High Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation
HHigh Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation
Miguel Duarte , , Justin Feng , Edgar Gasper´ın , , and David Hilditch CAMGSD, Departamento de Matem´atica, Instituto Superior T´ecnico IST,Universidade de Lisboa UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal, CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico IST,Universidade de Lisboa UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal, Institut de Math´ematiques de Bourgogne (IMB), UMR 5584, CNRS,Universit´e de Bourgogne Franche-Comt´e, F-21000 Dijon, France
A heuristic method to find asymptotic solutions to a system of non-linear wave equations nearnull infinity is proposed. The non-linearities in this model, dubbed good-bad-ugly, are known tomimic the ones present in the Einstein field equations (EFE) and we expect to be able to exploitthis method to derive an asymptotic expansion for the metric in General Relativity (GR) close tonull infinity that goes beyond first order as performed by Lindblad and Rodnianski for the leadingasymptotics. For the good-bad-ugly model, we derive formal expansions in which terms proportionalto the logarithm of the radial coordinate appear at every order in the bad field, from the secondorder onward in the ugly field but never in the good field. The model is generalized to wave operatorsbuilt from an asymptotically flat metric and it is shown that it admits polyhomogeneous asymptoticsolutions. Finally we define stratified null forms, a generalization of standard null forms, whichcapture the behavior of different types of field, and demonstrate that the addition of such terms tothe original system bears no qualitative influence on the type of asymptotic solutions found.
I. INTRODUCTION
Although the concept of null infinity has been presentin General Relativity (GR) for a long time, dating backat least to the works of Penrose, Newman, Bondi andSachs —see for instance [1–5], there are still questionsto be answered about the structure of spacetime nearby.From the point of view of mathematical relativity, nullinfinity plays a central role in the resolution of open prob-lems such as the weak cosmic censorship conjecture, the(non) peeling properties and global stability analysis ofspacetimes. From an astrophysical perspective, null in-finity should also play an important role since gravita-tional radiation is not localizable and hence it is onlywell defined at null infinity. The latest achievements ingravitational wave astronomy are coupled to advances innumerical relativity since the former rely on the accu-rate calculation of waveforms from astrophysically rele-vant scenarios. However, the waveforms that are rou-tinely computed in numerical relativity codes are evalu-ated at a large but finite radius and extrapolated to in-finity. Despite that the latter has proven to work, from amathematical point of view these wave forms should thusbe computed directly at null infinity. There have beenvarious approaches to include null infinity in the compu-tational domain. For instance, the work of H¨ubner [6, 7]and Frauendiener [8] makes use of the conformal Einsteinfield equations (CEFE) introduced by Friedrich [9, 10]building upon Penrose’s idea of bringing null infinity toa finite coordinate distance by means of a conformal com-pactification [1]. Although the CEFE provide a geomet-ric approach to the problem of the inclusion of null in-finity, the standard methods of numerical relativity thathave proven to work well for the strong field region ofspacetimes of physical interest cannot be trivially liftedover. In particular, this approach has not yet been used for compact binary evolutions.To overcome this situation a variety of approaches havebeen pursued, including using Cauchy-CharacteristicMatching [11] and the use of a suitable hyperboloidalinitial value problem. The latter involves providing ini-tial data on a hyperboloidal slice, a spacelike hypersur-face that intersects future null infinity. These slices arenot Cauchy hypersurfaces, as their domain of dependencedoes not cover the whole spacetime. The main techni-cal problem with this approach is that it results in for-mally singular terms which complicates significantly theirmathematical analysis as well as their numerical imple-mentation. Although challenging, this type of singularequation has been treated numerically in spherical sym-metry [12–15].In light of the above, a relevant problem to be solvedon the analytical side, with direct implications for nu-merical work, is the construction of an alternative to theCEFEs using more standard formulations of the Einsteinfield equations (EFE) but insisting on including null in-finity. A recent proposal to make inroads into the con-struction of such a formulation in the hyperboloidal setup is to use a dual frame approach [16], which consistsessentially of decoupling coordinates from the tensor ba-sis and carefully choosing each of them. This allows oneto write the EFE in generalized harmonic gauge (GHG)and then solve them in hyperboloidal coordinates. Var-ious different aspects of this proposal have been investi-gated [17–20]. Here we give just a brief overview. Anessential prerequisite for this to work is the satisfactionof the coordinate lightspeed condition [18]. As discussedin [18], the coordinate lightspeed condition is related tothe weak null condition [21]. The former is the require-ment that derivatives of the radial coordinate lightspeedshave a certain fall-off near null infinity, while the latter isexpected to be a sufficient condition on the non-linearities a r X i v : . [ g r- q c ] J a n of a quasilinear wave equation for establishing small dataglobal existence. Although it has not been shown in fullgenerality that the weak null condition implies small dataglobal existence, a recent work by Keir [22] proved thatif a system of quasilinear wave equations satisfies the hi-erarchical weak null condition , then small data global ex-istence is guaranteed. H¨ormander’s asymptotic system , aheuristic method that predicts the fall-off of solutions tosystems of quasi-linear wave equations, was used in [18]to show that through constraint addition, one can guar-antee that the resulting field equations satisfy the light-speed condition beyond the initial data.The work in [17] and [18] together shows that formallysingular terms can be avoided by using the dual foliationformalism [16] in combination with hyperboloidal coordi-nates and GR in GHG. However, even the simplest choiceof variables shows the existence of metric componentswith a fall-off of the type O ( R − log R ), with R a suit-ably defined radial coordinate. This can cause problemsin numerical evolutions. In [19], the authors use a toymodel composed of wave equations with non-linearitiesof the same kind as those present in the EFE to showthat these logarithmically divergent terms can be explic-itly regularized by a non-linear change of variables. Thistoy model is called the good-bad-ugly model as it splitsthe evolved fields into three categories according to theirfall-off near null infinity, and it is known to satisfy theweak null condition.In this work we generalize earlier results on the good-bad-ugly model, laying out a heuristic method to predictthe type of decay of terms beyond the leading ones in avery large class of systems of non-linear wave equationsnear null infinity. This provides us with the knowledgeof where log-terms may appear in asymptotic expansionsso that we are able to manage those terms appropriatelyin the numerics. The adjective heuristic in this contextis used to emphasize that the connection between theoriginal weak null condition introduced by Lindblad andRodnianski in [21] and small data global existence hasnot been proven yet, at least not in full generality. Toremove this adjective one would need to prove suitableestimates for our formal expansions. This goes beyondthe scope of the present article.In Sections II and III we outline our geometric setupand basic assumptions. In section IV we begin our anal-ysis proper by considering the same model used in [19].We show by induction that the bad field may have logsat every order in R − and the ugly field may have logsfrom second order onward under certain conditions, whilethe good field must have no logs at all. Additionally, wepresent a recursion relation that gives the coefficient as-sociated with any power of R − of the evolved