The Weak Null Condition in Free-evolution Schemes for Numerical Relativity: Dual Foliation GHG with Constraint Damping
TThe Weak Null Condition in Free-evolution Schemes for Numerical Relativity:Dual Foliation GHG with Constraint Damping
Edgar Gasper´ın and David Hilditch
Centro de Astrof´ısica e Gravita¸c˜ao - CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico - IST,Universidade de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (Dated: December 18, 2018)All strategies for the treatment of future null-infinity in numerical relativity involve some form ofregularization of the field equations. In a recent proposal that relies on the dual foliation formalismthis is achieved by the use of an asymptotically Minkowskian generalized harmonic tensor basis. Forthe scheme to work however, derivatives of certain coordinate light-speeds must decay fast enough.Presently, we generalize the method of asymptotic expansions for nonlinear wave equations to treatfirst order symmetric hyperbolic systems. We then use this heuristic tool to extract the expectedrates of decay of the metric near null-infinity in a free-evolution setting. We show, within theasymptotic expansion, that by carefully modifying the non-principal part of the field equations bythe addition of constraints, we are able to obtain optimal decay rates even when the constraintsare violated. The light-speed condition can hence be satisfied, which paves the way for the explicitnumerical treatment of future null-infinity. We then study the behavior of the Trautman-Bondimass under the decay results predicted by the asymptotic expansion. Naively the mass seems tobe unbounded, but we see first that the divergent terms can be replaced with a combination ofthe constraints and the Einstein field equations, and second that the Bondi mass loss formula isrecovered within the framework. Both of the latter results hold in the presence of small constraintviolations.
Contents
I. Introduction II. Basic set up III. The asymptotic system for first andsecond order systems
IV. A constraint damped hyperbolicreduction of GR in second order form
V. The asymptotic system for T ab = I T ab VI. The asymptotic system for first orderGHG with constraint damping
VII. The Trautman-Bondi mass VIII. Conclusions Acknowledgments References I. INTRODUCTION
The notion of gravitational radiation is intimately re-lated to the asymptotic behavior of the gravitationalfield. According to Penrose’s proposal [1–3] isolated sys-tems may be described by asymptotically simple space-times. Among these spacetimes those with vanishing cos-mological constant, also known as
Minkowski-like space-times , play a fundamental role for the current under-standing of gravitational radiation [4–7]. For these space-times, the conformal boundary I is a null-hypersurfacerepresenting idealized observers at infinity. It is in thissetting, in which the concept of gravitational radiation,originally introduced at the linear level, can be made rig-orous in the non-linear theory and results such as theloss of mass-energy due to outgoing gravitational radia-tion [8–10] properly formulated.Despite the advances achieved in numerical relativity(NR) in the last two decades, the inclusion of future nullinfinity in the computational domain is in general stillan open problem. In view of the first direct detectionof gravitational waves in 2016 [11] and the further devel-opment of gravitational wave astronomy in the years tocome, the most natural motivation to solve this problemis for the computation of astrophysical wave forms at fu-ture null infinity. Nevertheless there are other importantprinciple reasons to include the conformal boundary inthe numerical domain, not least the weak cosmic censor-ship conjecture.There are several approaches to include the conformalboundary in the computational domain. Possible avenuesinclude Cauchy-Characteristic Matching [12] and the useof a suitable hyperboloidal initial value problem. In thelatter approach, which we follow, initial data is given a r X i v : . [ g r- q c ] D ec on slices which are everywhere spacelike but which ter-minate at future null infinity. This is combined with asuitable radial compactification. Naively looking at thefield equations on such a slice however we get singularexpressions, for which some cure is needed, particularlyfor numerical applications. One option is to use the con-formal Einstein field equations introduced by Friedrichin [13]. Although this approach is well suited to study theasymptotic region of the spacetime, it poses difficulties tonumerically evolve realistic astrophysical scenarios, as itis not clear how to adapt those codes and techniques thatalready work well in the strong field region, for instancefor compact binaries. Recent progress on the numericalimplementation of the conformal Einstein field equationshas however been reported in [14]. Another proposalis to use a standard formulation of GR in generalizedharmonic gauge (GHG) but insist on performing a fullcompactification of spacetime. This leads to equationswhich are formally singular but which take a finite limitin a suitable gauge [15]. Following up on these ideas,Va˜n´o-Vi˜nuales and collaborators have shown successfulnumerical evolutions in spherical symmetry [16–18]. Seealso [19, 20] for other related approaches.An alternative proposal, using the coordinates of [21],was given in [22]. Here the strategy, broadly speaking, isto pose an hyperboloidal initial value problem and exploitthe dual foliation formalism [23]. In the dual foliationformalism two coordinate systems with two different foli-ations are employed. This allows us to use one coordinatesystem to construct a global tensor basis and another tocoordinatize the spacetime. In the current application, itis natural to take the first coordinate system adapted toa Cauchy hypersurface, and the other to a hyperboloidalslice [22]. In this proposal, one employs a first order re-duction of GHG. The regularization strategy is based onthe idea that, near null infinity, the metric looks trivialin the GHG tensor basis. It was found in [22] that forthe regularization to work certain derivatives of the co-ordinate light-speeds must have enough decay. We callthis the coordinate light-speed condition .In this article we analyze whether or not this conditioncan be realized and if so, in which circumstances. Inter-estingly, this is connected with the the weak null con-dition [24] and, consequently, to the concept of asymp-totic systems . The asymptotic system can be regardedas a heuristic tool to predict the asymptotics of the so-lutions to a general a system of quasilinear wave equa-tions. We view this tool as heuristic because, to the bestof our knowledge, it has not yet been established thatsolutions to the asymptotic system will always have thesame asymptotics as solutions to the original system.It has however been shown that if a system of waveequations satisfy the hierarchical weak null condition , acondition slightly stronger than the weak null condition,then, the original PDEs admit global solutions whoseasymptotics agree with the prediction of the asymp-totic system [25]. An important consequence is thatglobal existence for GR in harmonic gauge can be estab- lished without assuming that the constraints are satisfied.Moreover when the constraints are satisfied the formu-lation of [24] is expected to satisfy the coordinate light-speed condition, so naively the situation looks promising.In free evolution schemes however, constraint violationsare not only present but expected to grow rapidly, render-ing the solution unphysical [26–28]. Therefore the use ofthe field equations as in [24] is not appropriate for numer-ical free evolution as envisioned in our program. This isan old problem in NR, and various approaches have beensuggested to alleviate it. For instance in [27] a constraintaddition was proposed which effectively damps away highfrequency violations at the linear level. Unfortunately forour desired application the formulation of [24, 25] is not expected to satisfy the coordinate light-speed conditionin the presence constraint violations. Presently we solvethe problem with a GHG formulation, making a care-ful constraint addition to the field equations. With thisaddition the corresponding asymptotic system predictsthat constraint violations are strongly damped as oneapproaches null infinity, and we recover the same fall-offpresent in their absence. This in turn should ensure thatthe coordinate light-speed condition of [22] is satisfiedeven in the presence of small constraint violations.Although the calculation of the asymptotic system isstraightforward, most of the literature about asymptoticsystems [24, 29, 30] has been given in the context of sec-ond order equations. In view of the fact that the for-mulation of the Einstein field equations used in [22] isfirst order, we give a discussion of how to compute theasymptotic system for such systems. Finally we discussthe connection between the Trautman-Bondi mass andthe asymptotic system and recover the mass loss formulain this context. We also see that modifying the definitionof the Trautman-Bondi mass with the constraints, simi-lar results may be obtained even when small constraintviolations are present. II. BASIC SET UP
