New first-order formulation for the Einstein equations

Transcription

1 PHYSICAL REVIEW D 68, New first-order formulation for the Einstein equations Alexander M. Alekseenko* School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA Douglas N. Arnold Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455, USA Received 2 October 2002; published 23 September 2003 We derive a new first-order formulation for Einstein s equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the 3 decomposition with arbitrary lapse and shift. In the reduction to first-order form only eight particular combinations of the 8 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of a symmetric hyperbolic system in 4 unknowns, namely, the components of the extrinsic curvature perturbation and the eight new variables, from whose solution the metric perturbation can be computed by integration. DOI: 0.03/PhysRevD PACS number s : Ex, Dm I. INTRODUCTION Many ways have been proposed to formulate Einstein s equations of general relativity in a manner suitable for numeric computation. In this paper we introduce a new firstorder formulation for Einstein s equations. This system involves fewer unknowns than other first-order formulations that have been proposed and does not require any arbitrary parameters. In the simplest case of linearization around Minkowski space with constant lapse and vanishing shift, the system has the simple form & t ij l l ij, & t lji [l j]i. Here ij is the extrinsic curvature perturbation, a symmetric tensor, and ijk is a third-order tensor which is antisymmetric with respect to the first two indices and satisfies a cyclic identity, with the result that the system above is symmetric hyperbolic in 4 unknowns. Our approach applies as well to the full nonlinear Arnowitt-Deser-Misner ADM system with arbitrary lapse and shift. We work with the actual lapse, rather than a densitized version. In the nonlinear case the system involves 20 unknowns and can be written 0 h ij 2ak ij 2h s(i j) b s, & 0k ij ah mi h nj l f l mn, Here 0 ª t b l l is the convective derivative, indices are raised and lowered using the spatial metric components h ij, and the omitted terms are algebraic expressions in the h ij, their spatial derivatives l h ij, and the extrinsic curvature components k ij, and also involve the lapse a and shift components b i. The f lmn, which depend on the first spatial derivatives of the spatial metric, satisfy the same symmetries as in the linear case, and so represent eight unknowns. As is common, our derivation will start from the Arnowitt-Deser-Misner 3 decomposition ; see also 2. The ADM approach introduces a system of coordinates t x 0,x,x 2,x 3, with t a timelike variable and the x i spacelike for i,2,3, and encodes the four-metric of spacetime as a time-varying three-metric on a three-dimensional domain together with the lapse and shift, which are scalar-valued and three-vector-valued functions of time and space, respectively. Specifically, the coordinates of the four-metric are given by g 00 a 2 b i b j h ij, g 0i b i, g ij h ij. Here a denotes the lapse, the b i are the components of the shift vector b, and the h ij are the components of the spatial metric h. As usual, roman indices run from to 3 and (h ij ) denotes the matrix inverse to (h ij ). Let D i denote the covariant derivative operator associated with the spatial metric and set k ij 2a th ij a D (ib j), the extrinsic curvature. Then the ADM equations for a vacuum spacetime are & 0f lmn [l ak m]n. *Electronic address: alekseen@math.umn.edu Electronic address: arnold@ima.umn.edu t h ij 2ak ij 2D (i b j), t k ij a R ij k l l k ij 2k il k l j b l D l k ij k il D j b l k lj D i b l D i D j a, R i i k i i 2 k ij k ij 0, /2003/68 6 / /$ The American Physical Society

