aa r X i v : . [ g r- q c ] N ov Is the Bianchi identity always hyperbolic?
István Rácz ∗ Wigner RCPH-1121 Budapest, Konkoly Thege Miklós út 29-33. HungaryAugust 31, 2018
Abstract
We consider n + 1 dimensional smooth Riemannian and Lorentzian spacessatisfying Einstein’s equations. The base manifold is assumed to be smoothlyfoliated by a one-parameter family of hypersurfaces. In both cases—likewise itis usually done in the Lorentzian case—Einstein’s equations may be split into‘Hamiltonian’ and ‘momentum’ constraints and a ‘reduced’ set of field equa-tions. It is shown that regardless whether the primary space is Riemannian orLorentzian whenever the foliating hypersurfaces are Riemannian the ‘Hamilto-nian’ and ‘momentum’ type expressions are subject to a subsidiary first ordersymmetric hyperbolic system. Since this subsidiary system is linear and homo-geneous in the ‘Hamiltonian’ and ‘momentum’ type expressions the hyperbolic-ity of the system implies that in both cases the solutions to the ‘reduced’ set offield equations are also solutions to the full set of equations provided that theconstraints hold on one of the hypersurfaces foliating the base manifold. Consider a pair ( M, g ab ) , where M is an ( n +1) -dimensional ( n ≥ ) smooth, paracom-pact, connected, orientable manifold endowed with a smooth metric g ab with signaturewhich is either Euclidean or Lorentzian. Throughout this paper the geometry will be at the focus of our main concern. Inrestricting the geometry we shall assume that Einstein’s equations G ab − G ab = 0 , (1.1) ∗ email: [email protected] All of our other conventions will be as in [7]. G ab is assumed to have vanishing diver-gence. Note that whenever we have matter fields satisfying their field equations withenergy-momentum tensor T ab and with cosmological constant Λ the source term G ab = 8 π T ab − Λ g ab (1.2)suits to the above requirements.Concerning the topology of M we shall assume that the manifold M is foliatedby a one-parameter family of hypersurfaces, i.e. M ≃ R × Σ , for some codimensionone manifold Σ . In other words, then M possesses the structure of a trivial principalfiber bundle with structure group R .Note that this assumption is known to hold [3] for globally hyperbolic spacetimesbut we would like to emphasize that as the signature of the metric may not beLorentzian or even if it was, in deriving our key results, we need not to assume globalhyperbolicity of the pertinent spacetime. Our assumptions on topology of M areknown to be equivalent to the existence of a smooth function σ : M → R with non-vanishing gradient ∇ a σ such that the σ = const level surfaces Σ σ = { σ } × Σ comprisethe one-parameter foliation of M .Having the above generic setup it is natural to perform a n decomposition. Indoing so first a conventional n splitting of (1.1) will be done by generalizing con-ventional arguments (see, e.g. Section 2.4 of [2]). This n splitting can be performedon equal footing in both the Lorentzian and Riemannian cases yielding ‘Hamiltonian’and ‘momentum’ type expressions, along with a reduced set of equations referredas ‘evolutionary system’. By using the evolutionary system, a subsidiary system forthe constraint expressions is derived. A remarkable and unexpected property of thissubsidiary system is that regardless whether the metric of the imbedding manifoldis of Lorentzian or Euclidean signature—whenever the metric on the σ = const levelsurfaces is Riemannian—it comprises a first order symmetric hyperbolic system thatis linear and homogeneous in the constraint expressions. This guaranties then thatthe constraint expressions vanish identically throughout domains where solutions tothe evolutionary system exist provided that they vanish on one