Constraint propagation in the family of ADM systems

Constraint propagation in the family of ADM systems

Gen Yoneda*

Department of Mathematical Sciences, Waseda University, Shinjuku, Tokyo, 169-8555, Japan Hisa-aki Shinkai^†

Centre for Gravitational Physics and Geometry, 104 Davey Laboratory, Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802-6300

共Received 9 March 2001; published 23 May 2001兲

The current important issue in numerical relativity is to determine which formulation of the Einstein equa- tions provides us with stable and accurate simulations. Based on our previous work on ‘‘asymptotically constrained’’ systems, we here present constraint propagation equations and their eigenvalues for the Arnowitt- Deser-Misner共ADM兲evolution equations with additional constraint terms共adjusted terms兲on the right-hand side. We conjecture that the system is robust against violation of constraints if the amplification factors 共eigenvalues of the Fourier component of the constraint propagation equations兲are negative or purely imagi- nary. We show that such a system can be obtained by choosing multipliers of the adjusted terms. Our discussion covers Detweiler’s proposal and Frittelli’s analysis, and we also mention the so-called conformal- traceless ADM systems.

DOI: 10.1103/PhysRevD.63.124019 PACS number共s兲: 04.25.Dm, 04.20.Fy

I. INTRODUCTION

The effort to solve the Einstein equations numerically—

so-called numerical relativity—is now providing an interest- ing bridge between mathematical relativists and numerical relativists. Most of the simulations have been performed us- ing the Arnowitt-Deser-Misner 共ADM兲formulation关1兴or a modified version. However, the ADM formulation has not been proven to be a well-posed system, since its evolution equations do not present a hyperbolic form in its original or standard formulation.

Most simulations are performed using ‘‘free evolution’’

procedures: 共1兲 solve the Hamiltonian and momentum con- straints to prepare the initial data,共2兲integrate the evolution equations by fixing gauge conditions, and 共3兲 monitor the accuracy or stability by evaluating the constraints. Many tri- als have been made in the last few decades, but we have not yet obtained a perfect recipe for the long-term stable evolu- tion of the Einstein equations. Here we consider the problem through the form of the equations.

One direction in the community is to rewrite the Einstein evolution equations in a hyperbolic form and to apply it to numerical simulations关2兴. This is motivated by the fact that we can prove well-posedness for the evolution of several systems if they have a certain kind of hyperbolic feature. The authors recently derived 关3,4兴 three levels of a hyperbolic system of the Einstein equations using Ashtekar’s connection

variables 关5兴,¹ and compared them numerically 关6兴. We found that 共a兲 the three levels of hyperbolicity can be ob- tained by adding constraint terms and/or imposing gauge conditions, 共b兲there is no drastic difference in the accuracy of numerical evolutions in these three, and共c兲the symmetric hyperbolic system is not always the best for reducing nu- merical errors. Similar results in regard to 共a兲 and 共b兲 are reported by Hern关7兴based on the Frittelli-Reula formulation 关8兴.

What are, then, the criteria for predicting the stable evo- lutions of a system? Inspired by the ‘‘␭-system’’ proposal 关9兴, we have considered a so-called ‘‘asymptotically con- strained’’ system, that is, a system robust against the viola- tion of constraints 关10兴. The fundamental idea of the ‘‘␭ system’’ is to introduce artificial flow onto the constraint surface. However, we also found that such a feature can be obtained simply by adding constraint terms to the evolution equations which we named ‘‘adjusted systems’’ 关11兴. We explained the reason why this works by analyzing the evo- lution equations of the constraints 共the propagation of the constraints兲. We proposed that the stablity of the system can be predicted by analyzing the eigenvalues共amplification fac- tors兲 of the constraint propagation equations 共we describe this in detail in Sec. II兲. We confirmed that our proposal works in both the Maxwell and Ashtekar systems 关11兴.

The purpose of this article is to apply our proposal to the

*Email address: [email protected]

†Present address: Computational Science Division, Institute of Physical and Chemical Research 共RIKEN兲, Hirosawa 2-1, Wako, Saitama 351-0198, Japan. Email address: [email protected]

1We derived weakly, strongly (⫽diagonalizable兲, and symmetric hyperbolic systems. The mathematical inclusion relation is

weakly hyperbolic苹strongly hyperbolic 苹symmetric hyperbolic.

See details in关4兴.