2 A. M. ALEKSEENKO AND D. N. ARNOLD PHYSICAL REVIEW D 68, D j k ij D i k j j 0. 5 P i i 0, 8 Here R denotes the spatial Ricci tensor, whose components are given by second-order spatial partial differential operators applied to the spatial metric components. Also indices are raised and traces taken with respect to the spatial metric. We have used the notation of indices in parentheses to denote the symmetric part of a tensor: D (i b j) ª(D i b j D j b i )/2. Later we will similarly use bracketed indices to denote the antisymmetric part and sometimes bars to separate indices, so, as an example, u i j k l ª(u ijkl u kjil )/2. Equations 4 and 5, called the Hamiltonian and momentum constraints, do not involve time differentiation. The first two equations are the evolution equations. A typical approach is to determine the lapse and shift in some way, find relevant initial data for h and k satisfying the constraint equations, and then solve the evolution equations to determine the metric and extrinsic curvature for future times. The constraint equations may or may not be explicitly imposed during the evolution. For exact solutions of the evolution equation with initial data exactly satisfying the constraints, the constraints are automatically satisfied for future times. The system of evolution equations for h and k is first order in time and second order in space. They are not hyperbolic in any usual sense, and their direct discretization seems difficult. Therefore, many authors 3 6 have considered reformulations into more standard first-order hyperbolic systems. Typically these approaches involve introducing all the first spatial derivatives of the spatial metric components, or quantities closely related to them, as 8 additional unknowns. The resulting systems involve many variables, sometimes 30 or more. In the formulation proposed here, we introduce only eight particular combinations of the first derivatives of the metric components. In the next section of the paper we present our approach as applied to a linearization of the ADM system. This allows us to demonstrate the basic ideas with a minimal of technical complications and to rigorously establish the relationship between the new formulation and the ADM system. In the third and final section of the paper we carry out the derivation in the case of the full nonlinear ADM system. II. SYMMETRIC FORMULATION FOR THE LINEARIZED SYSTEM We linearize the ADM equations about the trivial solution obtained by representing Minkowski spacetime in Cartesian coordinates: h ij ij, k ij 0, a, b i 0. Consider a perturbation given by h ij ij ij, k ij ij, a, b i i, with the ij, ij,, and i supposed to be small. Substituting these expressions into the ADM system and ignoring terms which are at least quadratic with respect to the ij, ij,, and i and their derivatives, we obtain the linear system we shall study: t ij 2 ij 2 (i j), 6 j ij i j j 0. Here P is the linearized Ricci tensor, with components given by 9 P ij 2 i l lj 2 j l li 2 l l ij 2 i j l l. 0 In these expressions, and in general when we deal with the linearized formulation, indices are raised and lowered with respect to the Euclidean metric in R 3, so, for example, i and i are identical. In order to reduce the linearized ADM system to firstorder symmetric hyperbolic form, we first develop an identity for P valid for any symmetric matrix field. It is useful to introduce the notations S for the six-dimensional space of symmetric matrices and T for the eight-dimensional space of triply indexed arrays (w ijk ) which are skew symmetric in the first two indices and satisfy the cyclic property w ijk w jki w kij 0. We define the operator M: C (R 3,S) C (R 3,R 3 ) by (Mu) i l u il i u l l. Note that the linearized momentum constraint 9 is simply M 0.] From the definition 0, we have P ij l [l j]i 2 i M j. Now, for any vector-valued function v, we have i v j l v j li 2 l v [ j l]i l v l ij. Applying this identity to Eq. with v M we get P ij l [l j]i M [l j]i 2 l M l ij. 2 Define the operators L: C (R 3,S) C (R 3,T) and L*: C (R 3,T) C (R 3,S) by Lu lji [l u j]i, L*w ij l w l ij. One easily verifies that operators L and L* are formal adjoints to each other with respect to the scalar products u,v u pq v pq dx and z,w z pqr w pqr dx in the spaces C (R 3,S) and C (R 3,T) respectively. Introducing lji & L lji M [l j]i, we can then restate Eq. 2 as 3 t ij P ij i j, P ij & l lji 2 l M l ij.