fields interms of the previous order, ultimately in terms of the ini-tial data. In section V the model is generalized to allowthe wave operator to be built from a general asymptoti-cally flat metric whose components are allowed to dependanalytically on the evolved fields. An induction proofanalogous to the one shown in section IV is presented to assert that with such a wave operator, the equationsare much more deeply coupled and hence all fields mayinherit logs from one another, the main difference beingthe order at which they are allowed to first appear. Infact, the proof shows that the good-bad-ugly model allowsfor asymptotic solutions which can be written as polyho-mogeneous expansions, loosely speaking inverse power-law decay in R but with logarithmic obstructions, closeto null infinity. Finally, in section VI we generalize themodel even further by adding arbitrary linear combina-tions of what we call stratified null forms, a generalizationof the standard notion that knows about the behavior ofthe three different types of field, to the original systemand showing that these terms do not affect the the proof.Because we keep the metric general, naturally it is notpossible to find a final recursion relation for the evolvedvariables as is done in section IV, but once the exact de-pendence of the metric on the evolved fields is given, itshould be possible to find such a relation. Concludingremarks are given in section VII. II. GEOMETRIC SET UP
Representation of the metric:
Latin indices will beused as abstract tensor indices while Greek indices willbe used to denote spacetime coordinate indices. We as-sume the existence of a Lorentzian metric g ab with Levi-Civita connection ∇ and introduce the coordinate sys-tem X α = ( T, X i ), which we require to be asymptoti-cally Cartesian in a sense clarified below. We raise andlower indices with the spacetime metric g ab exclusively.Let ∂ α and dX α be the corresponding vector and co-vector bases. The covariant derivative associated to X α is ˚ ∇ and its Christoffel symbols are defined by,Γ[˚ ∇ ] abc = (˚ ∇ a ∂ bα )( dX α ) c . (1)Additionally we define shell coordinates X α (cid:48) =( T (cid:48) , X i (cid:48) ) = ( T, R, θ A ), where the radial coordinate R isrelated to X i in the usual manner as R = ( X ) +( X ) + ( X ) . Let ∂ α (cid:48) and dX α (cid:48) be the correspondingvector and co-vector bases. Shell coordinates have an as-sociated covariant derivative • ∇ with Christoffel symbols,Γ[ • ∇ ] bac = ( • ∇ b ∂ aα (cid:48) )( dX α (cid:48) ) c . (2)The transition tensor between the two covariant deriva-tives is defined by,Γ[˚ ∇ , • ∇ ] abc v c = ˚ ∇ a v b − • ∇ a v b , (3)where v a is an arbitrary vector field. We define outgoingand incoming null vectors according to, ψ a = ∂ aT + C R + ∂ aR ,ψ a = ∂ aT + C R − ∂ aR , (4)where C R + and C R − are fixed by the requirement that ψ a and ψ a are null vectors with respect to the metric g ab .We furthermore define two null co-vectors by, σ a = e − ϕ ψ a , σ a = e − ϕ ψ a , (5)where ϕ is fixed by requiring that σ a ∂ aR = − σ a ∂ aR = 1 , (6)so we can write, σ a = −C R + ∇ a T + ∇ a R + C + A ∇ a θ A ,σ a = C R − ∇ a T − ∇ a R + C − A ∇ a θ A . (7)We choose to write the inverse spacetime metric as, g ab = − τ − e − ϕ ψ ( a ψ b ) + /g ab , (8)where the null vectors satisfy, σ a ψ a = σ a ψ a = 0 ,σ a ψ a = σ a ψ a = − τ , (9)with τ := C R + − C R − . In (8), the normalization of the firstterm is carefully chosen so that, /g ab σ b = /g ab σ b = 0 , (10)and /g ba therefore serves as a projection operator orthog-onal to these two covectors. Note that /g ab is not theinverse induced metric on level sets of T and R , as it isnot orthogonal to ∇ a T or ∇ a R , but rather to σ a and σ a .At first sight this seems unsatisfactory geometrically, butsince we will be heavily using the method of characteris-tics it turns out that to be much more convenient to havea simple representation of the vectors ψ a and ψ a than thecovectors σ a and σ a . Our convention for /g ab follows fromthis fact. The metric can be written naturally as, g ab = − τ − e ϕ σ ( a σ b ) + /g ab . (11)This way we have ten independent metric functions,namely, C R ± , C ± A , ϕ , /g ab . (12)Tensors projected with /g ab will be denoted adding a slashto the kernel letter /T ab ≡ /g ac /g bd T (cid:48) cd . The covariantderivative associated to /g ab will be denoted as /D , so that, /D b v a := /g ca /g bd ∇ d v c , (13)where the vector satisfies v a = /g ba v b . We take the ob-vious extension for higher rank tensors. Similarly, wedefine the covariant derivative ˚ /D as,˚ /D b v a := /g ca /g bd ˚ ∇ d v c . (14) We define the vector field T a := ∂ aT and denote the co-variant derivative in the direction of T a as ∇ T . Anal-ogous notation will be used for directional derivativesalong other vector fields. Because we will have to dealwith terms proportional to R − n (log R ) m , the use of theterm ‘order’ might be confusing, as for the same powerof R − different values of m give rise to different decays.To clarify that, throughout this work, ‘order n ’ will de-note terms proportional to R − n . The good-bad-ugly system:
We introduce the follow-ing model, ˚ (cid:3) g = 0 , ˚ (cid:3) b = ( ∇ T g ) , ˚ (cid:3) u = R ∇ T u , (15)where g , b and u stand for good, bad and ugly fields, re-spectively, ˚ (cid:3) is called the reduced wave operator and itis defined by g ab ˚ ∇ a ˚ ∇ b . Because we will only be con-cerned with the large R regime, we are not concernedwith regularity at the origin. Therefore, for simplicity,we have adjusted the final equation of the model givenin [19] so that the source term appears with a simplecoefficient 2 /R . The metric g ab can be taken to dependon the evolved fields themselves in a manner we will ex-pand upon below. The leading order of the particularcase where the metric g ab is the Minkowski metric wasstudied in detail in [19]. III. ASSUMPTIONS
We need to place certain assumptions on the evolvedfields and metric functions that will allow us to formallyequate terms of the same order in (15) and retrieve sim-pler equations that are satisfied, order-by-order, by g , b and u close to null infinity. Evolved fields:
We define a null tetrad { ψ, ψ, X , X } ,where X and X are orthogonal to ψ a and ψ a and nor-malized respect to g ab , namely, g ab X aA ψ b = g ab X aA ψ b = 0and g ab X aA X bB = δ AB , with A = 1 ,
2. Let ω g,b be anyfield in { g, b } or any first derivative thereof. Based oninsight from [22] we will assume first derivatives of ω tohave the following behavior near null infinity, ω g,b = o + ( R − n ) ⇒ ∇ ψ ω g,b = o + ( R − n − ) ∇ ψ ω g,b = o + ( R − n ) ∇ X A ω g,b = o + ( R − n − ) , (16)with A ∈ { , } . Here, f = o + ( h ) as R → ∞ is definedas the condition, ∃ (cid:15) > R →∞ fhR − (cid:15) = 0 . (17)Note that this condition is a more restrictive versionof f = o ( h ), which can be informally stated as f falls-off faster than h (cid:15) as R goes to infinity . In particu-lar, o + ( h ) = o ( hR − (cid:15) ). The reason why we make thisslightly stronger assumption will become apparent oncewe start integrating error terms in the next section.Derivatives along ψ a and X A are called good deriva-tives, while the ones along ψ a are called bad derivatives.This naming convention is motivated by the fact that, forfields satisfying equations like ours, the former improvethe fall-off of the argument, whereas the latter do not.Let ω u be the field u or any first derivative thereof. Weknow from [19] that, in the case that ˚ (cid:3) is built from theMinkowski metric, derivatives of u have different asymp-totics from the other fields, so we assume, ω u = o + ( R − n ) ⇒ ∇ ψ ω u = o + ( R − n − ) ∇ ψ ω u = o + ( R − n − ) ∇ X A ω u = o + ( R − n − ) . (18)We make this set of assumptions using for exam-ple o + ( R − n ) instead of the more restrictive O ( R − n − ),because previous work has shown that similar equa-tions have asymptotic solutions proportional to, for in-stance, R − log R , and naively using big O notationwould not permit such solutions [19]. Furthermore, weare interested in physically relevant solutions, so we can-not allow fields which do not decay near null infinity.Therefore, we restrict our attention to a space of initialdata in which there is decay near null infinity, i.e., g = o + (1) , b = o + (1) , u = o + (1) . (19)Let S be a Cauchy surface defined by T = T , where T isa constant. In order to allow for nonzero ADM mass andlinear momentum, we choose initial data which decays atspacelike infinity as, φ = O S ( R − ) , ∇ T φ = O S ( R − ) , (20)where the subscript S is used to say that this fall-off is re-quired on a spatial slice, rather than at null infinity. Thisis not the most general choice of initial data which allowsfor nontrivial ADM mass and linear momentum, but itis broad enough to include practically all spacetimes ofinterest. Metric functions:
As derivatives of our metric func-tions will, in general, be present in the field equations,we must have a way to collect them in powers of R − .Therefore we require that the metric functions may bewritten as, C R ± = ± γ ± , C ± A = Rγ ± ,ϕ = γ ,/g ab = /η ab + R − γ ab , (21)where /η ab is the inverse metric on the round 2-sphereof radius R and the γ ’s are analytic functions of onlythe evolved fields in a neighborhood of null infinity, γ = γ ( g, b, u ). Moreover, we are interested in studying space-times with metrics that asymptote to the Minkowski met-ric as we approach I + , i.e. asymptotically flat metrics.This implies that the γ functions must go to zero as weapproach null infinity, that is, γ ( g, b, u ) | I + = 0 . (22) IV. FLAT METRIC
In this section we will study the asymptotics of the good-bad-ugly system (15) with a flat metric g ab = η ab near null infinity, where η ab is the inverse Minkowski met-ric. Based on our assumptions on the decay of the fieldsand their derivatives (16) and (18), we will equate termsof the same order to find simpler equations that will re-veal the asymptotics of g , b and u . Moreover, this sectionwill serve as a toy model for the next, where we analyzethe system for a general asymptotically flat metric. Re-quiring that the metric be flat implies, C R ± = ± ,ϕ = C ± A = 0 ,/g ab = /η ab , (23)meaning that all metric components are given, the onlyunknowns being the evolved fields themselves. The in-verse Minkowski metric can then be written in terms ofnull vectors in the following way, η ab = − ψ ( a ψ b ) + /η ab , (24)where ψ a and ψ a reduce to, ψ a = ∂ aT + ∂ aR ,ψ a = ∂ aT − ∂ aR . (25)The method we want to implement relies on integratingthe equations we get along different vector fields. Theseintegrations are made simpler if we work under a Bondi-like approach, rewriting the incoming null vector ψ a as afunction of the timelike vector ∂ aT and the outgoing nullvector ψ a , ψ a = 2 ∂ aT − ψ a . (26)Let φ be any field in { g, b, u } . We can expand the waveoperator in the following way,˚ (cid:3) φ = (cid:18) − ∇ ψ ∇ T − R ∇ T + 2 R ∇ ψ + ∇ ψ + (cid:0)(cid:0) ∆ (cid:19) φ , (27)where, (cid:0)(cid:0) ∆ φ := ˚ /D a ˚ /D a φ , (28)is the Laplace operator on the 2-sphere of radius R . Wetreat each equation in (15) separately, and starting withthe first. S I + cc ∗ ψ ∂ T FIG. 1. A schematic of our geometric setup. The methodproceeds first by integrating out along c , an integral curve ofthe outgoing null-vector ψ a and then up along integral curvesof ∂ aT . A. The Good field
Motivation for induction hypothesis:
We begin byrescaling g as, G := gR , (29)and plugging that into (27) to get the equation, − ∇ ψ ∇ T G + ∇ ψ G + (cid:0)(cid:0) ∆ G = 0 . (30)It is known from [22] that g and its derivatives sat-isfy (16), so there is a hierarchy among the different termsin (30) which allows us to neglect some of them and endup with a simpler equation that determines G to leadingorder. This is just H¨ormander’s first order asymptoticsystem for the wave equation. We know that the secondand third terms in (30) are of higher order than the firstone because they contain two good derivatives, so thefirst term must vanish by itself to leading order, ∇ ψ ∇ T G = o + ( R − ) . (31)We want to integrate this expression along integral curvesof ψ a and then ∂ aT . Since integrating error terms is notcompletely straightforward, we dedicate a paragraph af-ter the present one to proving that error terms remainsuitably small under integration. For now we will onlypay attention to the leading contributions. Let c ( s ) be anintegral curve of the vector field ψ a that passes throughthe point c ∗ ∈ S at a fiduciary value of s = s ∗ and inte-grate equation (31) along that curve to get, ∇ T G (cid:39) ˙ g ( c ∗ ) , (32)where ˙ g ( c ∗ ) = ∇ T g | c ( s ∗ ) . We use (cid:39) to represent equal-ity at large radius up to error terms that decay fasterthan those displayed in the expression. For example, wecan write f (cid:39) R − as short-hand for f = R − + o + ( R − ). As we have not specified the curve along which the in-tegration was performed, (32) is valid for any c and sowe can write ( ψ ∗ denoting the dependence on the initialdata at c ∗ ), ∇ T G (cid:39) ˙ g ( ψ ∗ ) , (33)where ˙ g ( ψ ∗ ) is fixed along any particular integral curveof ψ a . The exact same method will be used in the restof this work whenever integrating along integral curvesof ψ a . Integrating (33) in T we get, G (cid:39) (cid:90) TT ˙ g ( ψ ∗ ) dT (cid:48) + m g, , (34)where m g, is a scalar function that is independent of T and we choose it to be independent of R as well. In fact,our choice of initial data (20) requires that all of the m φ, functions throughout the rest of this work are indepen-dent of R . Moreover, because we impose (20), that choiceimplies that ˙ g ( ψ ∗ ) fall-off like O S ( R − ) because, ∇ T g (cid:39) ˙ g ( ψ ∗ ) R . (35)Let us now analyze the leading error terms. We definethe function G as, G R := G − G ( ψ ∗ ) = o + (1) , (36)where, G ( ψ ∗ ) := (cid:90) TT ˙ g ( ψ ∗ ) dT (cid:48) + m g, , (37)and assume that it also satisfies (16). Then from (30) weget,2 ∇ ψ (cid:18) R ∇ T G (cid:19) + ∇ ψ G R + (cid:0)(cid:0) ∆ (cid:18) G + G R (cid:19) = 0 . (38)Here it pays off to introduce the operator,˜ (cid:0)(cid:0) ∆ := R (cid:0)(cid:0) ∆ , (39)which makes the order in R − explicit, as (cid:0)(cid:0) ∆ amounts totwo good derivatives. Once again, collecting the lowestorder terms we get, asymptotically,2 ∇ ψ (cid:18) R ∇ T G (cid:19) + 1 R ˜ (cid:0)(cid:0) ∆ G = o + ( R − ) . (40)which gives, G (cid:39) − (cid:90) TT ˜ (cid:0)(cid:0) ∆ G dT (cid:48) + R (cid:90) TT ˙ g ( ψ ∗ ) dT (cid:48) + m g, . (41)In order to integrate along an integral curve of ψ a weparameterize c using the radial coordinate R so that weget, (cid:90) c R n dR = − n − R n − , (42)along any c . Note that the second term on the RHSof (41) grows like R , which would contradict (36). How-ever, if we were to write ∇ T g with what we know alreadyat first order and (41) as it is, we would get, ∇ T g = ∇ T G R + ∇ T G R (cid:39) R ( ˙ g + ˙ g ) + 12 ˜ (cid:0)(cid:0) ∆ G , (43)which implies that ˙ g can be absorbed into ˙ g . Thereforewe can choose solutions with ˙ g = 0 without any loss ofgenerality, so that, G (cid:39) G ( ψ ∗ ) , (44)where, G ( ψ ∗ ) := 12 (cid:90) TT ˜ (cid:0)(cid:0) ∆ G dT (cid:48) + m g, . (45)Equations (34) and (44) suggest that the field g may bewritten as, g = ∞ (cid:88) n =1 G n ( ψ ∗ ) R n , (46)and we prove that result shortly. However it is worthpausing here for a moment to take a closer look at howthe error terms behave under integration. Integration of error terms:
What we aim to show hereis that if f = o + ( R − n ) then, (cid:90) f dR = o + ( R − n +1 ) , (47)for all n ∈ N . In other words, we want to show,lim R →∞ R n − (cid:15) (cid:90) f dR = 0 . (48)Because (cid:15) >
0, and assuming f to be differentiable, wecan apply L’Hˆopital’s rule to the LHS of equation (48)in order to get,( − n + 1 − (cid:15) ) − lim R →∞ R n + (cid:15) f . (49)By definition we have that f = o + ( R − n ) = o ( R − n − (cid:15) ),which implies directly that (49) is zero. Therefore, (47)must be true. All error terms in this work are of theform o + ( R − n ), therefore this result will be used in everyintegration thereof. Note that if we had made the lessrestrictive assumption that f = o ( R − n +1 ) instead of f = o + ( R − n +1 ), for the case where n = 1 we would have hadto show that, lim R →∞ (cid:90) f dR = 0 . (50)This would not be possible using the same method andwe would not be able to ensure that error terms remainsmall. Induction proof:
We have seen that, g = G ( ψ ∗ ) R + G R , (51)with G = o + ( R ), so in order to prove our result (46), weonly need to show that if g can be written as, g = n − (cid:88) m =1 G m ( ψ ∗ ) R m + G n R n , (52)with G n = o + ( R ), then it can be written as, g = n (cid:88) m =1 G m ( ψ ∗ ) R m + G n +1 R n +1 , (53)with G n +1 = o + ( R ). We assume that each term in (52)satisfies (16), so we have a way to collect terms of thesame order. Then we plug (52) in (15) and formallyequate terms of order R − n − to find the PDE,2 R n ∇ ψ (cid:18) R n − ∇ T G n (cid:19) (cid:39) (cid:104) ( n − n −