of the level surfaces.These results are presented in Sections 2. Some useful relations are given in Section3 and in the Appendix.Having been a n type decomposition performed it is natural to ask whetheranalogous type of simplifications of the reduced equations in a succeeding n − splitting could also exist. The answer of this question requires—besides some obviousadditional restrictions on the topology of the base manifold—the identification ofthose conditions which guarantee that the covariant divergence of the new sourceterm ( n ) G ab vanishes. The corresponding analysis is carried out in Section 4. The mainconclusion here is that even though a formal n − splitting could be performed,2n general, there is no room to acquire additional new simplifications. What can bedone is nothing more than a redistribution of the simplifications associated with theprimary splitting of the original field equations.The paper is closed in Section 5 by remarks on some of the implications of thederived new results. n decomposition This section is to show that a reduced set of the equations can be deduced from (1.1)such that, regardless whether the metric of the imbedding manifold is of Lorentzian orEuclidean signature, the solutions to this reduced system are also solutions to the fullset (1.1) provided that the ‘constraints’ hold on one of the σ = const level surfaces.We are proceeding by separating the ‘evolution’ and ‘constraint’ equations byadopting the strategy of the conventional decomposition applied in spacetimeswith Lorentzian metric (see, e.g. [2]). In doing so denote by n a the ‘unit norm’vector field that is normal to the σ = const level surfaces. To allow the simultaneousinvestigation of both spaces with either Euclidean or Lorentzian signature and timelikeor spacelike level surfaces the sign of the norm of n a will not be fixed, i.e. it will beassumed that n a n a = ǫ , (2.1)where ǫ takes the value − or +1 .The induced metric h ab and the pertinent projection operator h ab on the levelsurfaces of σ : M → R are then given as h ab = g ab − ǫ n a n b , a nd h ab = g ab − ǫ n a n b , (2.2)respectively.Denote by E ab the left-hand-side of (1.1), i.e. E ab = G ab − G ab , (2.3)and define the ‘Hamiltonian’ E ( H ) and ‘momentum’ E ( M ) b expressions as E ( H ) = n e n f E ef , and E ( M ) b = n e h f b E ef , (2.4)respectively. Then, we have E ab = h ea h f b E ef + ǫ [ n a E ( M ) b + n b E ( M ) a ] + n a n b E ( H ) . (2.5)3hoose now as our ‘evolutionary’ system the combination E ( EVOL ) ab = h ea h f b E ef − κ h ab E ( H ) = 0 , (2.6)where κ is some constant. Then, by combining (2.5) and (2.6), we get E ab = ǫ [ n a E ( M ) b + n b E ( M ) a ] + [(1 − ǫ κ ) n a n b + κ g ab ] E ( H ) . (2.7)Taking now the ∇ a divergence of (2.7) and using our assumption concerning thevanishing of the covariant divergence ∇ a G ab , along with the twice contracted Bianchiidentity we get ǫ ( ∇ a n a ) E ( M ) b + ǫ ( n a ∇ a E ( M ) b ) + ǫ ( E ( M ) a ∇ a n b ) + ǫ n b ( ∇ a E ( M ) a ) (2.8) (1 − ǫ κ ) n [( ∇ a n a ) n b + ( n a ∇ a n b )] E ( H ) + n b ( n a ∇ a E ( H ) ) o + κ ∇ b E ( H ) = 0 . The ‘parallel’ and ‘orthogonal’ parts of (2.8) read then as n e ∇ e E ( H ) + ǫ h ef D e E ( M ) f = (1 − ǫ ) ( n e ∇ e n b ) E ( M ) b − ǫ (1 − ǫ κ ) ( ∇ e n e ) E ( H ) , (2.9) h af n e ∇ e E ( M ) f + ǫ κ h af D f E ( H ) = − h af E ( M ) f ( ∇ e n e ) − E ( M ) e ( ∇ e n f ) h fa (2.10) − ǫ (1 − ǫ κ ) h af ( n e ∇ e n f ) E ( H ) , where the relations ǫ = 1 and ∇ a E ( M ) a = D a E ( M ) a − ǫ ( n a ∇ a n b ) E ( M ) b (2.11)have been used, and D a denotes the unique torsion free covariant derivative operatorassociated with h ab .Although ( M, g ab ) may not have anything to do with time evolution we shall