ADM system共s兲. Especially, we consider the ‘‘adjusting pro- cess’’ 关adding constraints on the right-hand side 共RHS兲 of evolution equations兴and the resultant changes to the eigen- values of the constraint propagation systems. This adjusting process can be seen in many constructions of hyperbolic sys- tems in the references. In fact, the standard ADM for numeri- cal relativists is the version which was introduced by York 关12兴, where the original ADM system 关1兴 has already been adjusted using the Hamiltonian constraint共see more detail in Sec. III兲. The advantage of the standard ADM system is re- ported by Frittelli 关13兴 from the point of the hyperbolicity and the characteristic propagation speed of the constraints.

Our discussion extends her analysis to the amplification fac- tors.

One early effort of the adjusting mechanism was pre- sented by Detweiler关14兴. Our study also includes his system, and shows that this system actually works as desired for a certain choice of parameter 共Sec. IV兲. We also study the same procedure for the ‘‘conformal-traceless’’ ADM 共CT- ADM兲formulations关15,16兴which is recently the most popu- lar system in numerical simulations共Sec. V兲.

The analysis in the text is for perturbational violation on a flat background. Further applications are available, but we will discuss them in future reports. In the Appendix, we also give numerical demonstrations of the adjusted ADM systems discussed in the text.

II. CONSTRAINT PROPAGATION AND ‘‘ADJUSTED SYSTEM’’

We begin by reviewing the background of ‘‘adjusted sys- tems’’ and our conjecture.

The notion of the evolution equations of the constraints is often discussed from the point of whether they form a first class system or not. Fortunately, the constraints on the共origi- nal or standard兲ADM formulation are known to form a first- class system. Because of this fact, numerical relativists only need to monitor violation of the Hamiltonian and momentum constraints during free evolution of the initial data.

Our essential idea here is to feed this procedure back into the evolution equations. That is, we adjust the system’s evo- lution equations by characterizing the constraint propagation in advance. Let us describe the procedure in a general form.

Suppose we have a set of dynamical variables, u^a(xⁱ,t), and its evolution equations

⳵tu^a⫽f共u^a,⳵iu^a, . . .兲, 共2.1兲 which should satisfy a set of constraints C^␳(u^a,⳵iu^a, . . . )

⬇0. The evolution equation for C^a can be written as

⳵tC^␳⫽g共C^␳,⳵iC^␳, . . .兲. 共2.2兲 We can perform two main types of analysis on Eq.共2.2兲. 共1兲 If Eq. 共2.2兲 is in a first-order form 共that is, only in- cludes first-order spatial derivatives兲, then the level of hyper- bolicity and the characteristic speeds 共eigenvalues␭^l of the principal matrix兲will definitely determine the stability of the system. We expect mathematically rigorous well-posed fea- tures for strongly or symmetric hyperbolic systems, and the

characteristic speeds suggest to us satisfactory criteria for stable evolutions if they are real and under the propagation speed of the original variables u^a and/or within the causal region of the numerical integration scheme applied.

共2兲 On the other hand, the Fourier transformed equation 共2.2兲,

⳵tCˆ^␳⫽gˆ共Cˆ^␳兲, 共2.3兲 where C^␳(x,t)⫽兰Cˆ^␳(k,t)exp(ik•x)d³k also characterizes the evolution of the constraints independently of its hyperbolic- ity. As we have proposed and confirmed in 关11兴, the set of eigenvalues⌳ⁱof the coefficient matrix in Eq.共2.3兲provides a kind of amplification factor of the constraint propagation and predicts the increase or decrease of the violation of the constraints if it exists. More precisely, we showed in 关11兴 that if the eigenvalues of Eq. 共2.3兲 共a兲have a negative real- part or共b兲are nonzero共purely imaginary兲eigenvalues, then we see more stable evolutions than a system which does not.

This is because the constraints are damped if the eigen- values are negative and are propagating away if the eigen- values are purely imaginary. We found heuristically that the system becomes more stable共accurate兲 when the amplifica- tion factors,⌳’s, satisfy as much the above criteria and/or as large magnitude of⌳’s away from zeros.共Examples in 关11兴 are of the plane wave propagation in the Maxwell system and the Ashtekar system.兲 We remark that this eigenvalue analysis requires that we fix a particular background metric for the situation we consider, since the amplification factor depends on the dynamical variables u^a.

The above features of the constraint propagation, Eq.