2) + ˜ (cid:0)(cid:0) ∆ (cid:105) G n − . (54)Integrating equation (54) we find, G n (cid:39) − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT G n − dT (cid:48) + R n − (cid:90) TT ˙ g n ( ψ ∗ ) dT (cid:48) + m g,n . (55)It can be seen from (55) that, to leading order, G n doesnot satisfy (52) for a general ˙ g n . However, ˙ g n comes witha factor of R n − so, exactly like in (43), the behaviorof those terms is captured by ˙ g . We can then choosesolutions with initial data such that ˙ g n = 0 , ∀ n > T derivative of g to leading order as, ∇ T g (cid:39) R ( ˙ g + ... + ˙ g n ) + 12 ˜ (cid:0)(cid:0) ∆ G . (56)Notice that all the ˙ g n appear with the same prefactor. Sothe freedom that the integrations along ψ at each ordergive us in choosing initial data can in fact be expressedin the choice of one scalar function. This is expectedbecause the fact that we have n differential equations tosolve for the different orders in the field g is somewhatartificial, in the sense that they arise from a method tosolve a single differential equation. It is therefore naturalthat once we add all the terms, we are left with onlyone free function. The same approach will be taken forthe fields b and u whenever an integration along integralcurves of ψ a is done. This gives the result, G n (cid:39) − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT G n − dT (cid:48) + m g,n . (57)Clearly, (52) implies (53), so we conclude that g can bewritten as (46). Moreover, up to m g,n , we have a recur-sion relation that allows us to compute G n from G n − for any n > G n = − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT G n − dT (cid:48) + m g,n . (58) B. The Bad field
Motivation for induction hypothesis:
The case of b re-quires a different hypothesis. Following the same kind ofprocedure as in the g case, we begin by rescaling b in thefollowing way, B := bR , (59)and plugging it into (27). The b equation in (15) canthen be written as, − ∇ ψ ∇ T B + ∇ ψ B + (cid:0)(cid:0) ∆ B = R ( ∇ T g ) , (60)where g is now given. The bad field satisfies (16), socollecting terms of the lowest non-trivial order gives thefollowing, ∇ ψ ∇ T B (cid:39) − R ( ∇ T G ) . (61)We can integrate (61) to get, B (cid:39) −
12 log R (cid:90) TT ( ∇ T G ) dT (cid:48) + (cid:90) TT ˙ b ( ψ ∗ ) dT (cid:48) + m b, , (62)where ˙ b ( ψ ∗ ) is a scalar function. The behavior of b toleading order differs from that of g as it has a term thatgrows with log R . This result is in accordance with [19].We define the function B as, B R := B − B = o + (1) , (63)where, B := B , ( ψ ∗ ) + B , ( ψ ∗ ) log R ,B , ( ψ ∗ ) := (cid:90) TT ˙ b ( ψ ∗ ) dT (cid:48) + m b, ,B , ( ψ ∗ ) := − (cid:90) TT ( ∇ T G ) dT (cid:48) , (64)and assume that it also satisfies (16). The subscripts n and k in B n,k stand for the power of R − and the powerof log R associated with B n,k , respectively, and the samenotation will be used throughout this work except in thecase of g for a flat metric, where it is obvious that theassociated field vanishes when k is non-zero. While thisnotation seems needlessly cumbersome at this point, it will prove useful in the next section, where we find variouscombinations of powers of R − and log R . Replacing (63)in (60) and equating lowest order terms gives,2 ∇ ψ (cid:18) R ∇ T B (cid:19) + 1 R B , − (cid:0)(cid:0) ∆ B (cid:39) − R ∇ T G ∇ T G , which we can integrate to get, B (cid:39)
12 (1 − ˜ (cid:0)(cid:0) ∆) (cid:90) TT B , dT (cid:48) −
12 ˜ (cid:0)(cid:0) ∆ (cid:90) TT B dT (cid:48) + m b, + (cid:90) TT ∇ T G ∇ T G dT (cid:48) , (65)meaning we can write that, B (cid:39) B , ( ψ ∗ ) + B , ( ψ ∗ ) log R . (66)Equations (62) and (66) suggest that the field b may bewritten in the form, b = ∞ (cid:88) n =1 B n R n , (67)with B n = B n, ( ψ ∗ ) + B n, ( ψ ∗ ) log R and we prove thisresult in the following. Induction proof:
We have computed the bad field tofirst order, b = B R + B R , (68)with B = o + ( R ), so in order to prove our result (46), wemust show that if b can be written as, b = n − (cid:88) m =1 B m R m + B n R n , (69)with B n = o + ( R ), then it can be written as, b = n (cid:88) m =1 B m R m + B n +1 R n +1 , (70)with B n +1 = o + ( R ). We assume all terms in the sumin (69), as well as B n , satisfy (16) and we plug (69)into (15) to get,2 R n ∇ ψ (cid:18) R n − ∇ T B n (cid:19) (cid:39) − (2 n − B n − , + (cid:104) ( n − n −
2) + ˜ (cid:0)(cid:0) ∆ (cid:105) B n − − C n , (71)where C n is defined as, C n := i + j = n +1 (cid:88) i,j =1 ∇ T G i ∇ T G j . (72)Here, (cid:80) i + j = n +1 i,j =1 is meant as the sum over terms with anycombination of i and j as long as i, j ≥ and i + j = n +1.This can be integrated to get, B n (cid:39) (cid:34) − ˜ (cid:0)(cid:0) ∆( n − (cid:35) (cid:90) TT B n − , dT (cid:48) + m b,n (73) − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT B n − dT (cid:48) + 12( n − (cid:90) TT C n dT (cid:48) , as we wanted. As with ˙ g n above, we choose the func-tions ˙ b n = 0 , ∀ n >
1, effectively absorbing them into ˙ b in order to avoid a contradiction with assumptions (69).Finally we get a recursion relation for B n , B n = 12 (cid:34) − ˜ (cid:0)(cid:0) ∆( n − (cid:35) (cid:90) TT B n − , dT (cid:48) + m b,n (74) − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT B n − dT (cid:48) + 12( n − (cid:90) TT C n dT (cid:48) . This shows our hypothesis (67), with, B n, = 12 (cid:34) − ˜ (cid:0)(cid:0) ∆( n − (cid:35) (cid:90) TT B n − , dT (cid:48) + m b,n (75) − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT B n − , dT (cid:48) + 12( n − (cid:90) TT C n dT (cid:48) . and, B n, = − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT B n − , dT (cid:48) , (76)A closer look at (74) reveals that the only log R termcomes from B n − , and hence it is inherited by all ordersfrom B . C. The Ugly field
Motivation for induction hypothesis:
Once again werescale the field u by R as, U = uR , (77)and plug it into (27). The u equation in (15) can thenbe written as, − R ∇ ψ ( R ∇ T U ) + ∇ ψ U + (cid:0)(cid:0) ∆ U = 0 , (78)The ugly field has a somewhat different behavior fromthe other two. As can be seen in [19], both good and badderivatives improve the fall-off of u , as opposed to thecases of g and b , where only good derivatives improve.This means that all terms in (78) contribute to leadingorder and one would have to solve the whole equation at once. For this reason, we will focus on solutions that canbe decomposed as, U = m u, + U R , (79)where m u, is independent of T and R and U = o + ( R ).Plugging this into (78) we get, − R ∇ ψ ∇ T U + (cid:0) ∇ ψ + (cid:0)(cid:0) ∆ (cid:1) (cid:18) m u, + U R (cid:19) = 0 . (80)If we assume U to satisfy (16),2 ∇ ψ ∇ T U (cid:39) R ˜ (cid:0)(cid:0) ∆ m u, ⇒ (81) U (cid:39) (cid:90) TT (cid:104) log R ˜ (cid:0)(cid:0) ∆ m u, + ˙ u ( ψ ∗ ) (cid:105) dT (cid:48) + m u, . Equation (81) suggests that the ugly field can be writtenas, u = m u, R + ∞ (cid:88) n =2 U n R n , (82)with U n = U n, ( ψ ∗ ) + U n, ( ψ ∗ ) log R and we show thisresult in the following. Induction proof:
As was seen above, the first orderterm of the ugly field behaves differently from the rest,in that all of its derivatives improve. For this reason webegin the induction proof in the second order term, whichhas been computed in (79) and (81). To prove (82) wehave to show that if, u = m u, R + n − (cid:88) m =2 U m R m + U n R n , (83)with U n = o + ( R ), then, u = m u, R + n (cid:88) m =2 U m R m + U n +1 R n +1 , (84)with U n +1 = o + ( R ). We assume that all orders in (83)and U n satisfy (16) and we substitute that in (15) to get,2 R n − ∇ ψ (cid:18) R n − ∇ T U n (cid:19) = − (2 n − U n − , + (cid:104) ( n − n −
2) + ˜ (cid:0)(cid:0) ∆ (cid:105) U n − , (85)which we can integrate to get, ∀ n > U n (cid:39) (cid:34) − ˜ (cid:0)(cid:0) ∆( n − (cid:35) (cid:90) TT U n − , dT (cid:48) + m u,n (86) − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT U n − dT (cid:48) . As in the g and b cases, we consider the initial data arisingfrom the ψ a integration ˙ u n to be zero for all n > U n in terms of U n − , U n = 12 (cid:34) − ˜ (cid:0)(cid:0) ∆( n − (cid:35) (cid:90) TT U n − , dT (cid:48) + m u,n (87) − (cid:34) n − (cid:0)(cid:0) ∆ n − (cid:35) (cid:90) TT U n − dT (cid:48) . Our results are summarized by the following:
Theorem 1.