referto a vector field σ a on M as an ‘evolution vector field’ if the relation σ e ∇ e σ = 1 holds. Notice that this condition guaranties that σ a nowhere vanishes nor becomestangent to the σ = const level surfaces. The unit normal n a to these level surfacesmay always be decomposed as n a = 1 N [( ∂ σ ) a − N a ] , (2.12)where N and N a denotes the ‘laps’ and ‘shift’ of the ‘evolution’ vector field σ a = ( ∂ σ ) a defined as N = ǫ ( σ e n e ) and N a = h ae σ e , (2.13)4espectively. Taking these relations into account, equations (2.9) and (2.10)—whenwriting them out explicitly in some local coordinates ( σ, x , . . . , x n ) adopted to thevector field σ a and the foliation { Σ σ } —can be seen to take the form (cid:26)(cid:18) N N h ij (cid:19) ∂ σ + (cid:18) − N N k ǫ h ik ǫ κ h jk − N N k h ij (cid:19) ∂ k (cid:27) E ( H ) E ( M ) i ! = (cid:18) EE j (cid:19) , (2.14)where, in virtue of (2.9) and (2.10), E and E j are linear and homogeneous expressionsof E ( H ) and E ( M ) i . It follows immediately that the coefficient matrices of the partialderivatives are symmetric if κ = 1 and, in addition, the coefficient of ∂ σ is also positivedefinite provided that the induced metric h ij is positive definite.Hereafter we shall assume that κ = 1 and that h ij is positive definite. The latteroccurs if the σ level surfaces are spacelike allowing the signature of metric g ab to beeither Lorentzian, with ǫ = − , or Euclidean, with ǫ = +1 , respectively. In thesecases (2.14) comprises a first order symmetric hyperbolic linear and homogeneoussystem A µ ∂ µ v + B v = 0 (2.15)for the vector valued variable v = ( E ( H ) , E ( M ) i ) T . As these type of equations are guar-anteed to have identically vanishing solution for vanishing initial data the ‘Hamil-tonian’ and ‘momentum’ expressions will be guaranteed to vanish throughout thedomain of existence of solutions to the evolutionary system (2.6), with κ = 1 , pro-vided they vanish on one of the slides of the foliation { Σ σ } .By combining the above observations we have the following: Theorem 2.1
Let ( M, g ab ) as described in Section 1 such that the metric inducedon the σ = const level surfaces is Riemannian. Then, regardless whether g ab is ofLorentzian or Euclidean signature, any solution to E ( EVOL ) ab = 0 , with κ = 1 , is also asolution to the full set (1.1) provided that E ( H ) and E ( M ) a vanish on one of the levelsurfaces. It is a remarkable property of (2.14) that ǫ and κ do not show up in the coefficientof ∂ σ , and once κ = 1 is chosen all the coefficients A µ in (2.15) are guaranteed to besymmetric regardless of the value of ǫ . The spatial indices of the pull backs of geometrical objects to the σ = const slices yielded in theapplied n decomposition will be indicated by lowercase Latin indices from the second half andthey will be assumed to take the values , . . . , n . The explicit forms
In exploring some of the consequences of Theorem 2.1 we shall need the explicit formsof the constraint expressions and the evolutionary system. In spelling out them weshall refer to the extrinsic curvature K ab which is defined as K ab = h ea ∇ e n b = 12 L n h ab , (3.1)where L n stands for the Lie derivative with respect to n a .The ‘Gauss’ and ‘Codazzi’ relations take the form h ea h f b h kc h dj R efkj = ( n ) R abcd − ǫ (cid:8) K ac K db − K bc K da (cid:9) , (3.2) h ea h f b n k h dj R efkj = D b K da − D a K db , (3.3)where ( n ) R abcd stands for the n -dimensional Riemann tensor associated with h ab .The various projections of the full Ricci tensor—which can be derived either bycontractions of the above two relations or that of the third non-trivial projection ofthe full Riemann tensor, n a h f b n c h dj R efkj —read as h ea h f b R