In what follows, lower case Latin letters from the firsthalf of the alphabet will denote abstract indices whileGreek letters will be used to denote coordinate indices.In addition, capital Latin letters will be use to label theelements of a non-coordinate frame basis (the angularpart of a spacetime null frame). Given a 2-tensor T ab and arbitrary frame vector fields X a , Y b the componentsof T ab in this frame will be denoted as T XX = X a X b T ab , T XY = X a Y b T ab , T Y X = Y a X b T ab and T Y Y = Y a Y b T ab and similarly for higher-valence tensors. Let m ab and ˚ ∇ represent the Minkowski metric and its correspondingLevi-Civita connection. Let ( T, R, θ A ) denote sphericalpolar coordinates where θ A with A ∈ { , } representsome arbitrary coordinates on S . Let ω Aa with A ∈{ , } denote a frame with corresponding coframe ˆ ω Aa ,such that σ ab = δ AB ˆ ω Aa ˆ ω Bb , (1)where σ ab represents the standard metric on S and δ AB is the Kronecker delta. Observe that, ω Aa is not a coor-dinate frame. In fact, respect to the natural coordinatebasis one can writeˆ ω Aa = Ω BA (d θ B ) a , (2)where Ω BA is a matrix which only depends on the coor-dinates θ A . In this frame, the Minkowski metric can beexpressed in abstract index notation as m ab = − (d T ) a (d T ) b + (d R ) a (d R ) b + R σ ab . (3)Let ( L a , L a , S Aa ) denote a null frame respect to m ab withthe following normalization m LL = − , m S A S B = δ AB ,m LL = m LL = m LS A = m LS A = 0 . (4)The relation between the flat null frame ( L a , L a , S Aa )and (( ∂ T ) a , ( ∂ R ) a , ω Aa ) is given by L a = ( ∂ T ) a + ( ∂ R ) a , L a = ( ∂ T ) a − ( ∂ R ) a ,S Aa = 1 R ω Aa . (5)Similarly, let ( ˆ L a , ˆ L a , ˆ S Aa ) denote the associatedcoframe. Observe that in terms of the correspondingcoframe ((d T ) a , (d R ) a , ω Aa ) one hasˆ L a = − (d T ) a + (d R ) a , ˆ L a = − (d T ) a − (d R ) a , ˆ S Aa = R ˆ ω Aa . (6)Therefore, the Minkowski metric can be succinctly ex-pressed as m ab = − ˆ L ( a ˆ L b ) + δ AB ˆ S Aa ˆ S Bb . (7)More generally, observe that, given a symmetric ten-sor T ab = T ( ab ) one can express it as, T ab = T LL ˆ L a ˆ L b + T LL ˆ L ( a ˆ L b ) + T LL ˆ L a ˆ L b + T TF AB ⊥ ABab + T ∅ δ AB ˆ S Aa ˆ S Bb − T LS A ˆ L ( a ˆ S Ab ) − T LS A ˆ L ( a ˆ S Ab ) , (8)where we have defined the trace on the sphere, T ∅ ≡ δ AB T S A S B , (9)and the tracefree part of the projection, T TF AB = ⊥ abAB T ab . (10)For this we employ the tracefree projection operator, ⊥ abAB = ˆ S Aa ˆ S Bb − δ AB ( δ CD ˆ S Ca ˆ S Db ) , ⊥ ABab = ˆ S Aa ˆ S Bb − δ AB ( δ CD ˆ S Ca ˆ S Db ) . (11) The independent components of T TF AB are T TF11 = − T TF22 = 12 ( T S S − T S S ) ≡ T + ,T TF12 = T S S ≡ T × . (12)They correspond of course to the two gravitational wavepolarizations in the case of physical interest. III. THE ASYMPTOTIC SYSTEM FOR FIRSTAND SECOND ORDER SYSTEMSA. Second order systems
Consider a set of quasilinear wave equations of the form˚ (cid:3) u I = (cid:88) | α |≤| β |≤ , | β |≥ a JKI,αβ ∂ α u J ∂ β u K + G ( u, u (cid:48) , u (cid:48)(cid:48) ) , (13)where I, J, K ∈ { , ..., n } , u = ( u , ..., u n ), ˚ (cid:3) = − ∂ t + δ ij ∂ i ∂ j , x i with i, j ∈ { , , } , α , β are multi-indices and G ( u, u (cid:48) , u (cid:48)(cid:48) ) vanishes to third order as ( u, u (cid:48) , u (cid:48)(cid:48) ) → u (0 , x ) = v ( x ) ∈ C ∞ , ∂ T u (0 , x ) = w ( x ) ∈ C ∞ , (14)decaying fast as R → ∞ , where R = (cid:112) δ ij x i x j . Recallthat a quadratic form N ab ≡ N (˚ ∇ a w, ˚ ∇ a w ) where w is a tensor, is said to be a null form if it vanishes uponformal replacement of ˚ ∇ a w with v a w , where v a is null.Similar definitions are made for quadratic forms involv-ing higher derivatives. The set (13) is said to satisfy thenull-condition if the quadratic nonlinearity is a null-form.The value of this definition is that, in 3 + 1-dimensions,systems that satisfy the null-condition have global solu-tions when fed small data, and moreover have solutions ofthe same asymptotic behavior as a linear wave equationnear null-infinity [31]. The key subtlety is that the null-condition is sufficient for long-term existence, but notnecessary. For numerical applications the more relevantquestion concerns the asymptotic behavior of solutionsrather than small-data global existence, but because oftheir close relationship in the mathematics literature theyare often discussed in tandem. For example the weak nullcondition, originally introduced in [24], is crucially tiedto the notion of the asymptotic system . One says thatthe wave equation (13) satisfies the weak null conditionif the corresponding asymptotic system, ∂ s ∂ q U I = A JKI,nm ( (cid:36) ) ∂ mq U J ∂ nq U K , (15)where A JKI,nm ( (cid:36) ) = (cid:88) | α | = n, | β | = m a I,αβ ˆ (cid:36) α ˆ (cid:36) β , ˆ (cid:36) = ( − , (cid:36) ) , (cid:36) ∈ S , (16)and U i = Ru i , q = R − T , s = ln R , has solutions which,roughly speaking, exist for all s , and whose derivativesgrow at most exponentially in s for all initial data decay-ing sufficiently fast in q . In this language, the classicalnull condition states that A JKI,nm ( (cid:36) ) = 0. Thus, a systemof quasilinear wave equations of the form given above sat-isfying the classical null condition will trivially satisfy theweak null condition, see [24, 32, 33] for further details.It is conjectured that systems satisfying the weak nullcondition admit small-data global solutions with asymp-totics near null-infinity as predicted by the asymptoticsystem. Model equation in second order form:
An illustrativeexample is to consider the following model equation ˚ (cid:3) φ = ( ∂ T ˜ φ ) , ˚ (cid:3) ˜ φ = 0 . (17)where the flat wave operator ˚ (cid:3) is expressed in sphericalpolar coordinates ( T, R, θ A ). To derive the asymptoticsystem, define Φ = Rφ and ˜Φ = R ˜ φ and make the changeof coordinates ( T, R, θ A ) → ( q, s, θ A ) where q = R − T and s = ln R . Substituting the expression for φ and ˜ φ interms of the rescaled variables Φ and ˜Φ into equation (17)and formally equating the terms with coefficients R − ,one obtains the following asymptotic system ∂ s ∂ q Φ = ( ∂ q ˜Φ) , ∂ s ∂ q ˜Φ = 0 . (18)Equation (17) is an example of a system of equationsthat does not satisfy the classical null condition but sat-isfies the weak null condition. Observe that the secondequation in (18) simply states that ∂ q ˜Φ does not dependon s . Then, using the method of characteristics, onecan integrate in s the first equation in (18) to concludethat ∂ q Φ = s ( ∂ q ˜Φ) . The latter implies for the originalset of variables that ∂φ decays as R − ln R [24].The motivation for the introduction of the asymptoticsystem is related to the observation that one expects bet-ter decay rates for L -derivatives of the field comparedwith other derivatives. Therefore, the construction of theasymptotic system is essentially to rescale the raw vari-ables according to their expected fall-off, move to suitablecoordinates, and then to throw away all quadratic termscontaining derivatives containing at least one derivativetangential to the outgoing light-cone, along with cubicand higher order terms. We will see in the next sectionhow each of these ingredients is adjusted for first ordersystems. B. First order systems
To the best of our knowledge, the notion of the weaknull condition and the asymptotic systems has been ex-ploited only for systems of wave equations. In the contextof gravitation, in particular, for the Einstein field equa-tions in harmonic gauge in [34] and more general systemsof wave equations in [25]. Although the wave equationcan be recast straightforwardly as a first order symmetric hyperbolic system by the introduction of an order reduc-tion variable w = ∂u , the weak null condition and thenotion of the asymptotic system has not been discussedin this context yet. This is of importance for applicationssince numerical evolution schemes often make use of afirst order formulation of the field equations. Moreover,although mathematically, the relation w = ∂u connectsthe first and second order formulations, numerically, theuse of reduction variable requires the introduction of areduction constraint C ≡ w − ∂u , and violations to this re-duction constraint tends to grow exponentially, or worse,during numerical evolutions [26–28, 35]. Therefore, oneis forced to modify the evolution equations to control thisbad behavior. With this motivation in mind, in this sec-tion, the model equation (17) is written in a form thatis suitable for numerical implementations and then thecorresponding asymptotic system is derived. Model equation in first order form:
Let ( ˚ M , m ab ) de-note the Minkowski spacetime equipped with the objectsintroduced in section II and consider the model equa-tion (17). Let ˚ N a denote the time unit normal to thesurface determined by the condition T = 0 and de-fine ˚ γ ab = m ab + ˚ N a ˚ N b . Let φ a be a spatial covector,namely ˚ N a φ a = 0, so that φ R and φ S A denote the com-ponents of φ a respect to the spatial frame (( ∂ R ) a , S Aa ).To perform the order reduction, define the reduction con-straints as C a ≡ ˚ γ ab ˚ ∇ b φ − φ a , (19)and the time reduction variable π via π = − ∂ T φ . Withformally identical definitions for ˜ φ , the evolution equa-tions for these fields read, ∂ T ˜ φ = − ˜ π,∂ T ˜ φ R = − ∂ R ˜ π + γ ( − ˜ φ R + ∂ R ˜ φ ) ,∂ T ˜ φ S A = γ ( − ˜ φ S A + S Aa ∂ a ˜ φ ) − S Ab ∂ b ˜ π,∂ T ˜ π = − δ AB S Aa ∂ a ˜ φ S B − ∂ R ˜ φ R − R ˜ φ R − ˜ φ S A ˚ ∇ a S Aa ,∂ T φ = − π,∂ T φ R = − ∂ R π + γ ( − φ R + ∂ R φ ) ,∂ T φ S A = γ ( − φ S A + S Aa ∂ a φ ) − S Ab ∂ b π,∂ T π = ˜ π − δ AB S Aa ∂ a φ S B − ∂ R φ R − R φ R − φ S A ˚ ∇ a S Aa . (20)The evolution equations for φ and ˜ φ in (20) are just thedefinition of the time reduction variables. The evolutionequations for φ R , φ S A and the corresponding hatted vari-ables in (20) with γ = 0 arise as a consequence of theno-torsion condition [˚ ∇ a , ˚ ∇ b ] φ = 0 and [˚ ∇ a , ˚ ∇ b ] ˜ φ = 0.Nevertheless, to reduce the effect produced by constraintviolations in numerical applications, one modifies theseequations by adding the following multiples of the reduc-tion constraints: γ C S A , γ C R , γ ˜ C S A , and γ ˜ C R . Here γ is a freely prescribable scalar function of the coordinates.Notice that the introduction of these terms affect theprincipal part of the equation. Nonetheless, one can showthat this system is symmetric hyperbolic for any choice ofthe formulation parameter γ [28]. The evolution equa-tions for π and ˜ π in (20) arise from expressing the waveequations (17) using the reduction variables.Following the discussion of the asymptotic system ofsection III A, one needs to rescale the variables appropri-ately. Nevertheless, since in the first order reduction ofthe equations the frame (( ∂ T ) a , ( ∂ R ) , S Aa ) was used toexpress the components of the reduction variable insteadof the flat null frame, one needs to perform the followingchange of variables before rescaling. Defining σ + = − π + φ R , σ − = − π − φ R , (21)with formally identical definitions for the hatted vari-ables, substituting π , φ R , ˜ π and ˜ φ R written in termsof σ + , σ − , ˜ σ + and ˜ σ − into equation (20) one obtains aset of evolution equations for the variables, { φ, σ + , σ − , φ S A , ˜ φ, ˜ σ + , ˜ σ − , ˜ φ S A } . Observe that when the constraints are satisfied σ + and σ − correspond to the L and L derivatives of φ , re-spectively. Then, one defines the rescaled variables as,Φ = Rφ, Σ + = R σ + , Σ − = Rσ − , Φ S A = R φ S A , (22)along with the analogous expressions for the hatted vari-ables, and substitutes these definitions into the evolutionequations using the chain rule to express the derivativesin terms of ∂ q and ∂ s . Notice the important point thatthe rescaling here takes place after the derivative is ap-plied. It is easily seen that this is the natural construc-tion by considering spherically symmetric solutions to theflat-space wave equation. Solving for the derivatives ofthe rescaled variables we obtain, ∂ q ˜Σ + (cid:39) ˜Σ − + γ ( ˜Σ + + ˜Φ − ∂ s ˜Φ) ,∂ s ˜Σ − (cid:39) γ ( ˜Σ + + ˜Φ − ∂ s ˜Φ) ,∂ q ˜Φ S A (cid:39) − ω Aa ∂ a ˜Σ − + γ ( ˜Φ S A − ω Aa ∂ a ˜Φ) ,∂ q ˜Φ (cid:39) − ˜Σ − ,∂ q Σ + (cid:39) (4Σ − + ( ˜Σ − ) ) + γ (4Σ + + 4Φ − ∂ s Φ) ,∂ s Σ − (cid:39) − ( ˜Σ − ) + γ (Σ + + Φ − ∂ s Φ) ,∂ q Φ S A (cid:39) − ω Aa ∂ a Σ − + γ (Φ S A − ω Aa ∂ a Φ) ,∂ q Φ (cid:39) − Σ − , (23)where (cid:39) represents equality up to error terms which de-cay one order faster in R than the displayed expressions.In this case, it represents equality up to order O ( R − ).These expressions are derived in full in the mathematicanotebooks associated with this paper [36]. They requirexAct [37].Neglecting the error terms implicit in the last equa-tions, namely, formally replacing (cid:39) with = defines theasymptotic system for equation (20). To see that this ac-tually corresponds to the asymptotic system (18) one has to examine the relation between the rescaled variablesand the corresponding rescaled reduction constraints.Let, C R = R C R , C S A = R C S A , (24)with analogous definitions for the hatted reduction con-straints. A direct computation using the first and fifthequation in (20), equation (19), and the definitions forthe rescaled variables give,Σ + = − C R − Φ + ∂ s Φ , Σ − = − ∂ q Φ + C R R + Φ R − ∂ s Φ R , Φ S A = − C S A + ω Aa ∂ a Φ , (25)and identical expressions for the hatted rescaled vari-ables. Substituting (25) and the corresponding hattedversion of these expressions into equation (23) and rear-ranging one obtains,2 ∂ s ∂ q Φ (cid:39) ( ˜Σ − ) + γ C R , ∂ s ∂ q ˜Φ (cid:39) ˜ γ ˜ C R ,∂ q Φ (cid:39) − Σ − , ∂ q ˜Φ (cid:39) − ˜Σ − ,∂ q C R (cid:39) γ C R , ∂ q ˜ C R (cid:39) γ ˜ C R ,∂ q C S A (cid:39) γ C S A , ∂ q ˜ C S A (cid:39) γ ˜ C S A . (26)These expressions are valid for any choice of the dampingparameter γ . Observe that if this parameter is set tozero or, more generally are chosen to decay sufficientlyfast, say γ (cid:39) R − , then one has2 ∂ s ∂ q Φ (cid:39) ( ˜Σ − ) , ∂ s ∂ q ˜Φ (cid:39) ,∂ q Φ + Σ − (cid:39) , ∂ q ˜Φ + ˜Σ − (cid:39) ,∂ q C R (cid:39) , ∂ q ˜ C R (cid:39) ,∂ q C S A (cid:39) , ∂ q ˜ C S A (cid:39) . (27)Following the philosophy of the asymptotic system byformally replacing (cid:39) with = one realizes that the lastfour expressions in (27) can be regarded as the asymp-totic equations for the rescaled reduction constraints. In-tegrating these equations, along an integral curve of ∂ q ,from a fixed q (cid:63) to q reveals, C R = C (cid:63)R , C S A = C (cid:63)S A . (28)where C (cid:63)R = C R | q = q (cid:63) , C (cid:63)S A = C S A | q = q (cid:63) and similarly forthe hatted rescaled reduction constraints. On the otherhand, the first four equations in (27) simply encode ∂ s ∂ q Φ = ( ∂ q ˜Φ) , ∂ s ∂ q ˜Φ = 0 . (29)Integrating these equations, this time along an integralcurve of ∂ s , we find,Φ = s G Φ ( q, θ A ) , ˜Φ = G ˜Φ ( q, θ A ) , (30)where G Φ and G ˜Φ are regular functions of their argu-ments. Since we are only interested in the behavior ofthe fields for large R , to have a more compact notation,the symbol ∼ will be used to denote equality where thefunctional dependence on the other coordinates has beensuppressed. Consistent with this notation one writes,Φ ∼ ln R, ˜Φ ∼ . (31)Assuming C (cid:63)R , C (cid:63)S A , ˜ C (cid:63)R and ˜ C (cid:63)S A are uniformly bounded,then, using equations (25) one has that,Σ + ∼ ln R, Σ − ∼ ln R, Φ S A ∼ ln R, ˜Σ + ∼ , ˜Σ − ∼ , ˜Φ S A ∼ . (32)Therefore one concludes that, σ + ∼ ln RR , σ − ∼ ln RR , φ S A ∼ ln RR , ˜ σ + ∼ R , ˜ σ − ∼ R , ˜ φ S A ∼ R . (33)Clearly if we wished to perform numerical evolution ofthis system on a compactified domain including null-infinity this result tells us that the rescaling of variablesresulting in Σ − would be too aggressive, since we needthat the evolved variables are at least not divergent. IV. A CONSTRAINT DAMPED HYPERBOLICREDUCTION OF GR IN SECOND ORDER FORMA. Hyperbolic reductions of GR
Let ( M , g ab ) denote a 4-dimensional manifoldequipped with a metric g ab . Let m ab denote theMinkowski metric and let ∇ and ˚ ∇ denote the Levi-Civita connection of g ab and m ab , respectively. The rela-tion between ∇ and ˚ ∇ can be parameterized via, ∇ a v b = ˚ ∇ a v b − Γ cab v c , (34)where v a is any covector. This relation can be taken todefine Γ cab . Consequently Γ cab can be expressed in termsof ˚ ∇ -derivatives of g ab asΓ cab = g cd (˚ ∇ a g bd + ˚ ∇ b g ad − ˚ ∇ d g ab ) . (35)Defining the contracted Christoffel symbols via Γ c ≡ g ab Γ cab , the Ricci tensor can be compactly expressed as, R ab = − g cd ˚ ∇ c ˚ ∇ d g ab + ∇ ( a Γ b ) + g cd g hf (Γ bdf Γ cah + 2Γ dbf Γ ( ac ) h ) . (36)Let C a ≡ F a + Γ a where F a are smooth functions of thecoordinates X µ and the metric g ab but not its derivatives.These are known as the gauge source functions as a choiceof F a determines a coordinate system X µ . To see this,observe that Γ µ = −∇ ν ∇ ν X µ , thus requiring C µ = 0 isequivalent to ∇ ν ∇ ν X µ = F µ . (37) The equations C a = 0 will be called GHG or harmonicconstraints in the F a = 0 case. If they are satisfied then,using equation (36), one sees that the vacuum Einsteinfield equations reduce to a set of wave equations for g ab g cd ˚ ∇ c ˚ ∇ d g ab = 2 g cd g hf (Γ bdf Γ cah + 2Γ dbf Γ ( ac ) h )+ 2 ∇ ( a F b ) , (38)where Γ abc is expressed in terms of derivatives of themetric g ab using equation (35). This hyperbolic reductionprocess can be succinctly expressed as follows. Define thereduced Ricci tensor as R ab = R ab − ∇ ( a C b ) , (39)so that equation (38) is encoded in the condition R ab =0. Observe that if the constraint C a = 0 is satis-fied then R ab = R ab . Moreover, if R ab = 0 then, asa consequence of the contracted second Bianchi iden-tity ∇ a R ab − ∇ b R = 0, one has that C a satisfies thefollowing propagation equation ∇ a ∇ a C b = − C a ∇ ( a C b ) . (40)Since this is a wave equation homogeneous in C a , thelatter implies that if C a and ∇ a C b vanish on a spacelikehypersurface