3 NEW FIRST-ORDER FORMULATION FOR THE... The reason for the factor of & will become apparent shortly. Taking symmetric parts, the last equation becomes P ij & L* ij 2 l M l ij. 4 We are now ready to introduce our first-order symmetric hyperbolic formulation. The unknowns will be C (R 3,S) and C (R 3,T), so the system has 4 independent variables in all. Substituting Eq. 4 in Eq. 7 and noting that l (M ) l (P ) l l 0 by the linearized Hamiltonian constraint 8, we obtain an evolution equation for : t ij & L* ij i j. 5 To obtain an evolution equation for, we differentiate Eq. 3 with respect to time and substitute Eq. 6 to eliminate. Simplifying and using the linearized momentum constraint 9 we obtain t lji & L lji lji, can be determined from ( ) ij ª (i j) by lji & L lji M [l j]i & i [l j] m [m l] ij m m j] il. 6 Equations 5 and 6 constitute a first-order symmetric hyperbolic system this is clear, since L and L* are formal adjoints. It follows that see, e.g., 7, Sec , ifthe lapse and the shift are sufficiently smooth, then for arbitrary initial values 0 and 0 belonging to H (R 3 ), there exists a unique solution to the system 5 and 6 with components in H (0,T) R 3. The Cauchy problem for the original linearized ADM system consists of Eqs. 6 9 together with specific initial values 0 and 0. The foregoing derivation shows that, if and satisfy the ADM system and is defined by Eq. 3, then and satisfy the symmetric hyperbolic system 5, 6. Conversely, to recover the solution to the ADM system from 5, 6, the same initial condition should be imposed on and should be taken initially to be PHYSICAL REVIEW D 68, first-order symmetric hyperbolic system 5, 6. Finally, define by Eq. 8. Then the ADM system 6 9 is satisfied. Proof. Equation 6 follows from Eq. 8 by differentiation. Next we verify the momentum constraint 9. Todosowe will show that ªM satisfies a second-order wave equation, and that (0) t (0) 0. Indeed, (0) 0 by assumptions. To see that t (0) 0, we apply the operator M to Eq. 5 and use the fact that M annihilates the Hessian i j for any function. Therefore t &ML*. Using Eq. 4 and the assumption that satisfies the Hamiltonian constraint at the initial time, we find that t i 0 MP 0 i 2 i l M 0 l 0. To obtain a second-order equation for, first we differentiate Eq. 5 in time and substitute Eq. 6 to get a second-order equation for : t 2 ij 2 L*L ij i j. Here we have used the fact that L* 0. Apply M to the last equation. Using the identity (ML*L ) l (i (M ) l) /2 and the fact that M annihilates Hessians, we find that ªM satisfies the second-order hyperbolic equation t 2 i l (i l). This is simply an elastic wave equation. Since (0) t (0) 0, vanishes for all time, i.e., the momentum constraint is satisfied. Now (P ) i i i (M ) i 0. Moreover, P applied to is identically zero. Therefore if we apply P to Eq. 8 and take the trace, we find that (P ) i i P (0) i i, which vanishes by assumption. This verifies the Hamiltonian constraint 8. It remains to verify Eq. 7 which, in view of Eq. 5, comes down to showing that &L* P. Since we have verified the Hamiltonian constraint, this will follow if we can establish Eq. 4, which is itself a consequence of Eq. 3. We used Eq. 3 at time t 0 to initialize, so it is sufficient to show that lji 0 & L 0 lji M 0 [l j]i. 7 t lji & t L lji M [l j]i. Once and are determined, the metric perturbation is given by ij ij t ij (i j), 8 as follows from Eq. 6. Theorem. Let the lapse perturbation and shift perturbation be given. Suppose that initial data 0 and 0 are specified satisfying the constraint equations 8, 9 at time t 0. Define 0 by Eq. 7, and determine and from the This follows directly from Eqs. 6 and 6. We conclude this section by computing the plane wave solutions to the hyperbolic system. That is, we seek solutions of the form ij ij f st n a x a, lji lji f st n a x a, the real number s gives the wave speed and the unit vector n i the wave direction, the polarizations ij and lji are constant, and the profile f (t) is an arbitrary differentiable function. Substituting these expressions into Eq., weget

