Dual Foliation Formulations of General Relativity
aa r X i v : . [ g r- q c ] S e p Dual Foliation Formulations of General Relativity
David Hilditch
Theoretical Physics Institute, University of Jena, 07743 Jena, Germany (Dated: September 8, 2015)A dual foliation treatment of General Relativity is presented. The basic idea of the constructionis to consider two foliations of a spacetime by spacelike hypersurfaces and relate the two geometries.The treatment is expected to be useful in various situations, and in particular whenever one wouldlike to compare objects represented in different coordinates. Potential examples include the con-struction of initial data and the study of trapped tubes. It is common for studies in mathematicalrelativity to employ a double-null gauge. In such studies local well-posedness is treated by referringback, for example, to the generalized harmonic formulation, global properties of solutions beingtreated in a separate formalism. As a first application of the dual foliation formulation we findthat one can in fact obtain local well-posedness in the double-null coordinates directly , which shouldallow their use in numerical relativity with standard methods. With due care it is expected thatpractically any coordinates can be used with this approach.
I. INTRODUCTION
For their consideration as an initial value problem thefield equations of General Relativity are typically splitinto a set of evolution and constraint equations. This isdone by introducing coordinates x µ = ( t, x i ). The levelsets of the time coordinate t are taken to be spacelike hy-persurfaces which foliate the spacetime. The unit normalto the hypersurfaces is used to 3 + 1 decompose the fieldequations in the natural way. This results in the vacuumfield equations in the textbook form [1–3], ∂ t γ ij = − αK ij + L β γ ij ,∂ t K ij = − D i D j α + α [ R ij − K ki K jk + KK ij ] + L β K ij ,H = R − K ij K ij + K = 0 ,M i = D j K ij − D i K = 0 . (1)Given two sets of observers, one associated with x µ , an-other with coordinates X µ = ( T, X i ), what is the re-lationship between the solutions as expressed in each in the picture? Unfortunately a clear presenta-tion of the resulting formalism is not readily available,despite being straightforwardly obtained by space-timedecomposition of the four-dimensional Jacobians J µµ = ∂X µ /∂x µ . The first aim of this work is to give just such apresentation, which can be found in section II. This dualfoliation approach will be useful in numerical relativity,where one expects it will help in the construction of ini-tial data and in the comparison of solutions constructedwith different choices of lapse α and shift β i .Amongst the most powerful machinery in mathemati-cal relativity is that of the double-null coordinates. Withthis choice the field equations exhibit a particular struc-ture that allows the demonstration of the stability ofMinkowski spacetime [4], and that a certain special classof vacuum initial data will collapse to form blackholes [5].One would thus like to use these coordinates in numericalrelativity, preferably with standard methods. A numberof applications present themselves; the conjectured insta-bility of Cauchy horizons [6], the propagation of weak- null singularities [7], and the critical formation of black-holes [8]. Unfortunately from this perspective the proofsof these impressive results employ a different formalismfor long-term results and local existence. This is a seriousbugbear because, as painful experience has taught, a nec-essary condition for any numerical method to converge isthat the PDE problem is locally well-posed. Thereforethe second aim of this work is to find such a formulation.In section III this is attempted in a direct way. The nor-mal approach is to modify the equations by introducingnew constraints, coupled in a particular way to the gaugechoice, and insodoing uncover, say, a strongly hyperbolicformulation. But we find in a pure gauge analysis thatthe standard form of the double-null coordinates are onlyweakly hyperbolic, and so can not be used in this way [9–11]. With appropriate alterations, there may be such astraightforward formulation, but because a preferred di-rection is singled out the construction will in any case becomplicated. We thus abandon the search.In section IV we summarize the first order reduc-tion [12] of the generalized harmonic gauge (GHG) for-mulation [13–15] employed in the numerical relativitycodes SpEC [16] and bamps [17–19]. We use the dualfoliation formalism to derive equations of motion for theJacobian that maps from generalized harmonic to double-null coordinates. These equations are minimally coupledto the field equations, and so their hyperbolicity may betreated easily. Indeed the Jacobians satisfy a set of non-linear advection equations, and so are hyperbolic. Wemay consider the full set of fields to be solved for as theGHG system with the Jacobians tacked on. We can sub-sequently change independent variables from x µ → X µ ,with X µ the double-null coordinates. The punchline isthat since the system has at most first derivatives, we cando so without generating any derivatives of the evolvedJacobians . Therefore the PDE properties of the systemare unaffected and we end up with a formulation thatis symmetric hyperbolic directly in the double-null coor-dinates. Weaker notions of hyperbolicity are also con-sidered. Concluding remarks are collected in section V.Geometric units are used throughout. II. THE DUAL FOLIATION FORMALISM
In this section we work in the intersection of two co-ordinate patches x µ = ( t, x i ) and X µ = ( T, X i ), relatedby the Jacobian J µµ = ∂X µ /∂x µ . The two time coor-dinates define distinct foliations of the spacetime, andwith them different notions of spacelike tensors, intrin-sic and extrinsic curvatures. These quantities are relatedin the natural way with a 3 + 1 split of the Jacobian.Consequently the form of the gravitational field equa-tions in each foliation is related. Throughout latin in-dices a, b, c, d, e will be abstract. Greek indices stand forthose in coordinates x µ , or if underlined in X µ . Simi-larly latin indices i, j, k, l, m, p stand for spatial compo-nents in x µ , and when underlined for spatial componentsin X µ . Indices n and v denote contraction in that slotwith the vectors n a and v a respectively. The summationconvention is always employed. A. Coordinate freedom
Coordinate change under a decomposition:
Con-sider two sets of coordinates x µ and X µ defined in thesame region of spacetime. Each of the two time coordi-nates t and T naturally defines a foliation of the space-time which we will refer to as the lower case and uppercase foliations respectively. In the lower case foliationwe define the standard lapse, normal vector, time vector,projection operator, and shift vector, α = ( −∇ a t ∇ a t ) − , n a = − α ∇ a t ,t a ∇ a t ≡ , ⊥ ab = δ ab + n a n b ,β a = ⊥ ba t b , β i = − αn a ∇ a x i . (2)With both indices downstairs the projection operator ⊥ ab becomes the natural induced metric γ ab on the lower casefoliation. The covariant derivative associated with γ ab isdenoted by D with connection Γ. The same definitionsare made in the upper case foliation, A = ( −∇ a T ∇ a T ) − , N a = − A ∇ a T ,T a ∇ a T ≡ , (N) ⊥ ab = δ ab + N a N b ,B a = (N) ⊥ ba T b , B i = − AN a ∇ a X i . (3)The covariant derivative associated with (N) γ ab is denotedby (N) D with connection (N) Γ. The Lorentz factor and boost vector:
The unit normalvectors of the upper and lower case foliations are relatedby N a = W ( n a + v a ) , (4)where we have defined the Lorentz factor W and lowercase boost vector v a , W = − ( N a n a ) , v a = 1 W ⊥ ba N b . (5) TABLE I: A summary of the definitions of the various met-rics, time reduction variables the relationship between them.The fourth column gives the object used as a time reductionvariable when employing the given metric. The final columngives states the equation numbers relating the curvature ofthe given metric with that of the others. ‘GCM’ stands forthe Gauss-Codazzi-Mainardi equations.Metric Connection Defn. Time der. Curv. Note g ab ∇ , (4) Γ (4) R ab γ ab D, Γ ⊥ g ab K ab R ab ‘GCM’ g ab D , G ⊥ (N) γ ab K ab R ab (33) (N) γ ab (N) D, (N) Γ (N) ⊥ g ab (N) K ab (N) R ab (37) Since the normal vectors have unit magnitude theLorentz factor and boost vector satisfy, W = 1 √ − v i v i , W ≥ > γ ij v i v j ≡ v . (6)This is simply the requirement that the upper and lowercase normal observers travel subluminally relative to oneanother. Observe that we also have, n a = W ( N a + V a ) , (7)with the upper case boost vector, V a = 1 W (N) ⊥ ba n b = ( W − − W ) n a − W v a , (8)so that there is a natural reciprocity between the coordi-nate systems as expected. We thus also obtain, W = 1 p − V i V i . (9)Provided the spatial boost vector, a vector S a which isspacelike in the upper case foliation, S a N a = 0, can bereconstructed from its projection into the lower case fo-liation with, S a = ( ⊥ S ) b ( δ ba + v b n a ) , (10)and obviously a similar result holds for all tensor va-lences. Therefore we may restrict our attention of uppercase spacelike tensors to their projections into the lowercase foliation, and thus look only at the spatial compo-nents in the lower case tensor basis. Decomposition of the Jacobian J µµ : Upon 3 + 1 de-composition the Jacobian J µµ ≡ ∂X µ /∂x µ can be writ-ten, n α J αα N α = − W , n α J iα ≡ π i ,J αi N α = W v i , J ii ≡ φ ii . (11)The components π i are given in terms of the upper caselapse, shift and boost vectors by, π i = W V i − W A − B i , (12)although we rarely find that this is the most convenientform for the expression. In matrix form we therefore havethe representation, J = A − W ( α − β i v i ) α π i + β i φ ii − A − W v i φ ii ! . (13)Note that by introducing the variables A − W v i and φ ii to replace first order spatial derivatives of the coordinateswe have effectively introduced reduction constraints, D [ i ( A − W v j ] ) = 0 , D [ i φ ij ] = 0 , (14)which we will refer to as the hypersurface (orthogonality)constraints. Here and in what follows one must be care-ful to note that the upper case underlined index is to betreated as a simple label rather than an open slot whenworking in the lower case coordinates. It is straightfor-wardly checked that the transformation, J − = α − W ( A − B i V i ) A Π i + B i Φ ii − α − W V i Φ ii ! , (15)with the various auxiliary quantities defined in the nat-ural way, is indeed the inverse transformation. Time development of the Jacobian:
By the equalityof mixed partials we have the Hamilton-Jacobi like equa-tions, ∂ t ( A − W v i ) = − D i (cid:2) α ( A − W ) (cid:3) + L β ( A − W v i ) ,∂ t φ ii = D i ( α π i ) + L β φ ii , (16)for the components J T i and J ii . These expressions holdregardless of the upper case coordinate choice, but the re-maining four components can be determined only once aparticular coordinate choice is known. Perhaps the sim-plest useful example is the generalized harmonic gaugechoice (cid:3) X α = H α , which results in, ∂ t ( A − W ) = α ( A − W )( K + E ) − D i ( αA − W v i )+ L β ( A − W ) ,∂ t π i = D j ( αφ ij ) + αE i + L β π i , (17)where the gauge source functions are decomposed as E =( A/W ) H T and E i = − H i . The lower case extrinsiccurvature is defined by, K ab = − ⊥ ca ∇ c n b , (18)and likewise in the upper case foliation, except that aselsewhere we append a label N . On a given spacetimewith coordinates x µ the system (16), (17) can be viewedas a simple first order reduction of the four wave equa-tions (cid:3) X µ = H µ . Other choices will be more conve-niently expressed once the relationship between the twoinduced geometries are known. Equations of motion for projected upper case objects:
The equations of motion of the projection of upper casespacelike tensors S a and S ab projected into the lower casefoliation are, ∂ t S i = αW L N S i + L ( β − αv ) S i ,∂ t S ij = αW L N S ij + L ( β − αv ) S ij , (19)for vectors and symmetric tensors respectively. Similarexpressions hold for arbitrary valences but will not beused in what follows. The projected upper case inducedmetric defined by g ab = γ ca γ db (N) γ cd is, g ij = (N) γ ij = γ ij + W v i v j . (20)Note that as the sum of a symmetric positive definitematrix and and semi-positive definite combination of theboost vector, the projected upper case metric is itselfpositive definite and thus invertible, and can be consid-ered a metric on leaves of the lower case foliation in itsown right, if we so wish. When doing so we will refer tothis object as the boost metric, and denote the covariantderivative by D with connection G . The boost metric hasinverse, ( g − ) ij = γ ij − v i v j , (21)by the Sherman-Morrison formula. We now aim to relatethe geometry of the upper and lower case foliations interms of the lower case normal, Lorentz factor and boostvector. A summary of the relationships between the fourdifferent metrics g ab , γ ab , (N) γ ab and g ab is given in Table I. Connections and curvatures:
The Christoffel symbolassociated with the upper case induced metric is givenby the standard expression, (N) Γ γµν = (N) ⊥ (4) Γ γµν , (22)which holds in arbitrary coordinates, and where here andin what follows we use the projection operator withoutindices to denote the projection on every open slot. TheChristoffel symbol associated with the lower case inducedmetric is defined similarly. By the argument aroundequation (10) we need only compute the projection ofthe upper case Christoffel into the lower case foliation.Using (20) and (22) we find that, ⊥ (N) Γ kij = g kl g mi g pj Γ lmp − W g kl g m ( i v j ) K lm + W v k g mi g pj K mp − W v k g m ( i v j ) a m + W g kl v i v j a l + 2 W α − g kl g m ( i v j ) ∂ m β l + W α − g kl v i v j ∂ n β l − W v k v i v j L n ln α , (23)where here we use an index n to denote contraction withthe lower case unit normal vector n a , and where the accel-eration of lower case Eulerian observers is a i = D i ln α .The upper case induced Ricci tensor can be computedfrom, (N) R µν = (N) ⊥ ∂ κ (N) Γ κµν − (N) ⊥ ∂ µ (N) Γ κκν + (N) Γ κµν (N) Γ δκδ − (N) Γ κµδ (N) Γ δνκ , (24)and likewise for the lower case curvature. The rela-tionships between the upper and lower case connectionsabove (22) can be used to compute the relationship be-tween the two Ricci curvatures by brute force, but it ismore convenient to use the Gauss-Codazzi equations, asdescribed momentarily. The upper case extrinsic curva-ture projected into the lower case foliation is, (N) K ij = W K ij − D ( i W v j ) − W A ( i v j ) ≡ W ( K ij − A ( i v j ) ) , (25)where here we also define K ij which, for lack of a bettername, we call the boost extrinsic curvature. Note thatwe define the trace K ≡ ( g − ) ij K ij in a nonstandard wayby using the inverse boost metric. The projected uppercase acceleration is, A i ≡ A i = D i (ln A ) + W v i (cid:16) v j D j (ln A ) + L n (ln A ) (cid:17) . (26)It is most convenient to express the various equationsin terms of A i rather than using this expression, as wewish to suppress the explicit appearance of a L n (ln A )contribution. Constraints under the coordinate change:
Comparingthe Hamiltonian and momentum constraints in each fo-liation, we find that, (N) H = W H − W M v , (N) M i = W M i + 2 W M v v i − W Hv i . (27)An index v denotes contraction with the boost vector v a .It is thus obvious that the full set of constraints willbe satisfied in the lower case foliation if and only ifthey are satisfied in the upper case foliation. Expand-ing out the upper and lower case constraints and usingprojected upper case extrinsic curvature (25) in combi-nation with (10), we easily find the relationship betweenthe two spatial Ricci scalars. As it stands, equation (27)is really of a purely geometric nature and independentof the gravitational field equations, where we think ofthe symbols as mere shorthands according to (1), or theupper case foliation equivalent. Electric and Magnetic decomposition of the Weyl ten-sor:
Especially in General Relativity the decompositionof the Weyl tensor W abcd into two spatial tracefree ten-sors, Electric and Magnetic parts, E ab = n c n d W acbd , B ab = n c n d ∗ W acbd , (28)has special significance in encoding the propagating de-grees of freedom of the gravitational field. The dual Weyl tensor here is ∗ W abcd = ǫ ef cd W efab . Evidently this de-composition is foliation dependent, because of the pres-ence of the normal vector n a . The relationship betweenthe two decompositions is given by, (N) E ij = (2 W − E ij − W E v ( i v j ) + W E vv γ ij + 2 W ǫ kv ( i B j ) k , (N) B ij = W B ij − W ǫ lij E lv − W ǫ lvi E jl , (29)where ǫ bcd = n a ǫ abcd stands for the lower case spatial vol-ume form. Furthermore this equation shows trivially thatchanges of coordinates can not create nor destroy gravita-tional waves. Since in vacuum the electric and magneticparts also satisfy a closed evolution system [20], it is alsoclear that if we are given initial data with vanishing E ij and B ij this will be the case once and for all. The re-lationship (29) holds in general, but using the vacuumEinstein equations, we have that, E ab = R TF ab − ( K ca K bc ) TF + KK TF ab , (30)and likewise for the upper case Electric part. We canthus relate the tracefree part of the upper case and lowercase spatial Ricci tensors as we did for the Ricci scalars,namely by expanding out the projected upper case ex-trinsic curvature with (25) and using (10) to obtain thenon-spatial components. Boost metric connection and curvature:
We wouldlike to have the equations of motion for upper case ob-jects in the lower case coordinates. But as it is moreconvenient to work with spatial tensors in the lower casecoordinates we instead work with the boost metric andboost extrinsic curvature ( g ij , K ij ) to obtain the desiredresults. The time derivative of the boost vector is conve-niently encoded as, ∂ t ( W v i ) = α W (cid:2) A i − A v v i − D i ln( α W ) (cid:3) + L β ( W v i ) . (31)The relationship between the connection of the boostmetric and the lower case spatial curvature is C kij = G kij − Γ kij , with C kij = W v ( i D j ) v k − D k ( W v i v j ) + v k L v g ij = ( g − ) kl h v ( i D j ) ( W v l ) − D l ( W v i v j ) i + v k L v γ ij . (32)This result can be obtained from the standard expression,see Ch. 7 of [21], since the lower case spatial metric andboost metric act in the same tangent space; notice thatthis is not the case when we try to relate the upper andlower case spatial connections. We could now examinehow spatial geodesics are deformed in the boost metric,but since these are not often used in practice we elect notto do so here. The standard expression, R ij = R ij − D [ i C kk ] j + 2 C lj [ i C kk ] l = R ij − D [ i C kk ] j − C lj [ i C kk ] l , (33)similarly relates the two curvatures. Note that in thethree spatial dimensions of the spatial slice the Weyl ten-sor is identically zero, so we need only consider the Riccitensors rather than the full spatial Riemann tensors. Pro-jecting upper case covariant derivatives into the lowercase foliation immediately reveals the geometric mean-ing of the boost covariant derivative. Let S a and S ab denote upper case spatial tensors, related in the stan-dard way (10) to their projections s a and s ab into thelower case foliation. Then we have, ⊥ (N) D i S = D i S + W v i ( L N S ) , (34)for a scalar S , and, ⊥ (N) D i S j = D i s j + W v i ( L N s j ) − X kij s k , ⊥ (N) D i S jk = D i s jk + W v i ( L N s jk ) − X lij s lk − X lik s jl , (35)for the tensors, with, X kij = W ( g − ) kl h ( ⊥ (N) K ) ij v l − ⊥ (N) K ) l ( i v j ) i = W ( g − ) kl (cid:2) K ij v l − K l ( i v j ) + v i v j A l (cid:3) . (36)The general pattern can be read off from these equa-tions. The boost covariant derivative is the part of theprojected upper case derivative formed from lower casespatial derivatives and the boost vector. The remainderdepends on the Lie derivative along N a and the boostextrinsic curvature K ij . We can relate the projected up-per case spatial Ricci tensor and the boost curvature bystraightforward, albeit tedious, direct computation, R ij = ⊥ (N) R ij + D k X kij − D i X kjk − X kkl X lij + X kil X ljk − W v i ⊥ ( (N) D j (N) K + A j (N) K ) − W − v k ⊥ (cid:2) (N) D k (N) K ij − (N) D ( i (N) K j ) k (cid:3) − W − v k ⊥ (cid:2) A k (N) K ij − A ( i (N) K j ) k (cid:3) . (37)This result holds regardless of the gravitational fieldequations, but possible further manipulation is possibleby the addition of the hypersurface constraints. Theprojected upper case covariant derivatives here can bereplaced using (35), which results in a slightly longer ex-pression in terms of K ij . Dual foliation formulation of the wave-equation:
Asa simple example we consider a 3 + 1 decomposition ofthe wave equation (cid:3) φ = 0 using the dual foliation. Forthe Lie-derivative of the boost metric we define, P ij ≡ W L ( W − v ) g ij , (38)again with the convention that P ≡ ( g − ) ij P ij . Intro-ducing the reduction variable L N φ = W π . We then ob-tain, ∂ t φ = α π + L ( β − αv ) φ ,∂ t π = α ( g − ) ij (cid:2) D i D j φ − X kij D k φ + A i D j φ (cid:3) + α (cid:2) K + K vv + P + L v log( W α ) (cid:3) π + L ( β + αv ) π . (39) These equations serve as a prototype when looking at themore complicated systems that follow. In particular theLie derivative terms for π differs from what one mightnaively expect. Gravitational field equations:
We denote A ( i v j ) ≡ A ⊗ v ij . Moving now to write the field equations in terms ofthe boost metric we obtain, ∂ t g ij = − α K ij + 2 α A ( i v j ) + L ( β − αv ) g ij , (40)and after delicate use of the first hypersurface con-straint (14), ∂ t K ij = α R ij − W − D ( i (cid:2) W − α A j ) (cid:3) + v α A i A j − W α ( g − ) kl (cid:0) D k A l + (2 − v ) A k A l (cid:1) v i v j − α L ( W − v ) (cid:0) W A ⊗ v ij (cid:1) − W − D ( i (cid:2) W α K j ) v (cid:3) − W α ( g − ) kl ( g − ) m ( i v j ) A k K lm − W α A v K v i v j − α A ( i K j ) v + α (cid:0) K + K vv + P + L v log( W α ) (cid:1) K ij + α K P ij − α (cid:0) v K + 2 K vv + P + W − A v (cid:1) A ⊗ v ij − α ( K − A ⊗ v ) ki ( K − A ⊗ v ) jk − K v ( i D j ) α − α ( g − ) kl ( K − A ⊗ v ) k ( i P j ) l + L ( β + αv ) K ij . (41)Notice that we end up here with equations involving prin-cipal derivatives of the lower case shift but not the lowercase lapse. That is, in equation (40) first derivativesof β i appear, but in equation (41) no second derivativesof α appear, and instead we have derivatives of A i . Thisshould be compared with the form of the equation (1)in which second derivatives of α are present. Intuitivelythis happens because we are mixing the use of lower casespatial coordinates with upper case spatial objects. TheHamiltonian constraint can also be expressed in terms ofthese variables, giving, H = R + ( K + K vv ) − K ij K ij + 2 D i (cid:0) v i K (cid:1) − g − ) ij D i (cid:0) K jv + W − v k D [ k W v j ] (cid:1) + W − γ ij γ kl D [ i W v k ] D [ j W v l ] . (42)Likewise the momentum constraint becomes, M i = W ( g − ) jk D j ( W − K ki ) − D i ( K + K vv ) + L v K vi + W ( g − ) jk (cid:2) D j D [ k W v i ] − W v i D j ( W − v l D [ l W v k ] ) (cid:3) − g − ) jk v l D [ l W v j ] D [ k W v i ] + R iv + P K vi , (43)where, up to additions of the hypersurface constraints,we define H = H − M v and M i = M i . For readability inequations (42) and (43) we write D [ i W v j ] ≡ D [ i ( W v j ] ).No complications arise in including the stress-energy ten-sor for nonvacuum spacetimes. We note in passing thatby taking g ij and K ij as evolved variables the evolutionequation for the metric also looks natural, since the pro-jected upper case acceleration appears as it would in atetrad formulation when the timelike leg differs from thenormal vector n a . With this choice of variables, we alsosee that the constraints can be written so that there isno explicit appearance of the projected upper case accel-eration A i . Furthermore, there is no explicit appearanceof L N A i in equation (41). Nevertheless we may be moreinterested in how the projected upper case extrinsic cur-vature evolves. This slightly more compact expression istrivially obtained by combining (41) and (31). The equa-tions are given in both forms in mathematica notebooksthat accompany the paper [22]. One expects to be able toformulate an initial data construction strategy naturallyaround the boost metric. A natural starting point forthis would be to examine conformal flatness of the boostmetric in boosted Schwarzschild. See [23] for work alongthese lines. Observe that the notation in these last equa-tions is made slightly more cumbersome by continuingto insist on raising and lowering indices with the spatialmetric γ ij , but we prefer to do so to avoid confusion withthe surrounding calculations. The Generalized Jang equation:
The Jang equa-tion [24] is a quasilinear partial differential equation ofminimal surface type, originally introduced as a tool forthe proof of the positive energy theorem. Given initialdata ( γ ij , K ij ) for the initial value problem in GeneralRelativity it reads, (cid:18) γ ij − D i T D j T D k T D k T (cid:19) (cid:18) K ij − D i D j T (1 + D k T D k T ) / (cid:19) = 0 , (44)for the unknown scalar field T . This equation was moti-vated by the characterization of slices of the Minkowskispacetime, in which there exists a scalar function T suchthat the second bracket of (44) vanishes, and such thatthe boost metric, g ij = γ ij + D i T D j T , is flat. With the present formalism it is clear that thenatural curved space generalization to (44) should be,( g − ) ij ( K ij − A ( i v j ) ) = K − W − A v = 0 , (45)which, remarkably corresponds to the upper case foli-ation being maximal, because using the inverse boostmetric to trace projected upper case quantities revealsthe full trace of the original upper case tensor. Onefurthermore expects an analogous characterization ofgeneral asymptotically flat initial data sets to that offlat-space, in roughly the following terms: Considerdata ( (N) γ ij , (N) K ij ) extracted from a spacetime ( M, g ),written in coordinates X µ = ( T, X i ), on some Cauchyslice, not necessarily a level set of T . An initial dataset ( γ ij , K ij ) corresponds to the same data if and onlyif there exist vectors ( v i , A i ) satisfying the hypersurfaceconstraint (14), which we may write in the form,( D × W v ) i = ( D ln A × A − W v ) i , (46)and a projected Jacobian transformation ϕ ii with in-verse ( ϕ − ) ii such that, g ij = γ ij + W v i v j , K ij = K ij − W − D ( i W v j ) , (47) satisfy, (N) γ ij = ( ϕ − ) ii ( ϕ − ) j j g ij , (N) K ij = W ( ϕ − ) ii ( ϕ − ) j j ( K ij − A ( i v j ) ) , (48)everywhere on the constant- t slice. The relationship be-tween the projected and true Jacobian transformation isthat, ϕ ii = (N) ⊥ iµ J µi , ( ϕ − ) ii = ⊥ iµ ( J − ) µi + W v i V i . (49)The transformation ϕ ii has the property that it maps N-spacelike contravariant tensor indices in X i coordinatesto x i coordinates and simultaneously projects the objectinto the lower case slice, and vice-versa for covariant n-spacelike indices in coordinates x i . The inverse propertyis easily verified by direct computation. The equivalenttransformation Φ ii in the opposite direction is definedin the obvious way. In the special case that the uppercase coordinates are global inertial on Minkowski space-time this characterization reduces to that stated abovemotivating the Jang equation. We propose that this,rather than the conformal transformation, as is some-times claimed, constitutes the relation that should beused to build the natural equivalence class over phys-ically equivalent solutions to metric-based formulationsof GR. Discussion of the dual foliation initial value problem:
Given a boost metric g ij , projected extrinsic curva-ture K ij , boost vector v i , and acceleration A i satisfy-ing the hypersurface, Hamiltonian and momentum con-straints we have a suitable set of initial data for vacuumGR. None of these quantities are invariant under changesof the upper case time coordinate T , but the form ofthe field equations is nevertheless invariant under thischange. One way to view the resulting additional free-dom is that by breaking the correspondence between thespatial metric and the projection operator onto slices ofthe foliation, we gain the freedom to take the spatial met-ric of other foliations as the evolved variable. The sub-sequent projection into the foliation to obtain the boostmetric is the most convenient way to deal with the vari-able in the 3 + 1 language, and fits nicely with earlierwork such as the Jang equation. It is natural to com-pare this reformulation with the freedom in the Maxwellequations, whose gauge can be altered without changingthe coordinates on spacetime. The electric and magneticfields are invariant under such changes, as they dependonly on the Faraday tensor and the choice of coordinates.We have exactly the same status with the boost freedom;the electric and magnetic parts of the Weyl tensor are de-termined purely by the choice of coordinates, accordingto (28), and thus independent of the choice in the boostfreedom. But the form of the field equations is invariantunder changes to the boost. Working with the upper case spatial tensor basis:
Al-lowing the boost freedom decouples, in the highestderivatives, the lower case lapse from the evolved vari-ables. Therefore it is natural to ask whether such a de-coupling can also be obtained in the lower case shift bykeeping all tensors in the X i coordinate basis. We there-fore now wish to drop the Jang-equation style use of twotime coordinates with everything expressed in the coordi-nate basis x i vectors, and instead use the X i basis tensorcomponents, whilst computing derivatives in the x µ co-ordinates. The time derivative of the projected Jacobianis, ∂ t ϕ ii = L ( β − αv ) ϕ ii − α ( AW ) − ϕ ji ∂ j B i ,∂ t ( ϕ − ) ii = L ( β − αv ) ( ϕ − ) ii + α ( AW ) − ( ϕ − ) ij ∂ i B j . (50)Given the time derivative of an upper case spatial tensorin lower case coordinates we can now use (50) to computethe lower case time derivative in the upper case basis.Take for example S ab symmetric upper case spatial, againwith projection s ij into the lower case foliation. Supposewe have, ∂ t s ij = α X ij + L β s ij , (51)then it follows that, ∂ t S ij = α ( ϕ − ) ii ( ϕ − ) jj X ij + 2 α ( AW ) − S k ( i ∂ j ) B k + L ( β − αv ) S ij . (52)Taking S ab as the upper case spatial metric, and lookingat (40) we immediately see that indeed the lower caseshift does become decoupled in the highest derivatives.It is sufficient to consider only this evolution equationbecause this is the only place where the shift is coupledin the principal part. The relationship between the uppercase Christoffel symbol and that of the boost metric is, ϕ ii ϕ j j ( ϕ − ) kk (N) Γ kij = G kij + X kij + ( ϕ − ) kk (cid:2) ∂ ( i ϕ kj ) − A − W v ( i ϕ lj ) ∂ l B k (cid:3) , (53)with X kij defined as above (36). Note that (53) can berewritten as, D ( i ϕ kj ) + (N) Γ kij ϕ ii ϕ jj = X kij ϕ kk − A − W v ( i ϕ lj ) ∂ l B k , (54)which in the setting with vanishing boost vector can beinterpreted as the statement that the Levi-Civita connec-tion of the spatial metric is the gauge covariant deriva-tive associated with spatial diffeomorphisms. The fullfield equations are given in this mixed basis form in [25],together with a canonical Hamiltonian treatment. III. DOUBLE-NULL FORMULATION
In this section we work in coordinates x µ = ( t, x i ).Throughout latin indices a, b, c, d, e will be abstract as in the previous section, and likewise latin in-dices i, j, k, l, m, p stand for spatial components in x µ asbefore. We perform a 2 + 1 decomposition against r onthe spatial slice, and take x µ = ( t, r, θ A ) to be adaptedcoordinates. Thus upper case latin indices A, B, C, D stand for those in the level-sets of r . A. Decomposition
Motivation:
We now turn our attention towards find-ing coordinates suitable for studying the collapse of grav-itational waves. It is known that sufficiently small per-turbations of the Minkowski spacetime are long-lived inthe pure harmonic gauge [26, 27], but this class of initialdata presumably does not include every possible data setthat eventually asymptote to the Minkowski spacetime;for sufficiently strong data, or indeed sufficiently strongpure gauge perturbations, coordinate singularities are ex-pected to form. Indeed there are examples of this phe-nomenon [28], and it may be that some of the difficultiesin evolving strong Brill waves in [29] were caused by theuse of the pure harmonic slicing. Therefore we look else-where. Empirically the generalized harmonic gauges [30]have been found very robust in both binary blackholeand collapse scenarios. However, from the mathematicalpoint of view, the strongest results concerning collapse toa blackhole employ a double-null foliation [5]. It is knownthat a particular type of initial data will form an appar-ent horizon before any coordinate singularity forms. Itis not clear that close to the critical threshold of black-hole formation these coordinates are well-behaved. It isalso rather doubtful that the double-null foliation will beuseful in the strong-field region in binary-blackhole space-times. But since these coordinates naturally conform tothe causal structure of the spacetime, and there is like-wise no guarantee of nice behavior for any other coordi-nate system, they seem to be in the best shape for con-sideration. In particular, in both 3 + 1-spherical symme-try [31] and a 2 + 1 dimensional setting [32] such coordi-nates have been effectively used in studies of critical col-lapse, neatly sidestepping the need for mesh-refinement.We thus look at related gauges suitable for the initialvalue problem.