ef = ( n ) R ab + ǫ (cid:26) − L n K ab − K ab K ee + 2 K ae K eb − ǫN D a D b N (cid:27) , (3.4) h ea n f R ef = D e K ea − D a K ee , (3.5) n e n f R ef = − n L n ( K ee ) + K ef K ef + ǫN D e D e N o , (3.6)where ( n ) R ab stand for the Ricci tensor associated with h ab .Taking all the above relations into account we have E ( H ) = n e n f E ef = 12 n − ǫ ( n ) R + ( K ee ) − K ef K ef − e o , (3.7) E ( M ) a = h ea n f E ef = D e K ea − D a K ee − ǫ p a , (3.8) E ( EVOL ) ab = ( n ) R ab + ǫ n − L n K ab − ( K ee ) K ab + 2 K ae K eb − ǫN D a D b N o − [ S ab − e h ab ] − h ab (cid:26) (1 − ǫ ) ( n ) R − ǫ L n ( K ee ) + (1 − ǫ ) ( K ee ) − (1 + ǫ ) K ef K ef − N D e D e N (cid:27) , (3.9)where e = n e n f G ef , p a = ǫ h ea n f G ef and S ab = h ea h f b G ef .For certain cases (in particular, whenever ǫ = − ) it is rewarding to do somealgebra by which it can be verified that E ( EVOL ) ab − n − h ab (cid:16) E ( EVOL ) ef h ef (cid:17) = e E ( EVOL ) ab , (3.10)6here e E ( EVOL ) ab = h ea h f b (cid:20) R ab − (cid:18) G ab − n − g ab [ G ef g ef ] (cid:19)(cid:21) + 1 + ǫn − h ab E ( H ) . (3.11)In virtue of the above relations we have Lemma 3.1
The evolutionary system (2.6) holds if and only if either(i) the right hand side of (3.9), or(ii) that of (3.11)vanishes.
The right hand side of (3.11) can also be written as e E ( EVOL ) ab = ( n ) R ab + ǫ n − L n K ab − ( K ee ) K ab + 2 K ae K eb − ǫN D a D b N o (3.12) − (cid:18) S ab − n − h ab [ S ef h ef + ǫ e ] (cid:19) + 1 + ǫ n − h ab n − ǫ ( n ) R + ( K ee ) − K ef K ef − e o . Note that by making use of the contractions e , p a and S ab our source term G ab can be decomposed as G ab = n a n b e + [ n a p b + n b p a ] + S ab , (3.13)while its divergence ∇ a G ab take the form [see also (A.8)] ∇ a G ab = e ( K ee ) n b + ( K ee ) p b + p e K eb + n b ( D e p e ) + D e S eb − ǫ n b ( S ef K ef )+ ˙ n b e + n b L n e + L n p b − p e K eb − ǫ ( ˙ n e p e ) n b − ǫ ( ˙ n e S eb ) , (3.14)where ˙ n a := n e ∇ e n a = − ǫ D a ln N . (3.15)Taking then the ‘parallel’ and ‘orthogonal’ parts of (3.14), ∇ a G ab = 0 (3.16)we get [see also (A.9) and (A.10)] L n e + D e p e + [ e ( K ee ) − ǫ ( ˙ n e p e ) − ǫ K ae S ae ]= 0 , (3.17) L n p b + D a S ab + [ − ǫ S ab ˙ n a + ( K ee ) p b + e ˙ n b ]= 0 . (3.18)7otice that in deriving (3.17) and (3.18) only the vanishing of the divergence ∇ a G ab has been used. We may replace G ab , for instance, by E ab . Accordingly, asimultaneous replacement of e , p a and S ab by E ( H ) , ǫ E ( M ) a and E ( EVOL ) ab + κ h ab E ( H ) ,respectively, yields a system of equations which can be seen to be equivalent to (2.9)and (2.10) whenever E ( EVOL ) ab = 0 . Note also that if the term E ( EVOL ) ab is kept inthese latter equations they can be used to justify the following statement which iscomplementary to that of Theorem 2.1. Lemma 3.2
If the constraint expressions E ( H ) and E ( M ) a vanish on all the σ = const level surfaces then the relations K ab E ( EVOL ) ab = 0 , (3.19) D a E ( EVOL ) ab − ǫ ˙ n a E ( EVOL ) ab = 0 . (3.20) hold for the evolutionary expression E ( EVOL ) ab . Once a n splitting has been done one may be interested in performing a succeeding n − decomposition provided that the σ = const level surfaces are guaranteedto be foliated by a one-parameter family of ( n − -dimensional hypersurfaces in Σ σ .Note, however, that before automatically adopting Theorem 2.1 and the equationslisted in the previous section the validity of all the assumptions made in derivingthem have to be inspected. The key requirement to be checked is the vanishing of thecovariant divergence of G ab . Therefore, once a n decomposition had been done,before performing the succeeding n − splitting, we need