S ⊂ M then C a = 0 in D ( S ) ⊂ M [38, 39].Here the domain of dependence of an achronal set A , isdenoted as D ( A ). Observe that this hyperbolic reductionstrategy is not unique as one can define a reduced Riccitensor as R ab = R ab − ∇ ( a C b ) + T ab . (41)where T ab is any expression homogeneous in C a sothat C a = 0 implies R ab = R ab . The correspondingpropagation equation for C a is then, ∇ a ∇ a C b = − C a ∇ ( a C b ) + 2 ∇ a T TR ab + C a T ba , (42)where T TR ab ≡ T ab − g ab T cc . Observe that, the righthand side of the last equation is homogeneous in C a aslong as T ab is chosen to be homogeneous in C a . Althoughall the possible reduced equations R ab = 0 are equivalentif the GHG constraints are satisfied, a different choiceof T ab can be used to obtain equations of a particulardesired form. For instance in [27], the constraint additionis chosen as T ab = γ T ab where, γ T ab = γ Γ cab C c − γ g ab Γ e C e − γ ( n ( a C b ) − g ab n c C c ) , (43)here n a is a freely specifiable vector and γ , γ , γ are,in general, scalar functions depending on the coordi-nates. The parameter γ is included to damp away highfrequency constraint violations while the parameters γ and γ are included to modify ∇ ( a C b ) so the constraintaddition made in the construction of the formulation isdone either respect to ∇ or ˚ ∇ , or some combinationthereof.Other choices are designed instead to exhibit cer-tain structures of the equations. In particular, if onesets T ab = C T ab where C T ab = 12 C a C b − C c ˚ ∇ ( a g b ) c . (44)Then the reduced Einstein field equations R ab = 0 read g cd ˚ ∇ d ˚ ∇ c g ab = P ab [˚ ∇ g ] + Q ab [˚ ∇ g ] + F ab , (45)with P ab [˚ ∇ g ] = − g cd g fh ˚ ∇ a g cf ˚ ∇ b g dh + g cd g fh ˚ ∇ a g cd ˚ ∇ b g fh , (46) Q ab [˚ ∇ g ] = − g cd g fh (cid:0) ˚ ∇ b g fh ˚ ∇ d g ac − ˚ ∇ a g fh ˚ ∇ d g bc + ˚ ∇ a g bc ˚ ∇ d g fh + ˚ ∇ b g ac ˚ ∇ d g fh + ˚ ∇ b g dh ˚ ∇ f g ac − ˚ ∇ d g bh ˚ ∇ f g ac + ˚ ∇ a g dh ˚ ∇ f g bc + ˚ ∇ f g ac ˚ ∇ h g bd + ˚ ∇ d g ac ˚ ∇ h g bf − ˚ ∇ a g bc ˚ ∇ h g df − ˚ ∇ b g ac ˚ ∇ h g df (cid:1) , (47)and the gauge source functions F a appear in the form, F ab = 2 ∇ ( a F b ) − F a F b + 2 F c ˚ ∇ ( a g b ) c . (48)The relevant observation here is that Q ab is a null form.Therefore, taking F a = 0, the constraint addition (44)places the reduced field equations R ab = 0 in the formgiven in [32] where the global stability of the Minkowskispacetime in wave coordinates was proven for a set ofrestricted initial data, when the harmonic constraints arefulfilled. B. The asymptotic system and the weak nullcondition
In this section the asymptotic system for the Einsteinfield equations in harmonic gauge as discussed in [24, 32,33] is reviewed in our conventions and the problems thatthe above hyperbolic reductions present for free evolutionschemes in NR are discussed.Consider a perturbation of the Minkowski spacetime,and write g ab as g ab = m ab + h ab , (49)where h ab is a symmetric 2-tensor. For the discussion inthe remainder of this subsection all the indices are movedusing the flat metric m ab except for g ab for which we have g ab = m ab − h ab + O ab ( h ) , (50)where h ab = m ac m db h cd and O ab ( h ) vanishes to secondorder at h = 0 [34]. This convention for raising and low-ering indices will be used when discussing the derivationof the asymptotic system of a given set of equations butit will be avoided otherwise. Now, consider the reducedRicci operator R ab = R ab − ∇ ( a C b ) + T ab with T ab = C T ab as in equation (44) in harmonic gauge, in other words,with vanishing gauge source functions F a = 0. In thiscase, the reduced Einstein field equations R ab = 0 implythe following system of wave equations for g ab g cd ˚ ∇ d ˚ ∇ c g ab = P ab [˚ ∇ g ] + Q ab [˚ ∇ g ] . (51)Using equation (49) and (50) one obtains the followingevolution equations for the perturbation h ab g cd ˚ ∇ c ˚ ∇ d h ab = P ab [˚ ∇ h ] + Q ab [˚ ∇ h ] + O ab ( h (˚ ∇ h ) ) , (52)with P ab [˚ ∇ h ] and Q ab [˚ ∇ h ] as given in equations (46)-(47)where ˚ ∇ g is formally replaced by ˚ ∇ h and O ab ( h (˚ ∇ h ) )denotes a quadratic form in ˚ ∇ a h bc with coefficients de-pending on h ab which vanish for h ab = 0. Similarly, theharmonic coordinate condition reads m ac ˚ ∇ c h ab = 12 m ac ˚ ∇ b h ac + O b ( h ˚ ∇ h ) , (53)where O b ( h ˚ ∇ h ) is linear in ˚ ∇ a h bc with coefficients de-pending on h ab which vanish for h ab = 0. To derive theasymptotic system, let q = R − T, s = ln
R, H ab = R h ab . (54)To have a more compact notation for directional deriva-tives that will often appear, the following notation willbe used, ˚ ∇ T = ( ∂ T ) a ˚ ∇ a , ˚ ∇ R = ( ∂ R ) a ˚ ∇ a , ˚ ∇ q = ( ∂ q ) a ˚ ∇ a , ˚ ∇ s = ( ∂ s ) a ˚ ∇ a . (55)Substituting (49) into (45), re-expressing the equation interms of H ab and formally equating terms with coeffi-cients R − , one obtains the following asymptotic system(2˚ ∇ s − H LL ˚ ∇ q )˚ ∇ q H ab = ˆ L a ˆ L b P (˚ ∇ q H, ˚ ∇ q H ) , (56)where H LL = L a L b H ab and P (˚ ∇ q H ) = − m ab m cd ˚ ∇ q H ac ˚ ∇ q H bd + 14 m ab m cd ˚ ∇ q H ab ˚ ∇ q H cd . (57)Contracting equations (56) with the flat nullframe ( L a , L a , S Aa ), exploiting that v a w b ˚ ∇ c H ab (cid:39) ˚ ∇ c H vw ,v a w b ˚ ∇ s ˚ ∇ q H ab (cid:39) ∂ s ∂ q H vw ,v a w b ˚ ∇ q ˚ ∇ q H ab (cid:39) ∂ q ∂ q H vw , (58)for v a , w a ∈ { L a , L a , S Aa } , one observes that the onlyequation in (56) with a non-vanishing right-hand side is,(2 ∂ s − H LL ∂ q ) ∂ q H LL = − ∂ q H + ) − ∂ q H × ) − ∂ q H LL ∂ q H LL − ∂ q H LL )( ∂ q H ∅ )+ 4 δ AB ( ∂ q H LS A )( ∂ q H LS B ) , (59)while all the others satisfy,(2 ∂ s − H LL ∂ q ) ∂ q H T U = 0 , (60)where T ∈ {
L, S A } and U ∈ { L, L, S A } . Proceedingsimilarly for the harmonic coordinate condition m ab C b = m ab Γ b = m ab F b = 0 , (61)using equations (49) and (54), and formally equatingterms with coefficients R − , the asymptotic harmonic co-ordinate condition reads˚ ∇ q H La − ˆ L a ˚ ∇ q H = 0 , (62)where H La = L c H ca and H = m ab H ab .Expressing equation (62) in components respect to theflat null frame one has, ∂ q H L T = 0 , ∂ q H ∅ = 0 , (63)where H ∅ is defined according to equation (9). Theasymptotic harmonic condition (62) will play an impor-tant role in the subsequent discussion since in free evo-lution schemes in NR one cannot ensure that the con-straint equations C a = 0 are satisfied but only that suchviolations are small. Moreover, notice that the asymp-totic equation (60) implies that ∂ q H T U is constant alongthe integral curves of the vector field 2 ∂ s − H LL ∂ q . Ob-serving that constraint violations C a (cid:54) = 0 imply that( ∂ q H L T ) | Σ (cid:54) = 0 one concludes that, if constraint viola-tions are present, they will be preserved in the asymptoticregion along the integral curves of 2 ∂ s − H LL ∂ q . As anadditional remark, observe that, if the constraints C a = 0are satisfied, then one can exploit equations (63) to re-duce the asymptotic equation (59) to(2 ∂ s − H LL ∂ q ) ∂ q H LL = − ∂ q H + ) − ∂ q H × ) , (64)where H × and H + correspond to the two gravitationalwave polarizations as in equation (12). This further sim-plification in the only equation of the asymptotic sys-tem (56) with non-vanishing right-hand side cannot be di-rectly attained in free evolution form without modifyingthe hyperbolic reduction determined by the constraintaddition C T ab . Thus, one requires a hyperbolic reduc-tion for which one recovers the latter equation withoutassuming that the constraint equations (63) are fulfilledand, more importantly, a hyperbolic reduction for whichconstraint violations C a (cid:54) = 0 are damped close to nullinfinity. C. The coordinate light-speed condition
In this subsection asymptotic expressions for quantitiesrelevant for the dual foliation formulation are written interms of the variables of the asymptotic system. In par-ticular, we are interested in finding how the solution tothe asymptotic system determines the radial coordinate light-speed asymptotically near null-infinity. To connectthe current discussion with the dual foliation formalism,is convenient to perform a 2+1+1 split. Since in this sub-section we are not computing the asymptotic system wewill use the metric g ab —as usual— to raise and lower in-dices and not m ab . In the language of [23] let ( T, R, θ A )be the upper case coordinate system X µ and use T todefine the usual lapse, projection operator, normal andshift vectors A ≡ ( − g ab ∇ a T ∇ a T ) − / , N a ≡ − A∇ a T,γ ab ≡ g ab + N a N b , B a ≡ γ ab ∇ b T. (65)Similarly, we use the coordinate R to define the corre-sponding normal vector S a , projector q ab , length scalar L and slip vector b a L ≡ ( γ ab ∇ a R ∇ b R ) − / , S a ≡L γ ab ∇ b R,q ab ≡ γ ab − S a S b , b a ≡ q ab ∇ b R. (66)Using the above definitions, the metric g ab , written as aline element, reads [23],d S = − A d T + L (d R + L − B R d T ) + q AB (d θ A + b A d R + B A d T )(d θ B + b B d R + B B d T ) . (67)On the other hand, the split g ab = m ab + h ab impliesd s =( − h T T )d T + 2 h T R d R d T + (1 + h RR )d R + 2 /h RA d θ A d R + 2 /h T A d θ A d