The decomposition:
In a double-null foliationthere are two crucial coordinates, optical functions,whose level sets are incoming and outgoing null surfaces.In the 3 + 1 setting we have however only singled outthe time-coordinate for special treatment. A given pairof these null hypersurfaces intersect in a spacelike two-sphere. Given a spatial slice of constant t , complete withspatial metric γ ij and extrinsic curvature K ij let us de-fine a new coordinate r , which we will use to performa 2 + 1 split. The idea is that the level sets of r shouldbecome the spheres on which the null surfaces intersect.Obviously the calculations that follow are essentially thesame as those of the standard 3 + 1 split, as described indetail in [1]. The coordinate r defines a unit normal s i to a surface of constant r according to, L − = γ ij ( D i r )( D j r ) , s i = γ ij LD j r . (55)We will call L the length scalar. The normal vec-tor s i naturally defines the induced metric in the two-dimensional level set, q ij = γ ij − s i s j . (56)Likewise we have the extrinsic curvature, χ ij = q ki D k s j , (57)so the first derivatives of the spatial normal vector are intotal, D i s j = χ ij − s i D/ j ln L . (58)We denote the covariant derivative compatible with theinduced metric q ij by D/ i , and likewise use ∂/ i for the par-tial derivative projected into the surface. Note the rel-ative change in sign in the definition of this extrinsiccurvature and that of the slice K ij . Let the vector r i betangent to lines of constant θ A , the two spatial coordi-nates in the level set. We have r i = Ls i + b i , (59)with b i s i = 0. We call the two-dimensional vector b i the slip vector. Note the relation r i D i r = 1. With thisnotation we can express the spatial metric as,d l = L d r + q AB (d θ A + b A d r )(d θ B + b B d r ) . (60)When performing the 3 + 1 decomposition one finds thatthe lapse scalar and shift vector are freely specifiable,which is of course not the case with the analogous lengthscalar and slip vector. The four-dimensional metric is,d s = − α d t + L (d r + L − β s d t ) + q AB (d θ A + b A d r + β A d t )(d θ B + b B d r + β B d t ) . (61)The coordinate light speeds in the increasing and decreas-ing r directions are, c r ± = ( − β s ± α ) L − , (62)whilst in the transverse directions we have, c A ± = − β A ∓ b A α L − . (63)Although they may not necessarily be associated withspherical-polar coordinates we will call these transversedirections ‘angular’. An obvious choice is α = L ,and β s = 0, under which the coordinate light-speedsare c r ± = ±
1. With this choice the combinations u = t − r and v = t + r are the optical outgoing and incoming nullcoordinates alluded to earlier, and the coordinates arenaturally adapted to the causal structure of the space-time in the null n a ± s a directions. The extrinsic curvatures:
We immediately split theextrinsic curvature χ AB into a trace and tracefree part, χ AB = ˆ χ AB + q AB χ . (64)We will similarly use the notation q for the determinantof the two-metric. Finally we decompose the extrinsiccurvature of the spacelike surface as embedded in thespacetime by K ij = s i s j K ss + q ij K qq + 2 s ( i q Aj ) K A + ˆ K AB , (65)where in accordance with the previous notation ˆ K AB stands for the projected, tracefree part. We use in-dices s to denote contraction with s a , and qq for atrace taken with the two-dimensional metric q AB . Wewrite K AB = q iA q jB K ij for the projected extrinsic cur-vature, and occasionally still use K = K ss + K qq as ashorthand for the trace of the extrinsic curvature. The Christoffel symbol:
The spatial Christoffel sym-bol is readily decomposed as,Γ sss = L s (ln L ) , Γ ssi q iA = ∂/ A (ln L ) , Γ sij q iA q jB = − χ AB , Γ kij q iA q jB q kC = Γ / CAB , Γ kss q Ak = − ∂/ A (ln L ) + L (cid:0) L r b A − b B ∂/ B b A (cid:1) , Γ kis q iB q kA = χ AB + L q AC ∂/ B b C , (66)with Γ / denoting the Christoffel symbols of the two-metric q AB . The spatial contracted Christoffel sym-bols Γ i = γ jk Γ ijk are therefore,Γ s = L s (ln L ) − χ , Γ i q iA = Γ / A + L (cid:0) L r b A − b B ∂/ B b A (cid:1) − D/ A (ln L ) . (67) Curvature:
As in (65) the spatial Ricci curvature R ij can be decomposed according to, R ss = − L D/ A D/ A L − L s χ − χ AB χ BA ,R qq = − L D/ A D/ A L − L s χ + R/ − χ ,R sA = D/ B χ BA − D/ A χ , ˆ R AB = −L s ˆ χ AB − L D/ A D/ T FB L + 2 ˆ χ C A ˆ χ BC , (68)by the classical Gauss-Codazzi-Mainardi equations. Vacuum field equations:
The 2 + 1 + 1 decomposedEinstein equations are given first by the constraints, H = R/ − L D/ A D/ A L − L s χ − ˆ χ AB ˆ χ AB − χ − K qq − K qq K ss + 2 K A K A + ˆ K AB ˆ K AB ,M s = −L s K qq + D/ A K A + 2 K A D/ A (ln L ) + χK ss − χK qq + ˆ χ AB ˆ K BA ,M A = L s K A + D/ B ˆ K AB − D/ A K qq − D/ A K ss + χK A − K ss D/ A (ln L ) + K qq D/ A (ln L ) + ˆ K AB D/ B (ln L ) . (69)Since null geodesic expansions in the n a ± s a directions aregiven by χ ± K qq , we can view the H and M s constraintsas dictating how the expansions vary over the slice. Nextwe have evolution equations for the metric, ∂ t (ln L ) = − αK ss + β A D/ A (ln L ) + D s β s ,∂ t b A = − αLK A + L D/ A ( L − β s ) + L r β A + L / β b A ,∂ t q AB = − αK AB + 2 β s χ AB + L / β q AB , (70)and for the decomposed extrinsic curvature, ∂ t K ss = −L s L s α + α [ R ss + 2 K A K A + K qq K ss + K ss ] − D/ A (ln L ) D/ A α − LK ss D s ( L − β s ) − LK A D/ A ( L − β s ) + β s D s K ss + β A D/ A K ss ,∂ t K qq = − D/ A D/ A α + α [ R qq + K qq + 2 K A K A + K qq K ss ] − χ L s α + 2 LK A D/ A ( L − β s ) + β s D s K qq + β A D/ A K qq ,∂ t K A = − D/ A L s α + α [ R sA + K qq K A ] + χ BA D/ B α − LK A D s ( L − β s ) − LK BA D/ B ( L − β s )+ L / β K A + β s L s K A ,∂ t ˆ K AB = − D/ A D/ T FB α − ˆ χ AB L s α + 2 LK ( A D/ T FB ) ( L − β s )+ α [ ˆ R AB − K CA ˆ K BC + ( K A K B ) T F + K ss ˆ K AB ]+ β s L s ˆ K AB + L / β ˆ K AB . (71)Here we have defined the Lie-derivative in the level-setof r , in the obvious way, and where it is understood thatthe vector argument must be projected with q AB , whichallows us to write, for example, L / β ˆ K AB = q kA q lB (cid:0) q ji β i D/ j ˆ K kl + 2 ˆ K j ( k D/ j ) ( q ji β i ) (cid:1) . (72)Finally the equation of motion for the extrinsic curva-ture χ ij can be computed from the relation 2 χ ij = L s q ij .The result is, ∂ t χ AB = −L s ( αK AB ) + α L / K q AB + 2 K ( A D/ B ) α + αK ss χ AB + 2 αK ( A D/ B ) (ln L ) + ( L D/ A D/ B L ) β s − D/ A D/ B β s + β s L s χ AB + L / β χ AB . (73)If we want to treat r as a radial coordinate, and theremaining θ A as angular coordinates we obtain regularityconditions at r = 0. These conditions will be discussedelsewhere. B. Double-null formulation
We now look at the field equations imposing thedouble-null gauge explicitly. The aim here is first, topresent the simplified form of the field equations in thisgauge, and second, to examine whether or not hyperbol-icity of the full system can be obtained.
Pure gauge analysis:
Let us examine the behaviorof infinitesimal perturbations to coordinates satisfyingthe optical conditions α = L and β s = 0 above, ad-ditionally taking β A = b A . This sign is chosen as-suming that the gravitational wave is traveling mostlyin the minus r direction consistent with earlier work.This is Christodolou’s gauge choice in [5], rewritten in atime-space rather than a double-null form. The expres-sions [10] for the time development of the perturbationsto the time and space coordinates are, ∂ t θ = U − ψ i D i α + β i ∂ i θ ,∂ t ψ i = V i + αD i θ − θD i α + L β ψ i . (74)Then we have, U ≡ ∆[ α ] = ∆[ L ] = ∆[( γ ij D i rD j r ) − / ]= L L s ψ s − θLK ss + ψ A D/ A L ,V s ≡ s i ∆[ β i ] ,V A ≡ ∆[ β A ] = ∆[ b A ]= − θLK A + L D/ A ( L − ψ s ) + L r ψ A + L / ψ b A , (75)and obtain the pure gauge subsystem, ∂ t [ L − θ ] = LD s [ L − ψ s ] + b A D/ A [ L − θ ] ,∂ t [ L − ψ s ] = LD s [ L − θ ] + b A D/ A [ L − ψ s ] ,∂ t ( q · ψ ) A = ∂ r ( q · ψ ) A + L D/ A [ L − ( θ + ψ s )] − LK A ( θ + ψ s ) . (76)This first order PDE system is only weakly hyperbolic.The arguments presented in [10], building on thoseof [9, 33], can be used to show that no strongly hyper-bolic formulation can be built with this gauge condition,at least if the formulation is constructed under the stan-dard free-evolution approach. Therefore the double-nullgauge can not be directly used in numerical relativity inthe standard way, but requires a more subtle approach,or some modification. Note that the problem here comesfrom the choice β A = b A , and there are simple modifica-tions under which strong hyperbolicity of the pure gaugesubsystem can be obtained. Nevertheless, building a for-mulation of GR which is at least strongly hyperbolic withone of these good , modified, conditions will be more in-volved because here the s i direction is singled out forspecial treatment. Therefore in the following sections weinstead look for a simpler approach employing the dualfoliation formalism. Fixing the gauge:
Despite the shortcomings un-earthed by the pure gauge analysis, for completeness wepresent the full field equations with in the double-nullform. As above we choose α = L , β s = 0 and β A = b A .This choice has no effect on the constraints (69), but theevolution equations become, ∂ t (ln L ) = − LK ss + b A D/ A (ln L ) ,∂ t b A = − L K A + L r b A ,∂ t q AB = − αK AB + L / b q AB , (77)0for the metric components, and ∂ t K ss = −L s L s L + L [ R ss + 2 K A K A + K qq K ss + K ss ] − D/ A (ln L ) D/ A L + b A D/ A K ss ,∂ t K qq = − D/ A D/ A L + L [ R qq + K qq + 2 K A K A + K qq K ss ] − χ L s L + b A D/ A K qq ,∂ t K A = − D/ A L s L + L [ R sA + K qq K A ] + χ BA D/ B L + L / β K A ,∂ t ˆ K AB = − D/ A D/ T FB L − ˆ χ AB L s L + L [ ˆ R AB − K CA ˆ K BC + ( K A K B ) T F + K ss ˆ K AB ] + L / b ˆ K AB , (78)for the extrinsic curvature. Finally we have, ∂ t χ AB = −L s ( LK AB ) + L L / K q AB + 2 K ( A D/ B ) L + LK ss χ AB + 2 LK ( A D/ B ) (ln L ) + L / b χ AB . (79)Taking linear combinations of these variables one canrewrite so that all of the equations take the form of‘transport equations’ in the n a ± s a directions, but withtransverse derivatives appearing as sources. Particularlyrelevant are the combinations χ ± K qq , as they revealthe Raychaudhuri equations. Up to this trivial change ofvariables and the split into time-space derivatives, ratherthan the double-null choice, this is the same system pre-sented in Ch. 3 of [4], where it was also noted that thissystem is not hyperbolic. IV. COORDINATE SWITCHED FIRST ORDERGENERALIZED HARMONIC GAUGEA. Double-null Jacobians
Time and Radial coordinates:
Let us now abandonthe idea of evolving in double-null coordinates directly,and instead examine how the spacetime could be con-structed in these coordinates a posteriori, having con-structed the spacetime locally in the harmonic gauge, forexample. This is similar to the strategy employed in [5].It has also been used in numerically, in for example [34].Let us work in lower case coordinates x µ . We would likethe upper case coordinates to satisfy the double-null con-ditions. As elsewhere, N a = − A ∇ a T , S a = L ∇ a R . (80)with X µ = ( T, R, Θ A ). We choose A = L to impose thedouble-null gauge, regardless of the angular coordinates.Under this condition we define ingoing and outgoing nullvectors L a , K a , − N a + S a = L a = L ˆ L a , − N a − S a = K a = L ˆ K a . (81)The renormalized vectors ˆ L a and ˆ K a generate nullgeodesics in the L a and K a directions. It is natural to define two lower case spatial vectors v a ± and s a ± from theJacobian according to, v ± i = E ± s ± i = φ Ri ∓ L − W v i . (82)The vectors s a ± have unit magnitude. Indices s ± standfor contraction with these vectors. The scalars E ± arethe energy of the ingoing and outgoing congruences asmeasured by the Eulerian observers n a . In terms of theJacobian they are, E ± = L − W ∓ π R . (83)The null geodesic vectors are then,ˆ K a = − E − (cid:0) n a + s a − (cid:1) , ˆ L a = − E + (cid:0) n a − s a + (cid:1) . (84)Straightforward computation then reveals evolutionequations, ∂ t ln E ± = L ( β ± αs ± ) ln E ± + α (cid:0) K s ± s ± ± L s ± ln α (cid:1) ,∂ t v ± i = ± α D ( s ± ) v ± i ± E ± D i α + L β v ± i . (85)The hypersurface constraints were used freely to arrive atthis result. Interestingly the term appearing as a sourcein the first equation is essentially a characteristic variableof the (first order in time, second order in space) GHGformulation. Notice furthermore that, after adjusting thenormalization of v i ± , these can be compared with the re-sults of [35], and are of course compatible. Note alsothat the equations for E ± follow from γ ij v ± i v ± j = E ± .The equations (85) form a symmetric hyperbolic systemin ( E ± , v ± i ) which is equivalent to a system in the scalarcomponents L − W and π R and the vector parts φ Ri and L − W v i of the Jacobian. Angular coordinates:
To complete the equations ofmotion for the Jacobian we require a choice for φ Ai and π A . We have already seen that from the pure gaugepoint of view the choice ‘ β A = b A ’ is problematic. Tounderstand how this issue appears in the Jacobian for-mulation, let us assume that the component π A takes theform, π A = − m i φ Ai , (86)for some known lower case spatial vector m i . This cor-responds to determining the angular coordinates by Lie-dragging so that ( n a + m a ) ∇ a Θ A = 0. This family in-cludes ‘ β A = ± b A ’ by choosing m a = ± s a ± . Plugging therelation (86) into the Jacobian equation of motion (16)and using the hypersurface constraints gives the compactexpression, ∂ t φ Ai = L ( β − α m ) φ Ai . (87)If the vector m a is given a priori this equation is hy-perbolic. On the other hand if we wish to make m a dependent on the other components of the Jacobian,the derivative of m a in the Lie-derivative is problem-atic, as it is a one-way coupling, in the sense that the1equations of motion for the angular coordinates explic-itly depend upon the ( T, R ) coordinates in the princi-pal part, but not vice-versa. This type of coupling isdangerous because it can leave non-trivial Jordan blocksin the principal symbol if the speeds associated withthe