Let X α = ( T, X i ) be an asymptoticallyCartesian coordinate system with an associated covariantderivative ˚ ∇ . The good-bad-ugly system defined as, ˚ (cid:3) g = 0 , ˚ (cid:3) b = ( ∇ T g ) , ˚ (cid:3) u = R ∇ T u , (88) where ˚ (cid:3) := η ab ˚ ∇ a ˚ ∇ b and η is the Minkowski metric, ad-mits formal polyhomogeneous asymptotic solutions nearnull infinity of the type, g = ∞ (cid:88) n =1 G n ( ψ ∗ ) R n ,b = ∞ (cid:88) n =1 B n R n ,u = m u, R + ∞ (cid:88) n =2 U n R n , (89) where B n = B n, ( ψ ∗ ) + B n, ( ψ ∗ ) log R and U n = U n, ( ψ ∗ ) + U n, ( ψ ∗ ) log R and with initial data on S ofthe type, g | S = (cid:80) ∞ n =1 m g,n R n b | S = (cid:80) ∞ n =1 m b,n R n u | S = (cid:80) ∞ n =1 m u,n R n , ∇ T g | S = O S ( R − ) ∇ T b | S = O S ( R − ) ∇ T u | S = O S ( R − ) , (90) where m φ,n are scalar functions that are independentof T and R . This is valid outside a compact ball cen-tered at R = 0 . Additionally, the functions G n are givenby (37) and (58) , B n are given by (64) and (74) and U n by (81) and (87) . Remark 1.
Looking at equation (87) we see that the onlyway for U n to have a term with log R in it is if U n − does too. As this is valid for any n > , the orders of U n higher than U can only have a log R term if U doesas well. In other words, if we require (cid:0)(cid:0) ∆ m u, = 0 , thefield u will have no log R terms at any order. In fact, withthat requirement, it can easily be seen that U n satisfies ahypothesis analogous to that of the field g , namely, u = ∞ (cid:88) n =1 U n ( ψ ∗ ) R n . (91) Remark 2.
The good-bad-ugly system (15) admits astatic solution that is obtained with the following initialdata in S , g | S = m g, R b | S = m b, R u | S = m u, R (92) where the functions m φ, satisfy the condi-tion (cid:0)(cid:0) ∆ m φ, = 0 . In that case, looking at the recursionrelations (58) , (74) and (81) , we see that the series istruncated at n = 1 and hence all the higher order termsvanish. V. ASYMPTOTICALLY FLAT METRICS
In this section we follow the same procedure as in theprequel, but this time employing a more general metricwhose functions are allowed to depend analytically on thefields g, b and u . Since we want to maintain the general-ity of those functions, the final recursion relation for the good-bad-ugly system will have to be written as a functionof g , b and u . Nevertheless, we will see that an inductionproof analogous to that of the flat metric case can bemade for a general asymptotically flat metric under theassumptions given in section III. An asymptotically flatmetric that is simply given, rather than occurring as afunction of our unknown fields, could be treated similarly. Expansion of the reduced wave operator:
Let φ be anyfield in { g, b, u } and use (8) to expand the LHS of (15),˚ (cid:3) φ = (cid:20) − e − ϕ τ ∇ ψ ∇ ψ + 2 e − ϕ τ (˚ ∇ ψ ψ ) a ∇ a − τ ˚ /D a σ a ∇ ψ − τ ˚ /D a σ a ∇ ψ + (cid:0)(cid:0) ∆ (cid:21) φ . (93)We want to write expression (93) in terms of deriva-tives of metric functions and φ along the vector fields ψ a and ∂ aT . For clarity, let us treat each term individuallyand put everything together in the end. Using (4), thefirst term on the RHS turns into, ∇ ψ ∇ ψ φ = ∇ ψ (cid:18) τ C R + ∇ T φ + C R − C R + ∇ ψ φ (cid:19) . (94)From the second term on the RHS we get,(˚ ∇ ψ ψ ) a ∇ a φ = ( • ∇ ψ ψ ) a ∇ a φ = 1 C R + ∇ ψ C R − ( ∇ ψ φ − ∇ T φ ) , where the first equality comes from the factthat Γ[˚ ∇ , • ∇ ] ψaψ = 0. The third term can be ex-panded as,˚ /D a σ a ∇ ψ φ = /g ab ( • ∇ a σ b − Γ[˚ ∇ , • ∇ ] aσb ) ∇ ψ φ (95)= /g ab ( −∇ b T ∇ a C R + ∇ b θ A ∇ a C + A − Γ[˚ ∇ , • ∇ ] aσb ) ∇ ψ φ = (cid:18) − C A τ /D A C R + /D A C + A − /g ab Γ[˚ ∇ , • ∇ ] aσb (cid:19) ∇ ψ φ , C A := C + A + C − A and ∇ ψ φ should be written in termsof ∇ T φ and ∇ ψ φ with (4), whereas the fourth term reads,˚ /D a σ a ∇ ψ φ = (cid:18) − C A τ /D A C R + /D A C − A − /g ab Γ[˚ ∇ , • ∇ ] aσb (cid:19) ∇ ψ φ . Putting all of this together in (93) gives, C R + ˚ (cid:3) φ = − e − ϕ ∇ ψ ∇ T φ + ∇ T φ ( /g ab Γ[˚ ∇ , • ∇ ] aσb + X T )+ ∇ ψ φX ψ − e − ϕ C R − τ ∇ ψ φ + C R + (cid:0)(cid:0) ∆ φ , (96)where X T and X ψ are, τ X T := C A /D A C R − − τ /D A C + A + 2 e − ϕ C R − C R + ∇ ψ C R + ,τ X ψ := C A τ /D A ( C R − C R + ) − C R − /D A C + A − C R + /D A C − A (97)+ C R − /g ab Γ[˚ ∇ , • ∇ ] aσb + C R + /g ab Γ[˚ ∇ , • ∇ ] aσb + 2 e − ϕ C R − C R + ∇ ψ C R + . Motivation for induction hypothesis:
As the γ func-tions (see (21)) are analytic functions of the evolved fieldsat null infinity, we can Taylor expand them around g = b = u = 0, because the fields are assumed to have decaynear null infinity. That gives, γ ( g, b, u ) = ∞ (cid:88) i =0 ∞ (cid:88) j =0 ∞ (cid:88) k =0 g i b j u k i ! j ! k ! (cid:18) ∂ i + j + k γ∂g i ∂b j ∂u k (cid:19) (cid:12)(cid:12)(cid:12) I + = ∂γ∂g (cid:12)(cid:12)(cid:12) I + g + ∂γ∂b (cid:12)(cid:12)(cid:12) I + b + ∂γ∂u (cid:12)(cid:12)(cid:12) I + u + ... , (98)where the second equality uses the fact that γ | I + = 0,because the metric is asymptotically flat. Equation (98)then implies that γ = o + (1) and, ω γ = o + ( R − n ) ⇒ (cid:40) ∇ ψ ω γ = o + ( R − n − ) ∇ X A ω γ = o + ( R − n − ) , (99)where ω γ is any γ function or any derivative of it. Notethat we intentionally left out any bad derivatives becausein order to know the asymptotic behavior of those wewould have to specify the dependence of γ on g , b and u .Remarkably, one can easily check that the expanded formof the reduced wave operator (96) does not include anybad derivatives of metric functions. Let us rescale g , b and u as (29), (59) and (79), respectively. With (99) onecan count the order of each term in (96) and see that onlythe first two terms contribute to leading order. These areexactly the same terms that contribute to leading orderin the flat metric case, which means that none of theextra terms that arise from allowing the spacetime tohave curvature can possibly contribute to first order. Weget the equations, ∇ ψ ∇ T G (cid:39) , ∇ ψ ∇ T B (cid:39) − R ( ∇ T G ) , ∇ ψ ∇ T U (cid:39) R ˜ (cid:0)(cid:0) ∆ m u, . (100) Note that, to leading order, ˜ (cid:0)(cid:0) ∆ is the Laplacian on the2-sphere of unit radius,˜ (cid:0)(cid:0) ∆ φ (cid:39) R /η ab ˚ ∇ a (cid:16) /η cb ˚ ∇ c φ (cid:17) , (101)because /g ab approaches the inverse metric on the 2-sphereof radius R . Therefore we have, G (cid:39) G , ( ψ ∗ ) B (cid:39) B , ( ψ ∗ ) + B , ( ψ ∗ ) log R U (cid:39) (cid:90) TT (cid:104) log R ˜ (cid:0)(cid:0) ∆ m u, + ˙ u ( ψ ∗ ) (cid:105) dT (cid:48) + m u, ( T ) . (102)In (102) we kept U as a function of m u, because thereis a remark to be made about it at the end of this sec-tion. As the metric functions are free to depend uponthe evolved fields, the second order equations (third or-der in the case of u ) may be coupled to first order terms.This means that G − G , , for instance, could have aterm proportional to log R that is coming from B . Onthe other hand, non-linearities could give rise to termsproportional to higher powers of log R . This seems tosuggest that g , b and u are polyhomogeneous functionswhere each term can have up to n powers of log R in the b case, and up to n − g and u cases. Formally, wetherefore conjecture g = ∞ (cid:88) n =1 n − (cid:88) k =0 (log R ) k G n,k ( ψ ∗ ) R n b = ∞ (cid:88) n =1 n (cid:88) k =0 (log R ) k B n,k ( ψ ∗ ) R n (103) u = m u, R + ∞ (cid:88) n =2 n − (cid:88) k =0 (log R ) k U n,k ( ψ ∗ ) R n . We proceed by induction as in the the previous cases.From (102) we can already know that to first orderin g and u , log R terms are not allowed, and the con-jecture (103) incorporates this property by construction.Truncating at n = 1, we have seen g = G , ( ψ ∗ ) R + G R ,b = B , ( ψ ∗ ) + B , ( ψ ∗ ) log RR + B R , (104) u = m u, R + U , ( ψ ∗ ) + U , ( ψ ∗ ) log RR + U R , with G = o + ( R ), B = o + ( R ) and U = o + ( R ), so inorder to show (103), we have to show that if we can write1the evolved fields as, g = n − (cid:88) m =1 m − (cid:88) k =0 (log R ) k G m,k ( ψ ∗ ) R m + G n R n b = n − (cid:88) m =1 m (cid:88) k =0 (log R ) k B m,k ( ψ ∗ ) R m + B n R n (105) u = m u, R + n − (cid:88) m =2 m − (cid:88) k =0 (log R ) k U m,k ( ψ ∗ ) R m + U n R n , where G n = o + ( R ), B n = o + ( R ) and U n = o + ( R ), thenwe can also write them as, g = n (cid:88) m =1 m − (cid:88) k =0 (log R ) k G m,k ( ψ ∗ ) R m + G n +1 R n +1 b = n (cid:88) m =1 m (cid:88) k =0 (log R ) k B m,k ( ψ ∗ ) R m + B n +1 R n +1 (106) u = m u, R + n (cid:88) m =2 m − (cid:88) k =0 (log R ) k U m,k ( ψ ∗ ) R m + U n +1 R n +1 , where G n +1 = o + ( R ), B n +1 = o + ( R ) and U n +1 = o + ( R ).To do this, we must first find what the metric functions,and hence the γ functions, behave like if we assume (105). Behavior of γ functions: Let φ and φ be any of thefields in { g, b, u } . According to our assumption (105), φ i with i ∈ { , } can be written as φ i = n − (cid:88) m =1 m (cid:88) k =0 (log R ) k Φ m,k ( ψ ∗ ) R m + Φ R n , (107)for suitable scalar functions Φ m,k ( ψ ∗ ) and Φ = o + ( R ). Itis then straightforward to check that the product of anytwo evolved fields can also be written as, φ φ = n − (cid:88) m =1 m (cid:88) k =0 (log R ) k ¯Φ m,k ( ψ ∗ ) R m + ¯Φ R n , (108)once again for suitable functions ¯Φ m,k ( ψ ∗ ) and ¯Φ = o + ( R ), which is formally the same as (107). This meansthat no matter how many times we multiply any powersof the evolved fields, it is always possible to write theresulting product as (108). If we plug (105) into (98) weget in each term a product of powers of the fields g , b and u , so we can write any γ function as, γ = n − (cid:88) m =1 m (cid:88) k =0 (log R ) k Γ m,k ( ψ ∗ ) R m + Γ R n , (109)with Γ = o + ( R ). Induction proof:
We plug (105) and (109) into (96)and collect terms proportional to R − n − . In the g equa-tion, for instance, we see that the only terms in (96) thatmay contain G n are the first two, while none of the re-maining terms may contain B n or U n . Putting all terms with G n , B n and U n on the LHS and all the rest on theRHS we get the system, R n ∇ ψ (cid:18) R n − ∇ T G n (cid:19) (cid:39) Ω gn − ,R n ∇ ψ (cid:18) R n − ∇ T B n (cid:19) + ∇ T G n ∇ T G (cid:39) Ω bn − ,R n − ∇ ψ (cid:18) R n − ∇ T U n (cid:19) (cid:39) Ω un − , (110)where on the right hand sides Ω φn − depend on thefunctions { G m,k , B m,k , U m,k , m u, } , for m ∈ [1 , n − k ∈ [0 , m ], and their derivatives. Also, Ω φ := 0.At this point we need to establish the maximum powerof log R in the functions Ω φn − . Since we are collect-ing terms of order R − n − , naively we would say that aterm collected this way could have a maximum powerof n + 1. Although, any good derivative or factor of R − increases the order of the term without increasing thepower of log R . For instance, in the third term of equa-tion (96) applied to b , the maximum power is n , becauseit has one good derivative and no factors of R − . Withthis in mind, we can split the functions Ω φn − in the fol-lowing way, Ω gn − = n − (cid:88) p =0 (log R ) p Ω gn − ,p ( ψ ∗ ) , Ω bn − = n (cid:88) p =0 (log R ) p Ω bn − ,p ( ψ ∗ ) , Ω un − = n − (cid:88) p =0 (log R ) p Ω un − ,p ( ψ ∗ ) . (111)It is worth noting that the specific form of the Ω φn − ,p functions has no influence on the proof of our hypothesis,as long as it is possible to write (111). In fact (111) holdsfor a more general class of models than just (15), as willbe discussed in the next section. Equation (111) allows usto integrate (110) in order to get the asymptotic behaviorof G n , B n and U n in terms of { G m,k , B m,k , U m,k , m u, } .Let us begin with the first equation, R n ∇ ψ (cid:18) R n − ∇ T G n (cid:19) (cid:39) n − (cid:88) p =0 (log R ) p Ω gn − ,p ( ψ ∗ ) . We make use of the following integral, ∀ q (cid:54) = 1, (cid:90) (log R ) p R q dR = − (log R ) p ( q − R q − + pq − (cid:90) (log R ) p − R q dR = p (cid:88) i =0 − (log R ) i ( q − p − i +1 R q − p ! i ! , (112)2to get, G n (cid:39) n − (cid:88) p =0 p (cid:88) i =0 − (log R ) i ( n − p − i +1 p ! i ! (cid:90) TT Ω gn − ,p dT (cid:48) + m g,n ( T )= n − (cid:88) i =0 (log R ) i n (cid:88) p =0 − n − p − i +1 p ! i ! (cid:90) TT Ω gn − ,p dT (cid:48) + m g,n ( T )= n − (cid:88) i =0 (log R ) i G n,i ( ψ ∗ ) , (113)for some scalar functions G n,i ( ψ ∗ ) and for all n >
1. Asin the flat case, we choose ˙ g n = 0 for n >
1, and thesame applies for b (for n >
1) and u (for n > g can be written as (105), then it can also bewritten as (106). Therefore, we have, g = ∞ (cid:88) n =1 n − (cid:88) k =0 (log R ) k G n,k ( ψ ∗ ) R n , (114)as desired. The b equation likewise gives, R n ∇ ψ (cid:18) R n − ∇ T B n (cid:19) (cid:39) − ∇ T G n ∇ T G , (115)+ n (cid:88) p =0 (log R ) p Ω bn − ,p ( ψ ∗ ) , which we can integrate in order to get, B n (cid:39) n (cid:88) i =0 (log R ) i n (cid:88) p =0 − n − p − i +1 p ! i ! (cid:90) TT Ω bn − ,p dT (cid:48) ++ n − (cid:88) i =0 (log R ) i n − (cid:88) p =0 − n − p − i +1 p ! i ! (cid:90) TT n − (cid:88) k =0 ∇ T G n,k ∇ T G , dT (cid:48) + m b,n ( T )= n (cid:88) i =0 (log R ) i B n,i ( ψ ∗ ) , (116)for all n >