T + ( R /δ AB + /h AB )d θ A d θ B , (68)where /δ AB = Ω AC Ω BD δ CD , /h AB = Ω AC Ω BD h ω C ω D ,/h RA = Ω AB h Rω B , /h T A = Ω AB h T ω B . (69)Comparing expressions (67) and (68) gives − h T T = −A + ( B R ) + q AB B A B B , h RR = L + q AB b A b B ,h RT = q AB b A B B + LB R ,q AB = R /δ AB + /h AB ,/h RA = q AB b B ,/h T A = q AB B B . (70)Inverting the matrix encoded in the fifth equation in (70)to write q AB in terms of /δ AB and /h AB , the variables ofthe dual foliation formulation can be expressed as, A = (cid:18) ( h RT − q AB /h RA /h T B ) h RR + 1 − q AB /h RA /h RA − ( h T T − − q AB /h T A /h T A ) (cid:19) / , B R = h RT − q AB /h RA /h T B (cid:112) h RR + 1 − q AB /h RA /h RA , L = (cid:113) h RR + 1 − q AB /h RA /h RB ,b A = q AB /h RB , B A = q AB /h T B . (71)The last expressions can be written in terms of the com-ponents of h ab in the flat null frame using that h T T = 14 ( h LL + 2 h LL + h LL ) ,h RR = 14 ( h LL − h LL + h LL ) ,h RT = 14 ( h LL − h LL ) ,/h RA = 12 Ω AB ( h Lω B − h Lω B ) ,/h T A = 12 Ω AB ( h Lω B + h Lω B ) . (72)Recalling that h ab = R H ab and taking into account therelation between ω Aa and S Aa , as given in the expres-sions (5), gives h LL = 1 R H LL , h LL = 1 R H LL ,h LL = 1 R H LL , /h LA = Ω AB H LS B ,/h LA = Ω AA H LS A , /h AB = R Ω AC Ω BD H S C S D . (73)Now, recall that the radial and angular coordinate light-speeds in the dual foliation formulation [22, 23] are givenby C R ± = −B R ± AL − , C A ± = −B A ∓ b A AL − . (74)Thus, using equations (71), (72) and (73) we obtain C R + (cid:39) − H LL R , C R − (cid:39) − H LL R ,C A ± = O A ( R − ) . (75)For completeness of the discussion observe that the vari-ables of the dual foliation formulation are related to the variables of the asymptotic system via A (cid:39) ± ∓ H LL + 2 H LL + H LL R ,
L (cid:39) ± ± H LL − H LL + H LL R , B R (cid:39) ± H LL − H LL R ,b A = O A ( R − ) , B A = O A ( R − ) ,q AB (cid:39) Ω AA Ω AB ( R δ AB + RH S A S B ) . (76)Using the chain rule one has that ∂ T C R + (cid:39) R ∂ q H LL ,∂ R C R + (cid:39) R ( H LL − ∂ s H LL ) − R ∂ q H LL . (77)The coordinate light-speed condition [22] is that ∂ T C R + = ∂ i C + R = O ( R − − δ ) (78)with δ >
0. Consequently, the coordinate light-speedcondition is satisfied as long as H LL + ∂ s H LL = O ( R − δ ) , ∂ q H LL = O ( R − δ ) . (79) V. THE ASYMPTOTIC SYSTEM FOR T ab = I T ab In view of the remarks on made in subsection IV Babout free evolution schemes in NR, the following hy-perbolic reduction of the Einstein field equations will beconsidered. Let I T ab ≡ C T ab + W ab + W ab (80)here W ab and W ab are symmetric tensors expressed inthe flat null frame as in equation (8) where the only non-vanishing components are given by W LL = C c ˚ ∇ L g Lc − C c ˚ ∇ c g LL W LL = C L C L − ωR C L , W LS A = C L C S A − ωR C S A , W S A S B = δ AB (cid:16) C L C L − ωR C L (cid:17) . (81)In these expressions ω is a positive constant which willplay a similar role to the damping parameters γ , γ and γ in [27]. First notice that I T ab is homogeneous in C a so that C a = 0 implies I T ab = 0.The reduced Einstein field equations R ab = 0 implythe following set of wave equations, g cd ˚ ∇ d ˚ ∇ c g ab = P ab [ g ] + Q ab [ g ] + F ab + 2 W ab + 2 W ab , (82)0for the metric. Now, we consider g ab = m ab + h ab to com-pute the corresponding evolution equations for h ab andthen its asymptotic system. Here, since we are to discussthe derivation of the corresponding asymptotic system, itis understood, as in subsection IV B, that the metric m ab is used to raise and lower indices of all tensors exceptfor g ab for which one uses equation (50). Taking theseconsiderations into account, a direct computation usingequation (82) renders the following evolution equationsfor h ab , g cd ˚ ∇ c ˚ ∇ d h ab = P ab [ h ] + Q ab [ h ] + F ab + 2 W ab + 2 W ab + O ab ( h (˚ ∇ h ) ) , (83)where it is understood that F ab , W ab and W ab had beenexpanded up to order O ab ( h (˚ ∇ h ) ). For completenessobserve that the GHG constraint reads m ac ˚ ∇ c h ab = 12 m ac ˚ ∇ b h ac − F b + O b ( h ˚ ∇ h ) , (84)where O b ( h ˚ ∇ h ) is linear in ˚ ∇ a h bc with coefficients de-pending on h ab which vanish for h ab = 0.Recall that the asymptotic system is extracted fromthe terms corresponding to the leading order of R ab = 0,expressed in the asymptotic variables H ab and coordi-nates ( q, s ), which in the harmonic case discussed insubsection IV B is O ( R − ). Thus one can see that set-ting F a (cid:39) ˚ ∇ b F a = O ( R − ) ensures that the gauge sourcefunctions will not contribute to the asymptotic system.Furthermore, with such a choice of F a , the GHG asymp-totic constraint equations, i.e. the asymptotic expres-sions for C a , coincide with the harmonic case since thelatter are obtained from the asymptotic expression for Γ a whose leading order is O ( R − ). On the other hand, ob-serve that W ab and W ab do contribute to the asymptoticsystem since their components are either quadratic in C a or proportional to R − C a .Computing the asymptotic system for equation (82)one obtains (2˚ ∇ s − H LL ˚ ∇ q )˚ ∇ q H ab = T ab , (85)where T ab is expressed in the flat null frame as (8) wherethe only non-vanishing components are T LL = (cid:0) − ω + ∂ q H LL (cid:1) ∂ q H LL ,T LS A = (cid:0) − ω + ∂ q H LL (cid:1) ∂ q H LS A ,T S A S B = δ AB (cid:0) − ω + ∂ q H LL (cid:1) ( ∂ q H ∅ ) T LL = − ∂ q H + ) − ∂ q H × ) (86)On the other hand, as explained above, the asymptoticGHG constraint condition is identical to equation (62)for our choice of F a . A first observation is that contract-ing with L a L b equation (64) is of course recovered. Theterm W ab was included precisely for this purpose. Theadvantage of this approach is that the above succinct ex-pression for T LL is obtained directly from the evolution equations without assuming that C a = 0. As it will bediscussed in greater detail in the next subsection, theparameter ω in the term W ab was included so that theevolution equations ensure the small constraint violationsare damped away along the integral curves of then vectorfield 2 ∂ s − H LL ∂ q while the terms quadratic in C a in with W ab are introduced to be able to commute 2 ∂ s − H LL ∂ q and ∂ q when necessary. A detailed discussion on howto exploit these constraint additions in view of a futurenumerical implementation using the dual foliation for-malism is the content of the next subsection. A. Analysis of the asymptotic system
A direct computation then shows that the asymptoticsystem (85) can be expressed as( ∂ s − H LL ∂ q ) ∂ q H U = − ω∂ q H U + ∂ q H U ∂ q H LL , (87a)( ∂ s − H LL ∂ q ) ∂ q H V = 0 , (87b)( ∂ s − H LL ∂ q ) ∂ q H LL = − ∂ q H + ) − ∂ q H × ) , (87c)where V ∈ {
LL, LS A , × , + } and U ∈ {
LL, LS A , ∅ } .Observe the asymptotic equations for the componentsof H ab , split in three classes whereas in the asymptoticsystem of subsection IV B they split only in two. Thecomponents of H ab satisfying equation (87b) will be re-ferred as the “the good” components, on the other hand, H LL will be referred as “the bad” component and finally,those satisfying equation (87a) as the “the ugly” compo-nents. Observe that the asymptotic system (87) doesnot follow the hierarchical structure introduced in [25].In particular, the equation for H LL does not lie on thebottom level of the hierarchy and its equation containsquadratic non-linearities of H LL itself. Nevertheless wewill see in what follows that one can still integrate theseequations and obtain, under suitable assumptions on theinitial data, bounded solutions for H U and moreover,that ∂ q H U ∼ O ( R − ω ).For completeness, observe that, with the current de-cay assumption on the gauge source functions F a , theasymptotic GHG constraint equations are identical tothose given in equations (63) and can be compactly writ-ten as C U = 0 , (88)where C U ≡ ∂ q H U . Nevertheless, recall, that the condi-tions encoded in (88) are not expected to be fulfilled in afree evolution numerical implementation as one expectssmall violations to the GHG constraints. Consequently,the following analysis will be done without assuming thatequation (88) is satisfied. Solution for the uglies:
Observe that the equationsencoded in (87a) are the asymptotic equations for thecomponents of the metric perturbation associated withthe GHG constraints. In other words, equation (87a) can1be read as the following equation for the GHG constraints( ∂ s − H LL ∂ q ) C U = − ωC U + 12 C U C LL . (89)Let γ = ( s ( λ ) , q ( λ ) , θ A ( λ )) be a curve passing throughthe point γ (cid:63) = ( s (cid:63) , q (cid:63) , θ A(cid:63) ), where s (cid:63) = s ( λ (cid:63) ), q (cid:63) = q ( λ (cid:63) )and θ A(cid:63) = θ A ( λ (cid:63) ), satisfyingd s d λ = 1 , d q d λ = − H LL , (90)and θ A ( λ ) = θ A(cid:63) . Observe that integrating the first equa-tion in (90) and setting λ (cid:63) = s (cid:63) one has s = λ . Thus, onecan use s to parameterize the curve as γ = ( s, q ( s ) , θ A(cid:63) ).Then, considering equation (89) along γ one hasd C U ( γ )d s = − ωC U ( γ ) + 12 C U ( γ ) C LL ( γ ) . (91)Solving equation (91) for U = LL one obtains, C LL ( γ ) = 2 ω − e ω ( p +2 A (cid:63) ) , (92)where A (cid:63) is an integration constant. Using that ω > C LL ( s (cid:63) , q (cid:63) ) as C LL ( γ (cid:63) )then the solution can be written as C LL ( γ ) = 2 C LL ( γ (cid:63) ) ωe ωs (cid:63) C LL ( γ (cid:63) )( e ωs (cid:63) − e ωs ) + 2 ωe ωs . (93)This equation is dangerous since one could have blowup at finite value of s . Nevertheless, assumingthat C LL ( γ (cid:63) ) < ω the denominator in expression (93)is positive, and then C LL ( γ ) is bounded. Moreover, onehas that lim s →∞ C LL ( γ ) = 0 . Now, we can use expression (93) to solve equation (89)for C U with U ∈ { LS A , ∅ } . A direct calculation yields, C U ( γ ) = 2 C U ( γ (cid:63) ) ωe ωs (cid:63) C LL ( γ (cid:63) )( e ωs (cid:63) − e ωs ) + 2 ωe ωs . (94)Again observe that lim s →∞ C U ( γ ) = 0 . (95)Consequently, recalling that C U ≡ ∂ q H U , then, one con-cludes that, ∂ q H U ( γ ) is determined by equations (93)and (94). Now, to solve for H U observe that ∂ q (2 ∂ s − H LL ∂ q ) H U = (2 ∂ s − H LL ∂ q ) ∂ q H U − ∂ q H LL ∂ q H U . (96)Using the latter identity with equation (87a) we get ∂ q (cid:2) (2 ∂ s − H LL ∂ q ) H U (cid:3) = − ω∂ q H U . (97) ( s (cid:63) , q (cid:63) ) ( s (cid:5) , q (cid:5) ) − ∂ q − ∂ q ˙ γ = d d s ˙ γ = d d s γ ( s ; s (cid:63) ) s = s (cid:5) SS T • I + i FIG. 1: Schematic depiction of the integration of the com-ponents of H ab associated with the constraints. The angularcoordinates θ A(cid:63) have been suppressed to simplify the nota-tion in this figure and this caption. Given a point ( s (cid:63) , q (cid:63) ) onthe initial hypersurface S , the characteristic curve that passesthrough that point is denoted as γ ( s ; s (cid:63) ). Observe that vary-ing s (cid:63) , then one obtains a congruence of curves parameterizedby s (cid:63) . The diagram shows the hypersurface S T • determinedby the condition T = T • where, in turn, T • , is determinedby the intersection of the curve with constant s = s (cid:5) andthe characteristic γ ( s ; s (cid:63) ) as shown in the diagram. The inte-gral curves of ∂ q are the curves of constant s along which thefirst integration is performed. Then, the second integration isperformed along the characteristic curves γ ( s ; s (cid:63) ) Consequently, integrating from q to q we have(2 ∂ s − H LL ∂ q ) H U + 2 ωH U = G U ( s ) , (98)where G U ( s ) ≡ ((2 ∂ s − H LL ∂ q ) H U + 2 ωH U ) | S , (99)and S is the hypersurface determined by T = 0. Ob-serve that if p ∈ S , then p has coordinates ( s, q ( s ) , θ A )where q ( s ) = e s .Now, consider the curve γ intersecting the hyper-surface S at the point p (cid:63) with coordinates ( s (cid:63) , q (cid:63) , θ A(cid:63) ),where q (cid:63) = e s (cid:63) . Proceeding similarly as in equation (91)one can rewrite equation (98) asd H U ( γ )d s + ωH U ( γ ) = 12 G U ( s ) . (100)The latter equation can be solved to yield H U ( γ ) = e − ωs (cid:18) e ωs (cid:63) H U ( γ (cid:63) ) − (cid:90) ss (cid:63) e ω ¯ s G U (¯ s )d¯ s (cid:19) . (101)Thus, H U ( γ ) and ∂ q H U ( γ ) are determined by data on S via equations (101), (93) and (94). Additionally, observethat, from equation (98) one has ∂ s H U = 12 H LL ( γ ) ∂ q H U ( γ ) − ωH U + 12 G U ( s ) . (102)Therefore, ∂ s H U ( γ ) is determined by H U ( γ ), ∂ q H U ( γ )and G U ( s ). Finally, recalling that s = ln R , then, if the2initial data on S is such that H U | S ∼ O (1) , sup( ∂ q H LL | S ) < ω.∂ q H U | S ∼ O ( R − ) , ∂ s H U | S ∼ O ( R − ) , (103)Then G U ( s ) (cid:39) O (1) and, consequently, using equa-tions (101),(93) and (94) one concludes that ∂ q H U ( γ ) ∼ O ( R − ω ) , ∂ s H U ( γ ) ∼ O (1) ,H U ( γ ) ∼ O (1) . (104)Observe that, since ∂ R = R ∂ s then ∂ s H U ( γ ) ∼ O (1) im-plies that ∂ R H U ( γ ) ∼ O ( R − ). The latter decay ratesimply that the coordinate light-speed condition, as ex-pressed in equation (79), is satisfied if one sets ω ≥ δ . Solution for the goods:
Notice that in contrast withthe case of equation (87a), for equation (87b) thefields ∂ q H V are not associated to the constraints. Never-theless, along the curve γ equation (87b) simply readsdd s ∂ q H V ( γ ) = 0 . (105)Thus ∂ q H V ( γ ) = ∂ q H V ( γ (cid:63) ) . (106)Now, integrating the second equation in (90) one has that q = q (cid:63) − (cid:90) ss (cid:63) H LL (¯ s )d¯ s. (107)Observe that ∂ q H V ( γ (cid:63) ) = ∂ q H V ( s (cid:63) , q (cid:63) ) does not dependon s . Using q (cid:63) as a new coordinate. Namely, consideringa coordinate system ( s, u ) where u = q + (cid:82) ss (cid:63) H LL (¯ s )d¯ s then ∂ q H V ( γ (cid:63) ) is only a function of u . Moreover, since ∂ q H V = ∂ u H V then, integrating in u one has H V ( γ ) = (cid:90) uu (cid:63) ∂ q H V (¯ u )d¯ u. (108)Thus, H V ( γ ) does not depend on s , conse-quently, H V ( γ ) ∼ O (1). Solution for the bad:
To integrate equation (87c) ob-serve that along the curve γ one hasdd s ∂ q H LL ( γ ) = − (cid:2) ( ∂ q H + ) − ∂ q H × ) (cid:3) . (109)Using the previous results obtained for H V , one hasthat ∂ q H + and ∂ q H × do not depend on s . Thus, in-tegrating in s , one obtains ∂ q H LL ( γ ) = − (cid:2) ( ∂ q H + ( γ (cid:63) )) + ( ∂ q H × ( γ (cid:63) )) (cid:3) s. (110)Then, using the coordinates ( s, u ) and that ∂ q = ∂ u , onehas that H LL ( γ ) = − s (cid:90) uu (cid:63) (( ∂ q H + ( γ (cid:63) )) + ( ∂ q H × ( γ (cid:63) )) )d¯ u. (111)Observe that the integration of the asymptotic equationfor the goods and the bad is analogous to that of themodel equation (17). B. The eikonal equation
In this subsection the asymptotic form of the eikonalequation and its relation to the characteristics of theasymptotic system (85) are discussed. As emphasizedbefore, when dealing with asymptotic computations wewill use the Minkowski metric m ab to raise and lower in-dices in all tensors except for g ab in which case we useequation (50). The eikonal equation and its asymptotic system:
Thevectors L a and L a are null vectors in Minkowski space-time, namely m ab L a L b = m ab L a L b = 0. In what follows,the asymptotic form of vectors ξ a which are null respectto the perturbed metric g ab will be discussed.We start by recalling that the eikonal equation reads g ab ∇ a u ∇ b u = 0 , (112)and that if ξ a = g ab ∇ b u , then eikonal equation simplystates that ξ a is null respect to g ab . Now, let ξ a = ˚ ξ a + 1 R ˇ ξ a , (113)where m ab ˚ ξ a ˚ ξ b = 0 . (114)Using this notation one has that g ab ξ a ξ b = H ab ˚ ξ a ˚ ξ b R + 2 m ab ˚ ξ a ˇ ξ b R + 2 H ab ˚ ξ a ˇ ξ b R + m ab ˇ ξ a ˇ ξ b R + H ab ˇ ξ a ˇ ξ b R . (115)Thus, ξ a is a null vector respect to g ab to leading or-der R − , if the following condition is satisfied H ab ˚ ξ a ˚ ξ b + 2 m ab ˚ ξ a ˇ ξ b = 0 . (116)Naturally, one can algebraically impose that g ab ξ a ξ b = 0also to order R − and R − . Nevertheless, consistent withthe discussion of the asymptotic system, one is interestedonly in the leading order. In the following, it will beshown that, if equation (116) is satisfied on the initialhypersurface S then ξ a will be, to leading order, nullrespect to g ab in D ( S ) ⊆ M provided that the coordinatelight-speed condition of subsection IV C is satisfied. Thecondition imposed by the eikonal equation (112) can beformulated as an initial value problem as follows ∇ c ( g ab ξ a ξ b ) = 0 , ( g ab ξ a ξ b ) | S = 0 . (117)The initial condition ( g ab ξ a ξ b ) | S = 0 to leading or-der R − corresponds to equation (116) where the fieldsare restricted to the initial hypersurface S .Now observe that the condition ξ a = g ab ∇ b u is equiva-lent to the integrability condition ∇ [ a ξ b ] = 0. Using thisintegrability condition, the first equation in (117) reducesto the geodesic equation ξ b ∇ b ξ a = 0 . (118)3Using the integrability condition ˚ ∇ [ a ˚ ξ b ] = 0 and proceed-ing in analogous way with equation (114) one has,˚ ξ b ˚ ∇ b ˚ ξ a = 0 . (119)Notice that the geodesic equation (118) can be expressedmore explicitly, using the covariant derivative ˚ ∇ , as2 ξ a ˚ ∇ a ξ c + ξ a ξ b ( g ) cd (˚ ∇ a g bd + ˚ ∇ b g ad − ˚ ∇ d g ab ) = 0 . (120)Using equations (49), (54), (113), (118) and (119) oneobtains ξ a ∇ a ξ c (cid:39) R (cid:0) ξ a ˚ ∇ a ˇ ξ c + 2 ˇ ξ a ˚ ∇ a ˚ ξ c + m cd (˚ ξ a ˚ ξ b ˚ ∇ a H bd + ˚ ξ a ˚ ξ b ˚ ∇ b H ad − ˚ ξ a ˚ ξ b ˚ ∇ d H ab ) (cid:1) . (121)Thus, similarly as it was done for the Einstein field equa-tions, one says that the asymptotic system for equa-tion (118) is encoded in the leading order of the aboveexpansion. Recall that ˚ ξ a represents any null vector withrespect to m ab . Choosing ˚ ξ a = L a and contracting withthe flat null frame one obtains thatˆ L c ξ a ∇ a ξ c (cid:39) R ˚ ∇ q H LL , ˆ L c ξ a ∇ a ξ c = O ( R − ) , ˆ S Ac ξ a ∇ a ξ c = O ( R − ) . (122)Recalling from subsection V A that, along the character-istic curve γ , one has ˚ ∇ q H U (cid:39) O ( R − ω ) we observe thatsetting ω ≥ R − .Therefore, exploiting the discussion of subsection V Aand setting ω ≥