to check whether thenew source term, ( n ) G ab , in [ ( n ) R ab − h ab ( n ) R ] − ( n ) G ab = 0 , (4.1)does really have vanishing D a [ ( n ) G ab ] divergence. In doing so notice first that h ea h f b [ R ef − g ef R ] = h ea h f b R ef − h ab R (4.2)and—by substituting (1.1) to the left hand side, whereas (3.4) and (A.1) to the righthand side—the source term can be seen to read as ( n ) G ab = S ab − ǫ n − L n K ab − ( K ee ) K ab + 2 K ae K eb − ǫN D a D b N (4.3) + h ab (cid:20) L n ( K ee ) + 12 ( K ee ) + 12 K ef K ef + ǫN D e D e N (cid:21)(cid:27) . Relations analogous to (3.17) and (3.18) were derived first by York in context of the energy-momentum tensor T ab in [8] (see also [4]). Σ σ hypersurfaces thereby to proceed it suffices to ensure the existence of afoliation of Σ σ by a one parameter family of homologous codimension-two surfaces.Taking then the D a -divergence of this relation and by commuting Lie and covari-ant, as well as, covariant derivatives, by a tedious but straightforward calculation, itcan be verified that D a [ ( n ) G ab ] = L n p b + D a S ab + ǫ L n E ( M ) b + ǫ ( K ee ) h E ( M ) b + ǫ p b i + ˙ n a h − ǫ ( n ) R ab + L n K ab + ( K ee ) K ab − K ae K eb + ǫN D a D b N i − ˙ n b h L n ( K ee ) + K ef K ef + ǫN D e D e N i . (4.4)By inspecting (3.4) and (3.6), and the coefficients of ˙ n a and ˙ n b in (4.4) it can berecognized that they are equal to − ǫ h ea h f b R ef and n e n f R ef , respectively. Takingthen into account (1.1), along with G ab = R ab − g ab R , we get h ea h ea R ef = S ab − n − h ab [ S ef h ef + ǫ e ] (4.5) n e n f R ef = e − ǫn − S ef h ef + ǫ e ] . (4.6)These relations, along with (4.4), imply that D a [ ( n ) G ab ] = 0 (4.7)is equivalent to L n p b + D a S ab +[ − ǫ S ab ˙ n a + ( K ee ) p b + e ˙ n b ]+ ǫ h L n E ( M ) b + ( K ee ) E ( M ) b i = 0 . (4.8)In virtue of (3.18) and (4.8) the integrability condition (4.7) is guaranteed to holdwhenever h f b ∇ a G af = 0 and E ( M ) b = 0 on each of the σ = const level surfaces.In summarizing the above observations we have the following Proposition 4.1
The integrability condition (4.7) holds on Σ σ if h f b ∇ a G af , themomentum constraint expression E ( M ) b and its Lie derivative L n E ( M ) b vanish there. In interpreting this result recall first that—by our assumptions concerning thesource term for (1.1)—the projection h f b ∇ a G af vanish throughout Σ σ . In addition,in virtue of Theorem 2.1 the Lie derivative of both the Hamiltonian and momentumconstraint expressions vanish throughout Σ σ if they themselves vanish on Σ σ andthe evolutionary system holds. Thus, as far as we prefer to solve first both the9amiltonian and momentum constraints only on Σ σ we have to solve the reducedevolutionary system in M . In this case Proposition 4.1 has no use as it can guaranteethe integrability condition for the reduced system after the solution has been found.Note, however, that Proposition 4.1 allows a redistribution of the simplificationsguaranteed by Theorem 2.1. Namely, if we solve the momentum constraint on theentire base manifold in virtue of Theorem 2.1 and Proposition 4.1 besides solving theHamiltonian constraint on Σ σ and instead of solving the full reduced system on M it suffices to solve the second level of Hamiltonian and momentum constraints on acodimension-two surface in M whereas the corresponding new reduced evolutionarysystem (formally only) on Σ σ . By repeating this type of formal splittings and solvingalways the yielded new momentum constraints the entire process can be appliedinductively provided that product structure of the manifold allows it to be done.Applying this process, e.g. to the conventional Cauchy problem in the Lorentziancase one may get on a