4 A. M. ALEKSEENKO AND D. N. ARNOLD PHYSICAL REVIEW D 68, s ij f st n a x a &n l l ij f st n a x a, s lji f st n a x a &n [l j]i f st n a x a, and so the system reduces to the following linear eigenvalue problem: s ij &n l l ij, s lji &n [l j]i, 9 20 with the wave speed s as eigenvalues and the pairs ( ij, lji ) as eigenvectors. The eigenvalues of this system are 0 multiplicity 4, each multiplicity 3, and /& each multiplicity 2. To verify this and describe the eigenvectors we introduce a unit vector m i perpendicular to n i, and set l i i ab n a m b to complete an orthonormal frame. Then the following solution to the eigenvalue problem can be checked by direct substitution into Eqs. 9, 20 : s 0: 0,m [l l j] m i, 0,m [l l j] l i, n i n j,0, s : 0,2m [l l j] n i n [l l j] m i n [l m j] l i ; l i l j m i m j, & n [l l j] l i n [l m j] m i, l i m j m i l j, & n [l l j] m i n [l m j] l i, l i l j m i m j, & n [l l j] l i n [l m j] m i ; s /&: n (i l j), n [l l j] n i, n (i m j), n [l m j] n i. III. DECOMPOSITION OF THE FULL ADM SYSTEM In this section we develop a first-order formulation of the full nonlinear ADM system analogous to that developed for the linearized system in Sec. II. We continue to assume that the underlying manifold is topologically R 3 and view the ADM system as equations for the evolution of a Riemannian three-metric h on R 3. Thus h ij, i, j 3 are the components of the spatial metric. They form a positive-definite symmetric matrix defined at each point of R 3 and varying in time. Indices on other fields are lowered and raised using h ij and the inverse matrix field h ij. For the components of the Riemann tensor we have R ijkl j ikl i jkl h mn jkm iln ikm jln, the Cristoffel symbols are defined by ijk ( i h jk j h ki k h ij )/2. The components of the Ricci tensor are given by R ij h pq R piqj, which yields Pu ij 2 p h pq j u qi i h pq p u jq p h pq q u ji i h pq j u pq. Note that, in the case h ij ij, P coincides with the linearized Ricci operator introduced in Eq. 0. In the nonlinear case, P is related to the Ricci tensor by the equation R ij Ph ij c ) ij, 2 c ) ij ª 2 ph pq j h iq i h pq p h qj p h pq q h ij i h pq j h pq h pq h rs ips qjr pqs ijr is a first-order differential operator in h ij. Define an operator M: C (R 3,S) C (R 3,R 3 )by Mu i 2h pq [p u i]q. This formula extends the definition of the linearized momentum constraint operator introduced in the previous section. Up to lower order terms the momentum constraint 5 is given by the vanishing of Mk: D p k ip D i k p p 2h pq D [p k i]q Mk i 2h pq r q[p k i]r 22 r qp ªh rs qps ). Finally, we introduce operators L: C (R 3,S) C (R 3,T) and L*: C (R 3,T) C (R 3,S) by Lu ijl [i u j]l, L*v ij h qi h rj p v p qr. As in the linear case, the operators L and L* are formal adjoints to each other with respect to the scalar products u,w u ij w ij dx and v,z v ijl z ijl dx on the spaces C (R 3,S) and C (R 3,T), respectively. Finally, we introduce the new variables f lmn & Lh lmn Mh [l h m]n, 23 and develop the analogue of the identity 4. Lemma. The following identity is valid for the Ricci tensor: R ij & L*f ij 2 p h pq Mh q h ij c 2) ij, 24 R ij 2 h pq p j h iq i p h qj p q h ij i j h pq h pq h rs ips qjr pqs ijr. We define a second-order linear partial differential operator P: C (R 3,R 3 3 ) C (R 3,R 3 3 )by c 2) ) ij c ij h qi h rj p h qm h rn h pl Lh l mn 2 Mh [lh m]n 2 Mh [lh n]m

5 NEW FIRST-ORDER FORMULATION FOR THE... is first order in h ij. Proof. The formula 24 is a consequence of the identity & L*f ij Ph ij 2 p h pq Mh q h ij c 3) ij, 25 c 3) ij h qi h rj p h qm h rn h pl Lh l mn 2 Mh [lh m]n 2 Mh [lh n]m. To prove the identity 25 we note, first, that the operator P can be rewritten in terms of operators L and M as which yields Pu ij p h pq Lu qji 2 i Mu j, Pu ij p h pq Lu q ij 2 (i Mu j), and, second, that according to the definition of L*, L*v ij h qi h rj p h pl h qm h rn v l mn p h pq v q ij h qi h rj p h qm h rn h pl v l mn. 26 To derive the identity 25 we substitute & f qij for v qij in Eq. 26. The first term on the right-hand side of Eq. 26 then becomes p h pq Lh q ij 2 Mh qh ij 2 Mh (ih j)q 0 k ij &a L*f ij 2 a p h pq Mh q h ij c 4) ij, 27 c 4) ij a c 2) ij k l l k ij 2k il k l j k il