1, one can conclude that( g ab ξ a ξ b ) | S = 0 , (123)implies that, to leading order, g ab ξ a ξ b = 0 , in D ( S ) ⊆ M . (124)Additionally, observe that for ˚ ξ a = L a , using equa-tion (116), condition (123) reduces to( ˆ L a ˇ ξ a ) | S = − H LL | S . (125)By the argument above, to leading order, onehas g ab ξ a ξ b = 0 in D ( S ) ⊆ M . Using equation (116),this is equivalent toˆ L a ˇ ξ a = − H LL , in D ( S ) ⊆ M . (126) The characteristics of the asymptotic system and nullgeodesics:
We now show that the characteristics of theasymptotic system (85) correspond, to leading order, to null geodesics of the perturbed spacetime ( g ab , M ). Ob-serve that, using equations (5) and (54) one has that( ∂ q ) a = −
12 ( L a + L a ) , ( ∂ s ) a = RL a . (127)We define χ a = 1 R (cid:0) ( ∂ s ) a − H LL ( ∂ q ) a (cid:1) , (128)and observe that χ a is a tangent to the curve γ . Further-more, notice that χ a can be expressed as χ a = L a + 14 R H LL ( L a + L a ) . (129)Now, consistent with equation (113), consider thesplit χ a = ˚ χ a + R ˇ χ a with ˚ χ a = L a andˇ χ a = 14 H LL ( L a + L a ) . (130)Then, one readily observes thatˆ L a ˇ χ a = − H LL . (131)Consequently, using the results of subsection V B, oneconcludes that the vector χ a representing the tangentvector to the characteristics of the asymptotic sys-tem (85), is, to leading order, a null vector in the per-turbed spacetime ( g ab , M ). VI. THE ASYMPTOTIC SYSTEM FOR FIRSTORDER GHG WITH CONSTRAINT DAMPING
In this section, the first order GHG Einstein evolutionequations with the constraint additions introduced in sec-tion V are presented and its asymptotic system is derivedin analogous way as that of the model equation (17).In subsection VI A the general evolution equations aregiven, and, since this is not an asymptotic computation,the usual conventions for raising and lowering indices us-ing g ab is employed. In contrast, in subsection VI B thecorresponding asymptotic system is derived, and, conse-quently, the indices are handled in concordance with theasymptotic calculations of section IV B. A. First order GHG
Consider the reduced Ricci operator as given in equa-tion (41), where T ab denotes a generic constraint addi-tion. Then, using the foregoing conventions by express-ing the derivatives in terms of ˚ ∇ , the GHG evolutionequations [22], in abstract index notation, read N c ˚ ∇ c g ab = S ( g ) ab N f γ ad ˚ ∇ f φ dbc = − γ ad ˚ ∇ d π bc + γ γ ad ˚ ∇ d g bc + S ( φ ) abc N c ˚ ∇ c π ab = γ cd ˚ ∇ c φ dab + S ( π ) ab (132)4where S ( g ) ab = − π ab S ( φ ) abc = − γ φ abc + N d N h π bc φ adh + γ fh N d φ fbc φ adh S ( π ) ab = − ∇ ( a F b ) − ach Γ bdf g cd g hf − N c N d π ab π cd − g cd π ca π db − N d π df γ cf φ cab + 2 g cd g hf φ fdb φ hca + 2 T ab (133)andΓ abc = N ( c π b ) a − N a π bc + φ ( c | a | b ) − φ abc . (134)These evolution equations were written respect to normalderivatives to ease the subsequent discussion. Neverthe-less, observe that time derivatives and normal derivativesare related via ˚ ∇ T = A N a ˚ ∇ a + B a ˚ ∇ a . (135)See [28] for a discussion of these equations and the ap-pended mathematica notebooks for a detailed derivation.Notice that in the notation of [22] the formulation pa-rameters γ , γ have been set to zero and, ignoring thepotential parameters in the constraint addition term T ab , γ is the only parameter that has been left unspecified.The evolution system (132) is subject, as in the discus-sion of section V, to the GHG constraints C a ≡ Γ a + F a = 0 , (136)and, additionally, subject to the reduction constraints C abc ≡ γ ad ∇ d g bc − φ abc = 0 . (137) B. Evolution equations for the perturbation
To derive the asymptotic system we proceed as in pre-vious sections and set g ab = m ab + h ab . (138)Following the discussion of the asymptotic system in sub-section IV B, once the equations for the perturbations areobtained, as for example in equation (52), to derive theasymptotic system, the metric g ab is regarded just as asymmetric tensor and the indices are raised and loweredusing m ab . The latter implies that one has to be verycareful with the canonical position of indices before theexpansion. In particular recall that for the inverse metricone has g ab = m ab − h ab + O ab ( h ) (139)where the indices in the right hand side of the aboveexpansion are moved using m , in other words, h ab = m ac m db h cd . To simplify the notation and avoid length-ier expressions, let . = denote equality up to terms of or-der O ( h , h ˚ ∇ h, (˚ ∇ h ) ) so that one writes, g ab . = m ab − h ab . (140) The latter observations imply the following expansionsfor the normal and projector N a . = ˚ N a + ˇ N a ,γ ab . = ˚ γ ab + ˇ γ ab ,N a ≡ ( g − ) ab N b . = ˚ N a + ( ˇ N a − ˚ N b h ab ) ,γ ab ≡ ( g − ) cb γ ac . = ˚ γ ab + ( − h cb ˚ γ ac + ˇ γ ab ) ,γ ab ≡ ( g − ) ac ( g − ) bd γ cd . = ˚ γ ab + (ˇ γ ab − h ca ˚ γ cb − h db ˚ γ ad )(141)where,˚ γ ab ≡ m ab + ˚ N a ˚ N b ˇ γ ab ≡ N ( a ˇ N b ) + h ab ˚ N a ≡ − ( L a + L a ) ˇ N a ≡ − ˚ N a ˚ N b ˚ N c h bc (142)and all the indices in the right-hand side of equation (141)were moved using m ab . Since ˚ ∇ c m ab = 0 then, one hasthat the background values for π ab , φ abc and Γ abc vanish,namely, ˚ π ab = ˚ φ abc = ˚Γ abc = 0 . (143)To simplify the notation we will denote the correspond-ing perturbations to the latter quantities without addingˇ to each of these fields. Thus, hereafter, π ab , φ abc and Γ abc will represent the associated perturbations. Astraightforward computation renders the following evo-lution equations N a ˚ ∇ a h bc . = − π bc ,N a γ f b ˚ ∇ a φ bcd . = N a N b π cd φ fab + N a γ bh φ bcd φ fah + γ ( − φ fcd + γ f a ˚ ∇ a h cd ) − γ f a ˚ ∇ a π cd ,N c ˚ ∇ c π ab . = − m cd m hf Γ ach Γ bdf − N c N d π ab π cd − m cd π ca π db − N d π df γ cf φ cab + 2 m cd m hf φ fdb φ hca − γ cd ˚ ∇ c φ dab + 2 T ab − ∇ ( a F b ) , (144)where it is understood that N a , γ ab and γ ab have beensubstituted using equation (141) while T ab and ∇ ( a F b ) have been expanded out to order O ( h , h ˚ ∇ h, (˚ ∇ h ) ). Ob-serve that, in terms of the T and R coordinates onehas ˚ N a = (d T ) a . Thus, for completeness, let ˚ R a =(d R ) a . A direct computation shows that ˚ N a φ abc . = 0.The latter implies that one write φ abc . = ˚ R a φ Rbc + S Aa φ S A bc , (145)where φ Rab ≡ ˚ R c φ cab and φ S A ab ≡ S Ac φ cab . To derivethe asymptotic system, following the strategy describedin section III B, one defines σ + ab = π ab + φ Rab , σ − ab = π ab − φ Rab , (146)and expresses the evolution equations (144) in terms ofthe variables { σ + ab , σ − ab , φ abc , h ab } . + ab = R σ + ab , Σ − ab = Rσ − ab , Φ S A ab = R φ S A ab , H ab = Rh ab . (147)and defining T ab = R T ab , (148)one obtains, assuming as in section V that the gaugesource functions decay sufficiently fast F a ∼ R − , thefollowing expansions˚ ∇ q H ab + Σ − ab (cid:39) , − H LL ˚ ∇ q Σ − ab + γ ( H ab + Σ + ab − ˚ ∇ s H ab ) + ˚ ∇ s Σ − ab (cid:39) Σ − aL Σ − bL − ˆ L ( a Σ − b ) h Σ − L h + ˆ L a ˆ L b Σ − cd Σ − cd − T ab , ( H LL + H LL − H LL )˚ ∇ q Σ − ab + γ ( H ab − ˚ ∇ s H ab + Σ + ab ) + 2˚ ∇ q Σ + ab (cid:39) Σ − ab + Σ − ab (Σ − LL − − LL − − LL ) − Σ − aL Σ − bL + L ( a Σ − b ) h Σ − L h − L a L b Σ − cd Σ − cd + 2 T ab , ˚ ∇ q Φ S A ab = − ω Ac ˚ ∇ c Σ − ab − γ (Φ S A ab − ω Ac ˚ ∇ c H ab ) . (149)Now, to unwrap the connection of the asymptotic systemin first order and second order form, define the rescaledreduction constraints as follows C Rab = R C Rab , C S A ab = R C S A ab , (150)Then, a direct computation using equations (137), (144),(146) and (147), with the current decay assumptionsabout the gauge source functions F a , one obtainsΣ − ab (cid:39) − ∇ q H ab , Σ + ab (cid:39) − H ab − C Rab + ˚ ∇ s H ab + ( H LL − H LL − H LL )˚ ∇ q H ab , Φ S A ab = − C S A + ω Ac ˚ ∇ c H ab , (151)where, written in full, these quantities satisfy,Σ − ab + Σ + ab R = 2˚ ∇ q H ab + 2˚ ∇ s H ab R − H ab R − C Rab
R . (152)Substitution of the equations (152) and (151), into equa-tion (149) and some rearranging reveals,(2˚ ∇ s − H LL ˚ ∇ q )˚ ∇ q H ab (cid:39) T ab − γ C Rab − ˚ ∇ q H aL ˚ ∇ q H bL + 2 L ( a ˚ ∇ | q | H b ) c ˚ ∇ q H Lc − L a L b ˚ ∇ q H cd ˚ ∇ q H cd , ˚ ∇ q H ab + Σ − ab (cid:39) , ˚ ∇ q C Rab (cid:39) γ C Rab , ˚ ∇ q C S A ab (cid:39) γ C S A ab . (153) Setting γ ∼ R − and T ab = I T ab as given in equa-tion (80) and, following the philosophy of the asymptoticsystem by formally replacing the (cid:39) by = one obtains(2˚ ∇ s − H LL ˚ ∇ q )˚ ∇ q H ab = T ab , ˚ ∇ q H ab + Σ − ab = 0 , ˚ ∇ q C Rab = 0 , ˚ ∇ q C S A ab = 0 , (154)where T ab is the same tensor as that of equation (85). Fi-nally, from the last two equations in (154) one concludesthat C Rab = C (cid:63)Rab , C S A ab = C (cid:63)S A ab , (155)where C (cid:63)R = C R | q = q (cid:63) , C (cid:63)S A = C S A | q = q (cid:63) . Thus, assum-ing that the latter quantities are uniformly bounded, itfollows from the analysis of subsection V A and expres-sions (151) and (147) that σ −U ∼ O ( R − − ω ) , σ −V ∼ O ( R − ) ,σ + U ∼ O ( R − ) , σ + V ∼ O ( R − ) ,φ S A U ∼ O ( R − ) , φ S A V ∼ O ( R − ) ,h U ∼ O ( R − ) h V ∼ O ( R − ) σ − LL ∼ O ( R − ln R ) , σ + LL ∼ O ( R − ln R ) ,φ S A LL ∼ O ( R − ln R ) , h LL ∼ O ( R − ln R ) , (156)where U ∈ {
