) + f ( − x )) 2 , g − ( x ) := f 2 − ( x ) := ( f ( x ) − f ( − x )) 2 . If for generalized consta nt s c 1 , c 2 we w o uld hav e g + = c 1 , g − = c 2 , then for some generalized cons tants d 1 , d 2 we would hav e f + = d 1 , f − = d 2 , therefo re f ( x ) := f + ( x ) + f − ( x ) 2 = d 1 + d 2 2 , 92 4. POINT V ALUES & UNIQUENES S QUE STIONS that is, f is a constant, a nd we would b e done. W e may pro cee d no w in tw o diﬀeren t w ays: the ﬁrst is a v ariant of H. V erna eve’s ([ 48 ]) pro of o f the Lobster problem. First pr o of. W e distinguish the tw o p ossible cases , ’ f is constant in a neighbo rho o d o f 1’ or no t. Case 1 Assume ﬁrst, there ex is t a neig hborho o d Ω := (1 − δ, 1 + δ ) of 1, δ > 0 and c ∈ e R such that f = c o n Ω. Since f is inv ariant under pos itive standard scaling and symmetric, it follo ws that (i) F o r each h > 0, f = c o n ( h − hδ, h + hδ ). (ii) f ( x ) = f ( − x ) in G ( R ). Since G ( R ) is a sheaf, f = c on R \ { 0 } and w e hav e obtained a scaling inv ariant generalized function g := f − c with s upp( g ) ⊆ { 0 } . Applying Prop o sition 4.22 yields g = 0, that is , f is a constant and w e are done with the ﬁrst case. Case 2 If f | Ω ∈ G (Ω) is non-constant o n every standard neighbor ho od Ω = (1 − δ, 1 + δ ) ( δ > 0) o f 1, then we hav e for a ny repr esentativ e ( f ε ) ε of f : ( f ε | Ω − f ε (1)) / ∈ N (Ω) . Thu s there exists a r e presentativ e ( f ε ) ε of f along with a zero sequence ( ε k ) k , a sequence ( a k ) k ∈ [ 1 2 , 3 2 ] N and a n N suc h that for all suﬃciently large k w e ha ve (4.18) | f ε k ( a k ) − f ε k (1) | > ε N k . W e now fo llow the basic idea of H. V ernaeve in (theor em 7 in [ 48 ]). Let g k ( x ) := f ε k ( x ) − f ε k (1) for each k ≥ 1 . W e deﬁne A k := { x ∈ R : | g k ( x ) | < 1 3 ε N k } , B k := \ m ≥ k A m It is evident that for all k ∈ N g k (1) = 0, ther efore 1 ∈ B 1 . F ur thermore for each x ∈ R ∗ there e x ists ( n ε ) ε ∈ N suc h that for eac h ε ∈ I we hav e f ε ( x ) = f ε (1) + n ε . In particular g k ( x ) = f ε k ( x ) − f ε k (1) = n ε k . As a consequence ∀ x ∈ R ∗ ∃ k 0 ∀ k ≥ k 0 : x ∈ A k . This c le arly implies that fo r each x ∈ R ∗ there e x ists a k ≥ 1 such that x ∈ B k , therefore w e obtain (4.19) R ∗ ⊆ ( ∞ [ k =1 B k ) ⊆ R . In a similar way a s A k , B k we introduce the sets: C k := { x ∈ R : | g k ( xa k ) − g k ( a k ) | < 1 3 ε N k } , D k := \ m ≥ k C m . 4.3. SCALING INV ARIANCE 93 Again for each x ∈ R ∗ , x ∈ D k for some k , since b y the ass umption o f sca ling inv ar iance there exists an ( n ε ( y )) ε ∈ N ( R ) suc h that g k ( xa k ) − g k ( a k ) = f ε k ( xa k ) − f ε k (1) − f ε k ( a k ) + f ε k (1) = f ε k ( xa k ) − f ε k ( a k ) = n ε k ( a k ) . Therefore we hav e (4.20) R ∗ ⊆ ( ∞ [ k =1 D k ) ⊆ R . B k and D k are increasing sequences of L e bes gue measurable subsets of R . Let µ be the Lebesgue measur e on R and let B r ( x ) denote the o pe n ball with radius r and center x . F o r each ρ > 0 w e ha ve due to (4.19) and (4.20) (4.21) µ ( B ρ (0) \ B k ) → 0 , µ ( B ρ (0) \ D k ) → 0 , ( k → ∞ ) . Moreov er by constr uction B k ⊆ A k and D k ⊆ C k , therefore for e ach ρ > 0 w e also hav e (4.22) µ ( B ρ (0) \ A k ) → 0 , µ ( B ρ (0) \ C k ) → 0 , ( k → ∞ ) . Finally we deﬁne E k := { x ∈ R : | g k ( x ) − g k ( a k ) | < 1 3 ε N k } = a k C k . By the above we obtain for each ρ > 0 µ ( B ρ (0) \ E k ) = µ ( B ρ (0) \ a k C k ) = µ  a k ( B ρ | a k | (0) \ C k )  = | a k | µ ( B ρ | a k | (0) \ C k ) ≤ 3 2 µ ( B 2 ρ (0) \ C k ) → 0 , (4.23) whenever k → ∞ , since 1 2 ≤ | a k | ≤ 3 2 and due to (4.2 2). A consequence o f (4.2 2) and (4.23) is the following: µ ( B ρ (0) \ ( A k ∩ E k )) ≤ µ ( B ρ (0) \ A k ) + µ ( B ρ (0) \ E k ) → 0 , that is, for suﬃciently large k the in ters e c tion of A k and E k is not empt y , i. e., ∃ k 0 : ∀ k ≥ k 0 ∃ y k ∈ A k ∩ E k . Hence | g k ( y k ) | < 1 3 k N and | g k ( y k ) − g k ( a k ) | < 1 3 ε N k for all k ≥ k 0 . The triang le inequality yields for each k ≥ k 0 | g k ( a k ) | = | f ε k ( a k ) − f ε k (1) | < 2 3 ε N k . This contradicts line (4.1 8) and we are done. Alternative pr o of. First we c onsider the problem for f ∈ G ( R + ) i. e., ∀ λ > 0 : f ( λx ) = f ( x ) in G ( R + ) . This is equiv alent to the problem ∀ h ∈ R : g ( x + h ) = g ( x ) in G ( R ) , 94 4. POINT V ALUES & UNIQUENES S QUE STIONS where g := f ◦ exp. The r efore, b y ([ 40 ], Theorem 6) it follows that g = const. W e a re going to show that f is constant on R + as well. T o this end, note that the logarithm on e R + c is a w ell deﬁned mapping since it stems from ev alua tion of log ∈ G ( R + ). Assume that f is non-constant on the p ositive real n umbers , that is , there exis t x + c , y + c ∈ e R + c such that e f ( x + c ) 6 = e f ( y + c ). This is equiv alent to the fact that f ◦ e x p( x c ) 6 = f ◦ exp( y c ), where x c := log x + c , y c := log y + c , a con tra diction. By the symmetry of f we ha ve f = c = const on R \ { 0 } . Now we pro ceed as in Case 1 of the ﬁrst v aria nt of the pro of and w e are done.  4.3.4. Scaling inv ariance i n space. In the preceding section we establis hed that any generalize d function on the real line, which is inv ariant under po sitive standard sca ling, is a constant. An impo rtant information we used was that without loss of generality we may assume that f is symmetric. This help ed us to o vercome the obstac le that R \ { 0 } is not connected, and we were able to reduce the pro blem to scaling inv ariance of gener alized functions supp or ted at the origin. The analogous question in higher spa ce dimensions may be r educed to the one dimens ional case. In the following, d is an a rbitrary p ositive in teger. Theorem 4.24. Any gener alize d function f in R d which is i nvariant under st an- dar d sc aling is c onstant. Proof. Let f ∈ G ( R d ) b e inv ariant under p os itive (standard) scaling, that is, ∀ λ ∈ R , λ > 0 we hav e: f ( λx ) = f ( x ) . Fix a net ( a ε ) ε such that a ε ∈ L ⊂⊂ R d for all ε > 0 . Then the net ( g ε ) ε := ( f ε ( a ε t )) ε deﬁnes a g eneralized function g := [( g ε ) ε ] ∈ G ( R ). 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