T, R and Θ A blocks clash. Fortunately this dis-cussion suggests two alternatives. The first is to con-struct the angular coordinates using just vectors associ-ated with lower case coordinates. For example we couldchoose n a ∇ a Θ A = 0, or even t a ∇ a Θ A = 0, in which casethe angular coordinates could correspond to those builtdirectly from the lower case coordinates. One possibil-ity would be to use a reference metric to build a firstorder GHG formulation directly in spherical-polar coor-dinates. Then the upper case angular coordinates couldbe exactly those of the lower case system. The secondoption is to make sure that the speeds of the two sub-systems do not coincide, or that if they do the princi-pal symbol is nevertheless diagonalizable. For this wemight try N a ∇ a Θ A = W ( n a + v a ) ∇ a Θ A = 0, or in otherwords m a = v a , which is also equivalent to B A = 0 whenworking in upper case coordinates. Since, v i = ( E + + E − ) − (cid:0) v i − − v i + (cid:1) , (88)we arrive at, ∂ t φ Ai = − α v j D j φ Ai + α LW − φ Aj D i ( v j + − v j − ) − α LW − π A D i ( E + + E − ) + π A D i α + L β φ Ai , (89)after expanding the Lie-derivative. Hyperbolicity of the Double-Null Jacobians:
Thedouble-null Jacobian system with m i given a priori istrivially symmetric hyperbolic. Choosing instead m i = v i as in (89), the principal symbol for the subsystem withthe components ( E ± , v ± s , v ± A , φ As ) is, P s = α A s + β s , (90)with, A s = ± s s ± ± s s ± ± s s ± − L W π A ± L W φ As ± L W φ AB q AB − v s , (91)with an index ‘ s ’ denoting contraction with an arbitrarylower case unit spatial vector s i and q ab = γ ab − s a s b as elsewhere. The remaining block of the full principalsymbol, associated with φ AA , is decoupled, and has noaffect on the following discussion. The eigenvalues of theprincipal symbol are β s ± αs s ± and β s − αv s ± . For stronghyperbolicity we need that the symbol is diagonalizablefor every s i . Choose for example s i such that,( v i − s i − ) s i = 0 , (92) then the ‘ s s − ’ and ‘ v s ’ eigenvalues coincide, and the prin-cipal symbol is missing eigenvectors. Therefore the sys-tem is again only weakly hyperbolic. This type of degen-eracy occurs if m i is taken to be any vector constructedfrom v i ± , thus we must abandon the second alternativesuggested by the above discussion. This leaves the optionto fix m i a priori, or to choose a gauge of a different formfor the angular coordinates, such as the generalized har-monic option (17). At least for the choice ‘ β A = b A ’ thisresult was expected, because the pure gauge subsystemwe examined before is closely related to the system forthe Jacobians. B. First order generalized harmonic gauge
There is much work about formulations of general rel-ativity in first order form. For a review, see [36]. Ofparticular interest in recent years has been the first or-der reduction of the GHG formulation [12, 37]. To alarge extent this interest was driven by use inside theSpEC numerical relativity code. Here we summarize justthe relevant features. The first order GHG system is aquasilinear, first order symmetric hyperbolic system ofthe form, ∂ t u = A p ∂ p u + S . (93)The formulation has a set of constraints compatible withthe evolution equations. The key feature of the systemis that the principal part takes the form of a first or-der reduction of the wave equation. This is attained bycarefully coupling the coordinate choice (cid:3) x α = H α tothe field equations [13] and then reducing to first order.Gravitational radiation controlling, constraint preservingboundary conditions for the system have been studiedand implemented [29, 38, 39]. The evolution system isnow used frequently in the evolution of compact binaryspacetimes. Of crucial importance in the present workis that in the first order reduction, all first derivativesof the metric can be expressed in terms of evolved vari-ables without taking any derivatives. So, for example ifwe were to evolve the double-null Jacobians (85) along-side the GHG variables, we can formulate the system sothat no coupling occurs through derivatives. In otherwords the Jacobians are minimally coupled to the GHGformulation. C. Coordinate switch and hyperbolicity
Coordinate switch:
Suppose, without loss of general-ity, that we are given a first order quasilinear system ofthe form, ∂ T u = ( A A p + B p ) ∂ p u + A S . (94)Now consider the effect of a change of independent co-ordinates on the whole system, so that rather than us-ing the upper case X µ coordinates we employ the lower2case x µ ones. In the new coordinates the system of courseretains the same functional form, but now with, (cid:0) + A V (cid:1) ∂ t u = α W − (cid:16) A p ( ϕ − ) pp − (cid:0) + A V (cid:1) Π p (cid:17) ∂ p u + α W − S . (95)Recall here the various components of the inverse Jaco-bian defined by (15) and (49), and the shorthand A V = A i V i . It is easily confirmed that any closed constraintsubsystem remains closed. Symmetric hyperbolicity:
The original system is as-sumed to be symmetric hyperbolic, that is, there existsa symmetric positive definite symmetrizer H such that H ( A A p + B p ) , is symmetric for each p . In the present context, thissystem can be taken as a first order generalized har-monic formulation coupled, minimally, to the double-nullJacobians (85). Under a smallness assumption on V i symmetric hyperbolicity is unaffected by the change ofcoordinates. Taking the system in the form (95), thesymmetrizer is unaffected by the change of coordinates.If we insist on multiplying on the left by the inverseof (cid:0) + A V (cid:1) , then we pick up exactly a factor of thismatrix on the right of the modified symmetrizer. Butto obtain energy estimates for metric components usingnorms formed from the lower case coordinate basis com-ponents of tensors more effort may be necessary. Strong hyperbolicity:
The principal symbol of the sys-tem in X µ is, P SX = A A S + B S , (96)where on the right hand side superscript S denotes con-traction with an arbitrary unit upper case spatial covec-tor S i . Multiplying (95) by the inverse of (cid:0) + A V (cid:1) , theprincipal symbol in x µ can be read off, P sx = α W − (cid:16)(cid:0) + A V (cid:1) − A s − Π s (cid:17) . (97)On the right hand side superscript s denotes contractionwith an arbitrary unit lower case spatial covector s i , andunderlined superscript s denotes the same contraction,but pushed through the transformation ( ϕ − ) ii . Stronghyperbolicity is the requirement that for each s i there ex-ists a symmetrizer H s , uniformly symmetric positive def-inite in s i , such that the product of the symmetrizer withthe principal symbol is symmetric. For quasilinear prob-lems strong hyperbolicity, together with some smooth-ness conditions on the symmetrizer are sufficient to guar-antee local well-posedness of the initial value problem.These smoothness conditions are sometimes included inthe definition of strong hyperbolicity. The existence ofa symmetrizer for fixed s i is equivalent