suitable intermediate level a mixed elliptic-hyperbolic systemsfrom Einstein’s equations as it is done for a specific gauge choice in [1]. In answering the question raised in the title the main results of this paper makes itclear that some of the basic techniques developed for n splitting of Lorentzianspacetimes do also apply to spaces with Riemannian metric. The most remarkable as-pect here is that regardless whether the metric is of Euclidean or Lorentzian signaturethe subsidiary equations—these can be derived for the ‘Hamiltonian’ and ‘momen-tum’ type expressions, by making use of the Bianchi identity—are hyperbolic. Thisguaranties that in both cases the solutions to the ‘reduced’ set of equations are alsosolutions to the full set of Einstein’s equations (1.1) provided that the constraintshold on one of the hypersurfaces foliating the base manifold. Having been the first n type decomposition performed it is important to know if there may be room forfurther simplifications in a succeeding n − type decomposition. According toour findings there is no way to acquire new simplifications in a secondary splitting.It is remarkable that the new results apply regardless whether the primary space isRiemannian or Lorentzian.Having our results it would be useful to know whether they can be applied insolving some specific problems. To indicate that even in one of the simplest possiblesetup some non-trivial implications may follow let us consider the following example.Start with a four-dimensional Riemannian space foliated by a two-parameter familyof homologous two-surfaces. Then—by making use of a suitable gauge fixing—the This could be done at most n -times which is the number of the equations involved in the originalmomentum constraint. Σ σ hypersurfaces as a boundary-initial value problem withinitial data specified on one of the codimension-two surfaces foliating Σ σ . The otherelliptic system corresponding to the ‘evolutionary’ one in the present case has tobe solved on the entire of the base manifold M with boundary value yielded bythe aforementioned parabolic-hyperbolic boundary-initial value problem on Σ σ (seee.g. [5]).The results covered by this paper have also applications in the conventionalCauchy problem and in the initial boundary value problem. In [5] it is demonstratedthat the dynamics of four-dimensional spacetimes foliated by a two-parameter familyof homologous two-surfaces can be interpreted as a two-surface based ‘geometrody-namics’, whereas in [6]—by making use of Proposition 4.1, along with the fact thatthe results covered by Sections 3 and 4 did not require any restriction on signature ofthe metric induced on the Σ σ hypersurfaces—some of the unsettled issues such as thegeometric uniqueness in the metric based formulation of the initial boundary valueproblem will be addressed.It is worth emphasizing that concerning the metric only (1.1) had been used.This, besides the Riemannian or Lorentzian spaces satisfying Einstein’s equations,allows many other theories, as well. In particular, our assumptions are satisfied bythe ‘conformally equivalent representation’ of higher-curvature theories possessing agravitational Lagrangian that is a polynomial of the Ricci scalar. Note also that theinclusion of metrics with Euclidean signature may significantly increase the varietyof theories to be covered although no attempt has been made here to explore theseaspects.Let us finally mention that irrespective of the simpleness of the observations madehere they may have interesting applications elsewhere. It would be useful to knowwhether they could be used in string and brane theories, and also in various otheralternative higher dimensional Riemannian and Lorentzian metric theories of gravity.