1. Thus, by induction, we get, b = ∞ (cid:88) n =1 n (cid:88) k =0 (log R ) k B n,k ( ψ ∗ ) R n . (117)Finally, the u equation reads, R n − ∇ ψ (cid:18) R n − ∇ T U n (cid:19) (cid:39) n − (cid:88) p =0 (log R ) p Ω un − ,p ( ψ ∗ ) , and integrating it along integral curves of ψ a and ∂ aT gives, U n (cid:39) n − (cid:88) i =0 (log R ) i n − (cid:88) p =0 − n − p − i +1 p ! i ! (cid:90) TT Ω Un − ,p dT (cid:48) + m u,n ( T )= n − (cid:88) i =0 (log R ) i U n,i ( ψ ∗ ) , (118)for all n >
2. By induction, u = m u, R + ∞ (cid:88) n =2 n − (cid:88) k =0 (log R ) k U n,k ( ψ ∗ ) R n . (119)This concludes the proof. These results can be packagedin the following theorem. Theorem 2.
Let X α = ( T, X i ) be an asymptoticallyCartesian coordinate system with an associated covariantderivative ˚ ∇ . The good-bad-ugly system defined as, ˚ (cid:3) g = 0˚ (cid:3) b = ( ∇ T g ) ˚ (cid:3) u = R ∇ T u , (120) where ˚ (cid:3) := g ab ˚ ∇ a ˚ ∇ b and g is an asymptotically flat met-ric, admits a polyhomogeneous expansion near null infin-ity of the type, g = ∞ (cid:88) n =1 n − (cid:88) k =0 (log R ) k G n,k ( ψ ∗ ) R n b = ∞ (cid:88) n =1 n (cid:88) k =0 (log R ) k B n,k ( ψ ∗ ) R n (121) u = m u, R + ∞ (cid:88) n =2 n − (cid:88) k =0 (log R ) k U n,k ( ψ ∗ ) R n , with initial data on S of the type, g | S = (cid:80) ∞ n =1 m g,n R n b | S = (cid:80) ∞ n =1 m b,n R n u | S = (cid:80) ∞ n =1 m u,n R n , (122) g | S = (cid:80) ∞ n =1 m g,n R n b | S = (cid:80) ∞ n =1 m b,n R n u | S = (cid:80) ∞ n =1 m u,n R n , ∇ T g | S = O S ( R − ) ∇ T b | S = O S ( R − ) ∇ T u | S = O S ( R − ) , (123) where m φ,n are scalar functions that are independent of T and R . This is valid outside a compact ball centeredat R = 0 . Remark 3.
As can be seen from (102) , a sufficient con-dition to make the log R term in U vanish, is that, (cid:0)(cid:0) ∆ m u, = 0 , (124)3 which is exactly the same requirement as in the flatcase. Although, with a general metric we cannot ex-pect the log R terms to vanish at all orders, because or-der n = 3 in u is already coupled to the b equation andmight therefore inherit up to one power of log R , depend-ing on the form of the metric functions. Remark 4.
Once again one can see that the good-bad-ugly system admits a static solution given by, Rg = m g, ,Rb = m b, ,Ru = m u, , (125) as long as the requirement (cid:0)(cid:0) ∆ m φ, = 0 is fulfilled. If allfurther initial data are set to zero, then the series trun-cates at n = 1 and all orders vanish except the first one.Note that this static solution differs from (92) becausehere the operator (cid:0)(cid:0) ∆ is not necessarily the Laplace opera-tor on the 2-sphere of radius R , but an analogous oper-ator constructed from /g ab that coincides with the formerto leading order. VI. STRATIFIED NULL FORMS
We can generalize this proof to encompass models morecomplicated than the standard good-bad-ugly system . Infact there is a large class of terms that, added to the RHSof (15) require no significant changes in the inductionproof. These terms are a generalization of the classicalnull forms, see [23, 24], that know about the differenttypes of field. Let us define stratified null forms as termsthat involve up to one derivative of the evolved fields andfall-off faster than R − close to null infinity. For example,a term which has one good derivative, one bad derivativeand no explicit dependence on coordinates, say, ∇ ψ g ∇ ψ b , (126)is necessarily o + ( R − ), and is therefore a stratified nullform. Another type of term that fulfills this requirementis one which is quadratic in bad derivatives, but has onepower of R − , say, 1 R ∇ ψ g ∇ ψ b . (127)Finally, a term where any derivative hits an ugly fieldand a bad derivative hits any field, say, ∇ a u ∇ ψ b , (128)is also a stratified null form, because any derivative hit-ting an ugly field, necessarily improves its decay. We willneed this definition because it distinguishes the termsthat significantly change our proof from those that donot. Let us replace our earlier system with˚ (cid:3) g = N g , ˚ (cid:3) b = ( ∇ T g ) + N b , ˚ (cid:3) u = R ˚ ∇ T u + N u , (129) where N φ are arbitrary linear combinations of stratifiednull forms. As stratified null forms are at least of or-der o + ( R − ), regardless of any of these terms we addto the RHS of the good-bad-ugly system, the first orderequations (102) remain the same, as they are the resultof collecting terms proportional to R − ( R − in the u case), ∇ ψ ∇ T G (cid:39) , R ∇ ψ ∇ T B (cid:39) − ( ∇ T G ) , R ∇ ψ ∇ T U (cid:39) ˜ (cid:0)(cid:0) ∆ m u, . (130)Stratified null forms will, in general, contribute to thenext order, however they will not contain derivativesof G , B or U . This is true for all n . At each step wecollect terms of order R − n − to find equations for G n , B n or U n and stratified null forms will only containderivatives of { G m,k , B m,k , U m,k , m u, } , for m ∈ [1 , n − k ∈ [0 , m ]. This implies that any terms arising fromstratified null forms can be absorbed into Ω φn − so thatwe get (cf. (110)), R n ∇ ψ (cid:18) R n − ∇ T G n (cid:19) (cid:39) Ω (cid:48) gn − ,R n ∇ ψ (cid:18) R n − ∇ T B n (cid:19) + ∇ T G n ∇ T G (cid:39) Ω (cid:48) bn − ,R n − ∇ ψ (cid:18) R n − ∇ T U n (cid:19) (cid:39) Ω (cid:48) un − , (131)where Ω (cid:48) φn − are just Ω φn − , as defined earlier, plus anyextra terms coming from N φ . Naturally, (111) is stillvalid and hence g , b and u can be written as polyhomo-geneous functions (113), (116) and (118). Although wecan expect the final recursion relations of the g , b and u fields to change in general, the induction proof remainsunchanged. Therefore, regardless of the addition of anystratified null forms to the good-bad-ugly system, we havethe following result, g = ∞ (cid:88) n =1 n − (cid:88) k =0 (log R ) k G n,k ( ψ ∗ ) R n b = ∞ (cid:88) n =1 n (cid:88) k =0 (log R ) k B n,k ( ψ ∗ ) R n (132) u = m u, R + ∞ (cid:88) n =2 n − (cid:88) k =0 (log R ) k U n,k ( ψ ∗ ) R n , This implies that we can generalize
Theorem
Theorem 3.