LL, LS A , ∅ } , V ∈ {
LL, LS A , × , + } .For completeness notice that the generalized harmonicgauge condition (136) implies, to leading order, the fol-lowing constraint equation m ab Γ cab + F a . = 0 . (157)Assuming as before that F a ∼ R − a direct calculationshows that the asymptotic form of these equations readΣ −U = 0 . (158)Naturally, this implies that when the constraints are notviolated we have σ −U = 0. Nonetheless, recall that, inthe analysis of subsection V A does not rely on the sat-isfaction of the asymptotic GHG constraints. In otherwords, the analysis of subsection V A shows that evenif small violations of the GHG constraints are presentone has σ −U ( γ ) ∼ O ( R − − ω ) so that the coordinate light-speed condition can be satisfied close to I . VII. THE TRAUTMAN-BONDI MASS
In this section we discuss the implications of thepresent analysis for the definition of the Trautman-Bondimass. The main issue to be analyzed here is if the H LL component enters into the expression defining the Bondimass and whether or not this affects its boundedness.6Let H be an hyperboloidal slice in ( M , g ab ) andlet S ⊂ H denote a surface of constant R in H . Recallthe definitions introduced in section IV C for the 2+1+1split and define k ≡ q ab ∇ a R b . (159)where q ab = g ac g bd q cd . Additionally, let ˚ q ab ≡ ˚ γ ab − ˚ R a ˚ R b and ˚ k ≡ ˚ q ab ∇ a ˚ R b , where ˚ q ab = m ac m bd ˚ q cd , with ˚ γ ab and ˚ R a as defined in section VI. Using the above defini-tions, the Trautman-Bondi mass can be expressed as M = − π (cid:90) S ∞ ( k − ˚ k )d S (160)where d S denotes the area element in ( S , q ab ) and S ∞ represents a cut of null-infinity [40]. Consistent with thenotation of subsection (IV C), some introducing arbitrarycoordinates θ A on S , one has d S = (cid:112) det[ q AB ]d θ . Adirect computation using equation (76) givesd S (cid:39) R (cid:112) det[ σ AB ]d θ (161)where σ AB denotes the standard metric on S in the θ A coordinates. To have a more compact notation we denotethe area element of S as d S . Observe that, proceedingin analogous way as for the vector N a in section VI, onehas the following expansions S a . = ˚ R a + ˇ R a , S a . = ˚ R a + ( ˇ R a − h ab ˚ R b ) (162)where, ˇ R a ≡ ( − ˚ N a ˚ N b ˚ R c + ˚ R a ˚ R b ˚ R c ) h bc . (163)Then, a direct computation shows that k − ˚ k (cid:39) K + C L − δ AB ˚ ∇ a (cid:16) R H LS A ω Ba (cid:17) , (164)where K = − H LL R − H ∅ R + ∂ s H ∅ R + 18 R (2 ∂ s − H LL ∂ q ) H LL . (165)and C L ≡ ˆ L a C a where C a encodes the GHG con-straints (136). Observe that, the last term in (164)denotes the sum of the divergence of the vectorfields R H LS A ω Ba which can be regarded as vectorson S . Consequently, the last term in (164) drops outafter integration on S .Recall that in the calculation of the asymptotic con-straint conditions (63) were obtained from the coefficientof the leading order term R − . Nonetheless, computedto second order C L reads C L (cid:39) ∂ q H ∅ R + 1 R (cid:20) H LL − H LL + H ∅ + H LS A ˚ ∇ a ω Aa + δ AB ω Aa ˚ ∇ a H LS B + H LL ∂ q H ∅ + ∂ s ( H ∅ − H LL ) − ∂ q (cid:16) ( H × ) + ( H + ) + ( H ∅ ) − H LL H LL (cid:17)(cid:21) (166) where it was assumed as usual that F a ∼ R − so thatthe gauge source functions do not appear in the latterexpansion. To simplify the subsequent discussion, fromthis point on-wards it will be assumed that the GHGconstraints are satisfied to all orders so that C a = 0.Taking into account the latter observation one has M = − π (cid:90) S ∞ K d S. (167)To verify that the derived expression for K is correctobserve that the mass loss formula can be recovered asfollows. Using that ∂ T = − ∂ q one obtains ∂ T M = 18 π (cid:90) S ∞ ∂ q K d S. (168)Then, a direct computation using C a = 0 to re-place ∂ q H LL and ∂ q H ∅ and using equation (161), renders ∂ T M = 164 π (cid:90) S (2 ∂ s − H LL ∂ q ) ∂ q H LL d S . (169)Using the asymptotic equation (87c), one recovers themass loss formula ∂ T M = − π (cid:90) S (cid:18) ( ∂ q H + ) + ( ∂ q H × ) (cid:19) d S . (170)As a side remark it is observed that the Hamiltonianand momentum constraints an be written in term of theGHG constraints and the asymptotic equations for H LL and H LS A .Having verified that one can recover the mass loss for-mula, the main observation to be made is that K con-tains H LL and, as discussed in subsection V A, H LL ∼ ln R . Nevertheless, the Trautman-Bondi mass is welldefined. To see why this is the case, observe that,since C a = 0 and hence ∂ q H LL = 0, equation (87c) canbe written as(2 ∂ s − H LL ∂ q ) H LL = − (cid:90) qq (cid:63) ( ∂ q H + ) + ( ∂ q H × ) d¯ q. (171)Substituting the latter expression into equation (165),one finds that K can be rewritten as K = 14 R (cid:18) − H LL − H ∅ + ∂ s H ∅ (cid:19) − R (cid:90) qq (cid:63) ( ∂ q H + ) + ( ∂ q H × ) d¯ q. (172)Thus, we observe that, despite that at first instance onewould conclude that M is diverging as it contains H LL ,by virtue of the Einstein field equations, in this case,in the form of the asymptotic equation for H LL , theTrautman-Bondi mass is well defined.Our final remark is that one can formally recover theseresults for the case in which there are small violations7to the constraints C a (cid:54) = 0 simply by redefining theTrautman-Bondi mass as M = − π (cid:90) S ∞ ( k − ˚ k ) − C L d S (173)and exploiting the results of subsection V A, touse ∂ q H U ∼ O ( R − ω ) instead of ∂ q H U = 0, in each ofthe previous computations. VIII. CONCLUSIONS
The dual foliation formalism [22, 23, 41–43] is an ap-proach to GR in which the tensor basis and choice of co-ordinates are left uncoupled. In [22] a proposal was givento use the formalism to help in the numerical treatmentof future null-infinity via a suitably posed hyperboloidalinitial value problem. Nevertheless, in order for this pro-posal to work, one of the requirements, found in [22],is that certain derivatives of the coordinate light-speedhave enough decay. The latter condition is called thecoordinate light-speed condition. Here we have studiedwhether or not one can expect the coordinate light-speedcondition to be satisfied. This was done with the use ofasymptotic expansions originally introduced in [29, 30]and employed in [24] to define the weak null condition.We have shown that the coordinate light-speed conditionis related to the asymptotic harmonic constraints. Asdiscussed in the main text, if the harmonic constraintcondition is satisfied then the coordinate light-speed istrivially fulfilled. Nevertheless, numerical errors are in-herent in free evolution schemes as one expects, albeitsmall, violations to the constraint equations. Moreover,constraint violations are expected to grow during the nu-merical evolution. Consequently we must analyze thesystem without assuming that the constraints are satis-fied. It turns out that one cannot expect to satisfy thecoordinate light-speed condition without modifying thefield equations in such a way that one damps away con-straint violations. Therefore we proposed a constraintaddition such that resulting asymptotic system impliesconstraint damping in outgoing null directions. In otherwords we have shown that by adding specific multiplesof the constraints, the asymptotic system predicts that the the coordinate light-speed condition will be satisfied.This paves the way for the explicit numerical treatmentof future null-infinity. Although the constraint additionproposed was tailored for the purposes of the applica-tion in mind, it could be easily generalized and modified.In the original discussion of the weak null condition forthe Einstein field equations in [24] one can classify thecomponents of the metric perturbation h ab into “good”components and “bad” components. The good compo-nents are those whose equations satisfy the classical nullcondition of [44–46] while the “bad” component satisfiesan equation that fails to satisfy the null condition. Aslower fall-off is hence expected this component. In ouranalysis we found that the price to pay to force dampingof constraint violations close to I + , and subsequentlyfulfillment of the light-speed condition, is to add anotherlayer to this structure. The components of h ab are nowbe classified across three categories, which we call “thegood, the bad and the ugly”. The equations for the com-ponents lying in the new “ugly” category are preciselythose associated with the constraints.Having established fall-off rates within the asymptoticsystem, we turned to the Trautman-Bondi mass, and re-covered the mass loss formula. Although at first glancethe Trautman-Bondi mass formula contains terms thatcould potentially blow up at I + , by virtue of the Ein-stein field equations it turns out to be well defined. Fi-nally, in concordance with the outlook of the rest of thework, we discussed how to modify the definition of theTrautman-Bondi with constraints in such a way that onecan reproduce the above remarks, even when small con-straints violations are present. Acknowledgments
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