Acknowledgments
The author is grateful to Lars Anderson, Bobby Beig, Gábor Zs. Tóth and Bob Waldfor helpful comments and suggestions. This research was supported by the EuropeanUnion and the State of Hungary, co-financed by the European Social Fund in theframework of TÁMOP-4.2.4.A/2-11/1-2012-0001 “National Excellence Program”. The Note also that the n -dimensional source term in (4.3) cannot be directly connected to matterfields. A Appendix:
This section is to provide some useful relations. These had been applied in derivingseveral relations in Section 3, and their adopted form will also be applied in ourupcoming papers [5, 6]. As our generic results are applicable in arbitrary dimensionand to spaces with metric of Euclidean or Lorentzian signature we believe that theserelations will find several applications.It has been used implicitly in deriving (3.6) and it plays some role elsewhere so itis useful to give the generic relation of the scalar curvatures which reads as R = ( n ) R + ǫ (cid:26) − L n ( K ee ) − ( K ee ) − K ef K ef − ǫN D e D e N (cid:27) . (A.1)Consider now a co-vector field L a on M foliated by the σ = const hypersurfaces.Then L a can be decomposed in terms of n a and fields living on the σ = const levelsurfaces as L a = δ ea L e = ( h ea + ǫ n e n a ) L e = λ n a + L a (A.2)where λ = ǫ n e L e and L a = h ea L e . (A.3)Making use of this decomposition the covariant derivative ∇ e L a and the divergence ∇ e L e can be decomposed as ∇ e L a = [ D e λ + ǫ n e L n λ ] n a + λ ( K ea + ǫ n e ˙ n a ) + D e L a − n e n a ( ˙ n f L f ) (A.4) + ǫ (cid:8) n e L n L a − n e L f K f a − n a L f K f e (cid:9) , ∇ e L e = ( h ea + ǫ n e n a ) ∇ e L a = L n λ + λ ( K ee ) + D e L e − ǫ ( ˙ n f L f ) . (A.5)Consider now a symmetric tensor P ab defined on M . Note first that P ab can bedecomposed in terms of n a and fields living on the σ = const level surfaces as P ab = π n a n b + [ n a p b + n b p a ] + P ab , (A.6)where π = n e n f P ef , p a = ǫ h ea n f P ef and P ab = h ea h f b P ef .12hen, the covariant derivative ∇ e P ab can be decomposed as ∇ e P ab = π [ ( K ea + ǫ n e ˙ n a ) n b + ( K eb + ǫ n e ˙ n b ) n a ] + [ D e π + ǫ n e L n π ] n a n b + ( K ea + ǫ n e ˙ n a ) p b + ( K eb + ǫ n e ˙ n b ) p a + n a (cid:2) D e p b + ǫ (cid:8) n e L n p b − n e p f K f b − n b p f K f e (cid:9) − n e n b ( ˙ n f p f ) (cid:3) + n b (cid:2) D e p a + ǫ (cid:8) n e L n p a − n e p f K f a − n a p f K f e (cid:9) − n e n a ( ˙ n f p f ) (cid:3) + (cid:2) D e P ab + ǫ (cid:8) n e [ L n P ab − P fb K f a − P af K f b ] − n a P fb K f e − n b P af K f e (cid:9) − n e n b ( ˙ n f P af ) − n e n a ( ˙ n f P bf ) (cid:3) , (A.7)while the contraction ∇ a P ab reads as ∇ a P ab = π ( K ee ) n b + ( K ee ) p b + n b ( D e p e ) + D e P eb − ǫ n b ( P ef K ef )+ ˙ n b π + n b L n π + L n p b − ǫ ( ˙ n e p e ) n b − ǫ ( ˙ n e P eb ) . (A.8)The parallel and orthogonal parts of (A.8) simplify as ( ∇ a P ae ) h eb = ( K ee ) p b + D e P eb + ˙ n b π + L n p b − ǫ ˙ n e P eb , (A.9) ( ∇ a P ae ) n e = ǫ [ π ( K ee ) + D e p e − ǫ P ef K ef + L n π − ǫ ˙ n e p e ] . (A.10)It also follows from (A.7) that ∇ a P ee = ǫ [ D a π + ǫ n a L n π ] + D a ( P ee ) + ǫ n a L n ( P ee ) , (A.11)with parallel and orthogonal parts ( ∇ f P ee ) h f a = ǫ D a π + D a ( P ee ) (A.12) ( ∇ f P ee ) n f = ǫ L n π + L n ( P ee ) . (A.13) References [1] Andersson L and Moncrief V:
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