j b l k lj i b l D i D j a. Here we used the fact that the Lie derivative b l D l k ij k il D j b l k lj D i b l b l l k ij k il j b l k lj i b l. We treat the second term on the right-hand side of Eq. 24 using the Hamiltonian constraint 4. Now, R i i 2 hij h pq p j h iq i p h qj p q h ij i j h pq h ij h pq h rs ips qjr pqs ijr h ij h pq i p h qj i j h pq h ij h pq h rs ips qjr pqs ijr i h ij h pq p h qj j h pq i h ij h pq p h qj j h pq h ij h pq h rs ips qjr pqs ijr i h ij Mh j c 5), PHYSICAL REVIEW D 68, c 5) i h ij h pq p h qj j h pq h ij h pq h rs ips qjr pqs ijr. Hence, due to the Hamiltonian constraint, p h pq Mh q h ij p h pq Mh q h ij h pq Mh q p h ij k pq k pq k p p 2 c 5) h ij h pq Mh q p h ij. Combining all lower order terms into an expression B ij, first order in h, we reduce Eq. 27 to p h pq Lh q ij 2 p h pq Mh q h ij 0 k ij &a L*f ij B ij p h pq Mh (i h j)q p h pq Lh q ij 2 p h pq Mh q h ij 2 (i Mh j) Ph ij 2 p h pq Mh q h ij. The substitution of & f lmn for v lmn into the second term of Eq. 26 gives the term c 3) ij precisely. The rest of the proof follows from the identity 2. We now proceed to the derivation of the new formulation of the ADM system. In Eq. 3 we substitute 0 b l l for t and replace R ij with the right side of Eq. 24 to get This is the first evolution equation of our system. The second evolution equation will be obtained by applying 0 to the definition of f 23 : 0 f lmn & 0 Lh lmn 0 Mh [l h m]n. First, we note that 0 Lh lmn L 0 h lmn 2 lb s s h mn m b s s h ln. Using the fact that (Mu) l 2h pq (Lu) plq, we then get

6 A. M. ALEKSEENKO AND D. N. ARNOLD PHYSICAL REVIEW D 68, Mh l M 0 h l 2 0 h pq Lh plq h pq p b s s h lq l b s s h pq. If we use this formula to compute 0 (Mh) l h mn and then antisymmetrize in l and m, we obtain 0 f lmn & L 0h lmn M 0 h [l h m]n & 0 h pq Lh p[l q h m]n & Mh 6) [l 0 h m]n c lmn, 29 6) c lmn 2& lb s s h mn m b s s h ln & h pq p b s s h q[l h m]n s h pq [l b s h m]n. Next we use Eq. 2 to relate the terms in Eq. 29 involving 0 h to the extrinsic curvature k. For the Lie derivative of the metric, we have Using this, Eq. 2 becomes 2D (i b j) b s s h ij 2h s(i j) b s. 0 h ij 2ak ij 2w ij, 30 w ij ªh s(i j) b s. Using the Leibniz rule we can then verify that 0 h ij 2ak ij 2w ij. Substituting these expressions in Eq. 29 we obtain 7) c lmn 7) 0 f lmn & L ak lmn & M ak [l h m]n c lmn, 3 6) c lmn & Lw lmn Mw [l h m]n 2& ak pq w pq Lh p[l q h m]n & Mh [l ak m]n & Mh [l w m]n. The final step is to invoke the momentum constraint to simplify the second term on the right-hand side of Eq. 3. Indeed, since the right-hand side of Eq. 22 vanishes, M ak l a Mk l 2h pq [p a k l]q 2ah pq s q[p k l]s 2h pq [p a k l]q. Substituting this in Eq. 3, we obtain the desired second evolution equation: 7) C lmn c lmn 0 f lmn & L ak lmn C lmn, & ah pq s q[p k l]s h mn ah pq s q[p k m]s h ln h pq [p a k l]q h mn h pq [p a k m]q h ln. 32 The two equations 28 and 32 constitute a first-order system for the unknowns k ij and f lmn. This system is coupled to the ordinary differential equation 30 through the terms B ij and C lmn which are algebraic combinations of h ij, l h ij,k ij, the lapse a and the shift b, and their spatial derivatives. The foregoing derivation shows that if h and k satisfy the ADM system 2 5, then h, k, and f satisfy the system 30, 28, 32. ACKNOWLEDGMENT D.A. was supported by NSF grant DMS R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation: An Introduction to Current Research, edited by L. Witten Wiley, New York, J. W. York, Jr., in Sources of Gravitational Radiation, edited by L. L. Smarr Cambridge University Press, Cambridge, England, S. Frittelli and O. A. Reula, Phys. Rev. Lett. 76, A. M. Abrahams and J. W. York, Jr., in Astrophysical Sources of Gravitational Radiation, Proceedings of the Les Houches Summer School of Theoretical Physics, edited by J.-A. Marck and J.-P. Lasota Cambridge University Press, Cambridge, England, A. Anderson and J. W. York, Jr., Phys. Rev. Lett. 82, L. E. Kidder, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. D 64, L. C. Evans, Partial Differential Equations American Mathematical Society, Providence, RI,