to the require-ment that the principal symbol has a complete set ofeigenvectors with real eigenvalues. Assuming that thesystem is strongly hyperbolic in the upper case coordi-nates X µ this necessary condition can be seen to hold in lower case coordinates x µ as follows. This discussion isadapted from [40]. Fix s i . The characteristic polynomialof P SX has real eigenvalues, and thus we have a hyper-bolic polynomial with respect to N µ . It follows that ifthe boost velocity is sufficiently small then P sx also hasreal eigenvalues [41]. Estimates of the range over whichthis condition is satisfied can be given. In the presenceof a simple block structure of A s , the eigenvalues willsimply be multiplied by those of ( + A V ). Take one ofthese eigenvalues λ . Strong hyperbolicity in X µ impliesthe existence of symmetric positive definite H with, H L ≡ H (cid:16) A s − ( + A V ) ( λ + Π s ) (cid:17) , (98)symmetric. Suppose that u is a non-vanishing vectorin the nullspace of L , and thus either an eigenvector orgeneralized eigenvector of P sx . Suppose that it is a gen-eralized eigenvector, so that u = Lv for some v . Then, u T Hu = u T H Lv = v T H Lu = 0 . (99)The second equality holds by symmetry of H L and thethird because Lu = 0. Since H is symmetric positivedefinite we then have that u = 0. Therefore if u is in thenullspace of L it is a true eigenvector of P sx . Thus P sx has a complete set of eigenvectors. Details concerningthe remaining uniformity condition in s i can be foundin [40]. Discussion:
The advantage of the dual foliation isnow clear. In section III B we were unable to find ahyperbolic formulation using the double-null coordinatesdirectly. Using the dual foliation however the construc-tion of such a formulation is essentially trivial. Sincethe whole system is symmetric hyperbolic we do notlose regularity when mapping between the two coordi-nate systems. Regularity will however need to be lookedat carefully if we are to use the double-null radial coor-dinate all the way to the origin, but this issue is onlythat of using spherical polar coordinates, and not relatedto the dual foliation strategy. With a little more book-keeping we can instead work from a standard first orderin time, second order in space [42, 43] formulation of GRand avoid the first order reduction, but postpone anypresentation thereof. One expects that this formulationcould be treated according to [44] to give an economicaldemonstration of a ‘neighborhood theorem’ for the GRcharacteristic initial value problem.
Physically what hashappened is that the coordinate and gauge degrees offreedom GR were decoupled as much as possible. Notethat in earlier work a symmetric hyperbolic formulationusing a double-null gauge was constructed using framevariables coupled to the Bianchi equations [45]. In con-trast here the aim was to arrive at such a formulationwith as few changes as possible to standard formulationsused in numerical work. From this practical point of viewthere is always the danger that the matrix (cid:0) + A V (cid:1) be-comes singular. We can mitigate against this by simplyevolving for a short but finite coordinate time, and thenresetting the Jacobian to the identity by transforming3all the fields into the upper case tensor basis. For suffi-ciently regular data hyperbolicity guarantees short timeexistence so that this procedure can be performed iter-atively. Coupled with strong theorems on continuationof solutions, this strategy could perhaps even be usedto guarantee that numerical calculations progress suc-cessfully into extreme regions of spacetime, although thedouble-null gauge might have to be replaced with someother suitable choice. For the numerical relativist, proba-bly the simplest summary of the double-null Jacobian re-sult is that it is the generalization of the dual coordinateframes approach [46] employed in the SpEC code [16] tothe situation in which two differing time coordinates areconsidered, and where the Jacobians satisfy dynamicalequations rather than arising as derivatives of algebraicrelationships between the coordinates. V. CONCLUSION
Making a dual foliation approach, we have shown thatit is possible to effectively decouple the choice of coor-dinates from local well-posedness of the field equationsof general relativity. This was done by evolving a firstorder reduction of the generalized harmonic formulationalongside Jacobians mapping from the desired coordinatesystem to the generalized harmonic one. The importantexample of the double-null gauge was considered. Butin fact the set of coordinate choices resulting in equa-tions of motion for Jacobians that are minimally coupledto the remaining field equations is extremely large , andlocal well-posedness inside this class follows from well-posedness of the coordinate choice alone. It is thus ex-pected that with due care, this observation will allow usto evolve the generalized harmonic formulation using say,the Maximal slicing, and quasi-isotropic spatial coordi-nates, for example. We hope that this would allow for direct comparison of the results of [47] in a modern nu-merical relativity code with minimal changes. See [48–52]for recent work on the problem. It is important to real-ize that the role of the generalized harmonic formulationin the construction is completely auxiliary. In fact anywell-behaved formulation of GR is amenable to the sametrick, perhaps after a suitable reduction to first order, ifwe wish to avoid complicated book-keeping. The general-ized harmonic formulation is simply the most convenientexample.For numerical applications it will be necessary totranslate the constraint preserving boundary condi-tions [53] of the harmonic system into the new set ofvariables and coordinates, but no major difficulty isexpected in doing so. The obvious next step is to sys-tematically implement and test the approach, preferablyin a simple context. Afterwards we expect to use thedouble-null coordinates to study the critical collapse ofgravitational waves using the bamps code [17–19].
Acknowledgments
I am grateful to Bernd Br¨ugmann, Sascha Husa, RonnyRichter, Andreas Weyhausen and especially Juan Anto-nio Valiente-Kroon for helpful discussions. Thanks goalso to Tim Dietrich, David Garfinkle and Milton Ruizfor their feedback on the manuscript. Finally I wish tothank the entire UIB relativity group for their warm hos-pitality during a recent visit in which part of the work wascompleted. Many of the calculations presented here wereperformed using mathematica with the package xTensor by Jos´e-Mar´ıa Mart´ın-Garc´ıa [54]. The notebooks areavailable at the website of the author [22]. This workwas supported in part by DFG grant SFB/Transregio 7‘Gravitational Wave Astronomy’. [1] Eric Gourgoulhon. 3+1 formalism and bases of numericalrelativity. 2007.[2] Miguel Alcubierre. Introduction to 3+1 NumericalRelativity. Oxford University Press, Oxford, 2008.[3] Thomas W. Baumgarte and Stuart L. Shapiro.Numerical Relativity: Solving Einstein’s Equations onthe Computer. 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