Let X α = ( T, X i ) be an asymptoticallyCartesian coordinate system with an associated covariantderivative ˚ ∇ . The good-bad-ugly system defined as, ˚ (cid:3) g = N g , ˚ (cid:3) b = ( ∇ T g ) + N b , ˚ (cid:3) u = R ˚ ∇ T u + N u , (133)4 where N φ are arbitrary linear combinations of stratifiednull forms, ˚ (cid:3) := g ab ˚ ∇ a ˚ ∇ b and g ab is an asymptoticallyflat metric, admits a polyhomogeneous expansion nearnull infinity of the type, g = ∞ (cid:88) n =1 n − (cid:88) k =0 (log R ) k G n,k ( ψ ∗ ) R n b = ∞ (cid:88) n =1 n (cid:88) k =0 (log R ) k B n,k ( ψ ∗ ) R n u = m u, R + ∞ (cid:88) n =2 n − (cid:88) k =0 (log R ) k U n,k ( ψ ∗ ) R n , (134) with initial data on S of the type, g | S = (cid:80) ∞ n =1 m g,n R n b | S = (cid:80) ∞ n =1 m b,n R n u | S = (cid:80) ∞ n =1 m u,n R n , ∇ T g | S = O S ( R − ) ∇ T b | S = O S ( R − ) ∇ T u | S = O S ( R − ) , (135) where m φ,n are scalar functions that are independent of T and R . This is valid outside a compact ball centeredat R = 0 . VII. CONCLUSIONS
In this paper we laid out a heuristic method to pre-dict the decay of terms beyond first order in R − in the good-bad-ugly system. In its most general form the modelconsists of a set of coupled nonlinear wave equations inwhich the three different classes of fields have differentasymptotic properties near null-infinity. We began withthe simplest form of this system, as introduced in [19],built from the Minkowski metric and found that nearnull infinity there exist formal solutions to this model inwhich the bad field may have log R terms at every orderin R − , the ugly field may have logs from second orderonward, whereas the good field has no logs at all. Weshowed furthermore a recursion relation that allows usto find each order in R − from the previous one to arbi-trary order. The method is, however, heuristic becausewe have not shown that all physically relevant solutions ofthe good-bad-ugly system admit expansions of this form.This was used as a warm-up for a more general systembuilt from a general asymptotically flat metric. Keepingthe metric functions fairly general, insisting essentiallyonly that they be analytic functions of the evolved fields,we showed by induction that there is a class of asymptoticsolutions near null infinity characterized by polyhomoge-neous functions, the main difference between the threetypes of fields being the order at which log terms arefirst allowed to appear. As the metric components wereintentionally left free, a full recursion relation for a gen-eral metric was not possible. However, we anticipate noreason why this method would not be straightforwardlyapplicable to any metric with these requirements in or-der to find such relations. In a final generalization to the model we considered the effect of non-linearities of a spe-cial class that we call stratified null forms. By definitionthese are precisely the terms involving up to one deriva-tive of the evolved fields that fall-off faster than O ( R − ).All of our results are subsumed within Theorem 3, whichsays that the same type of expansion also works out inthe presence of arbitrary stratified null forms.The restriction of having just one field of each typein our model is purely for simplicity. A more generalsetup with sets of fields of each type just requires morebook-keeping. In fact, in future work, we aim to ap-ply this method to the EFE in GHG to predict that itsasymptotic solutions can be written as polyhomogeneousfunctions near null infinity. Due to the complexity ofthe full field equations, the asymptotic system will pre-sumably be very long, but we anticipate that the thenon-linearities studied in the good-bad-ugly system al-ready capture the subtleties of those in GR. By findingthe first few orders of a polyhomogeneous expansion ofasymptotic solutions to the EFE, we expect to be ableto recover the peeling properties, or a polyhomogeneousgeneralization, of the gravitational field.Similar polyhomogeneous behavior of the gravitationalfield close to spatial and null infinity has been obtained bymeans of the conformal Einstein field equations in [25, 26]—see also [27–30] for further discussion of peeling. Thepolyhomogeneous expansions described in [25, 26] are for-mal in the sense that the appropriate energy estimatesneeded to rigorously prove that these expansions arise asan actual solution from some given initial data are stilllacking. The polyhomogeneity result we potentially ex-pect to obtain by exploiting the methods presented abovefor the EFE in GHG would be formal in the same sense.Ultimately we aim to make contact with the expansionsgiven in [31] in harmonic gauge, in which no log termsare present. It is worth mentioning that the logarithmicterms appearing in the expansions described in [25, 26]have a very different origin from those analyzed here. Inthe case of the good-bad-ugly model and the EFE in GHGthe logarithmic terms appear in the asymptotic expan-sion due to the form of the non-linearities in the equa-tions, while the logarithmic terms of [25, 26] appear evenin a linear context, such as the spin-2 field equations, in aMinkowski background in the framework of the cylinderat spatial infinity as discussed in [32].We furthermore hope that this work will be a step-ping stone towards a full regularization of GR in GHGat null-infinity. Knowing from the outset where the logsmay appear up to arbitrary order, one can employ a‘subtract-the-logs’ strategy as the one used in [19] in or-der to treat these divergent terms, or indeed attemptto carefully choose gauge source functions that eradicatethem all together.5 ACKNOWLEDGMENTS
The Authors wish to thank Alex Va˜n´o-Vi˜nualesfor helpful comments on the manuscript. MD ac-knowledges support from FCT (Portugal) programPD/BD/135511/2018, DH acknowledges support fromthe FCT (Portugal) IF Program IF/00577/2015,PTDC/MAT- APL/30043/2017. JF acknowledges sup-port from FCT (Portugal) programs PTDC/MAT- APL/30043/2017, UIDB/00099/2020. EG gratefully ac-knowledges support from the European Union’s H2020ERC Consolidator Grant “Matter and Strong-FieldGravity: New Frontiers in Einstein’s Theory,” GrantAgreement No. MaGRaTh-646597. EG also acknowl-edges support from the European Union (through thePO FEDER-FSE Bourgogne 2014/2020 program) andthe EIPHI Graduate School (contract ANR-17-EURE-0002) as part of the ISA 2019 project. [1] R. Penrose, Phys. Rev. Lett. , 66 (1963).[2] E. Newman and R. Penrose, Journalof Mathematical Physics , 566 (1962),https://doi.org/10.1063/1.1724257.[3] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner,Proceedings of the Royal Society of London. Series A,Mathematical and Physical Sciences , 21 (1962).[4] R. Sachs, Phys. Rev. , 2851 (1962).[5] R. Sachs, Proc. Roy. Soc. Lond. A , 103 (1962).[6] P. H¨ubner, Class. Quantum Grav. , 2823 (1999).[7] P. H¨ubner, Class. Quantum Grav. , 1871 (2001).[8] G. Doulis and J. Frauendiener, (2016), arXiv:1609.03584[gr-qc].[9] H. Friedrich, Proc. Roy. Soc. London A 375 , 169 (1981).[10] H. Friedrich, Proc. Roy. Soc. London
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