共2.2兲, will change when we modify the original evolution equations. Suppose we adjust the RHS of Eq.共2.1兲by adding the constraints,

⳵tu^a⫽f共u^a,⳵iu^a, . . .兲⫹F共C^␳,⳵iC^␳, . . .兲, 共2.4兲 then Eq. 共2.2兲will also be modified as

⳵tC^␳⫽g共C^␳,⳵iC^␳, . . .兲⫹G共C^␳,⳵iC^␳, . . .兲. 共2.5兲 By taking the characteristic speed of Eq. 共2.5兲and the am- plification factor of the Fourier transformed equation 共2.5兲, the predicted stability of the system 共2.4兲becomes different to that of the original system, Eq.共2.2兲.

Our proposed ‘‘adjusted system’’ is obtained by finding a certain functional form of F(C^␳,⳵iC^␳, . . . ) in Eq.共2.4兲so as to get a more stable prediction in the analysis of the eigen- values␭^l and⌳ⁱ. In the following discussion, we show two eigenvalues ␭^l and ⌳ⁱ for each ADM system. We remark again that the term ‘‘characteristic speed’’ here is not for the dynamical equation 共2.1兲, but for the constraint propagation equations 共2.2兲.

III. STANDARD ADM SYSTEM

A. Standard ADM system and its constraint propagation We start by analyzing the standard ADM system. By

‘‘standard ADM’’ we mean here the most widely adopted system, due to York 关12兴, with evolution equations

⳵t␥i j⫽⫺2␣^Ki j⫹ⵜi␤j⫹ⵜj␤i, 共3.1兲

⳵tK_{i j}⫽␣^Ri j

(3)⫹␣^KKi j⫺2␣^KikK^k_j⫺ⵜiⵜj␣

⫹共ⵜi␤^k兲K_{k j}⫹共ⵜj␤^k兲K_ki⫹␤^kⵜkK_{i j}, 共3.2兲 and constraint equations

HªR⁽³⁾⫹K²⫺K_{i j}K^{i j}, 共3.3兲 MiªⵜjK^j_i⫺ⵜiK, 共3.4兲 where (␥i j,K_{i j}) are the induced three-metric and the extrin- sic curvature, (␣^,␤i) are the lapse function and the shift covector, ⵜi is the covariant derivative adapted to␥i j, and R_{i j}⁽³⁾is the three Ricci tensor.

The constraint propagation equations, which are the time evolution equations of the Hamiltonian constraint 共3.3兲 and the momentum constraints 共3.4兲, can be written as

⳵tH⫽␤^j共⳵jH兲⫺2␣␥^{i j}共⳵iMj兲⫹2␣^KH⫹␣共⳵l␥mk兲

⫻共2␥^ml␥^{k j}⫺␥^mk␥^{l j}兲Mj⫺4␥^{i j}共⳵j␣兲Mi, 共3.5兲

⳵tMi⫽⫺共1/2兲␣共⳵iH兲⫹␤^j共⳵jMi兲⫹␣^KMi⫺共⳵i␣兲H

⫺␤^k␥^jl共⳵i␥lk兲Mj⫹共⳵i␤k兲␥^{k j}Mj. 共3.6兲 The simplest derivation of Eqs. 共3.5兲 and 共3.6兲 is by using the Bianchi identity, which can be seen in Frittelli 关13兴. 关Note that C in 关13兴 is half our H, and we have corrected typos in Eq. 共11兲in关13兴.兴

The characteristic part of Eqs.共3.5兲and共3.6兲can be ex- tracted as

⳵t

冉

^MHi

冊

^⯝

冉

^{⫺共1/2␤l兲␣␦il ⫺␤2l␣␥␦jiil}

冊

^⳵l

冉

^MHj

冊

^⫽..Pl⳵l

冉

^MHj

冊

^,

共3.7兲 which indicates that the characteristic speeds共eigenvalues of the characteristic matrix, P^l) are

␭^l⫽共␤^l,␤^l,␤^l⫾␣

冑

␥^ll兲 共no sum over l兲. 共3.8兲 Since rank( P^l⫺␤^l)⫽2, the matrix P^l is diagonalizable, but not the symmetric.

Simply by inserting共1/2兲in front ofHabove, we obtain

⳵t

冉

^HM/2i

冊

^⯝

冉

^{⫺␤␣␦l il ⫺␤␣␥l␦jiil}

冊

^⳵l

冉

^HM/2j

冊

^{; 共3.9兲}

the characteristic matrix becomes symmetric共with the same eigenvalues兲. This is a feature of the standard ADM system that was pointed out by Frittelli. 共ActuallyH/2 is the form originally given by the Lagrangian formulation.兲

B. Amplification factors on the Minkowskii background As a first example, we consider the perturbation of Minkowskii spacetime: ␣⫽1, ␤ⁱ⫽0, ␥i j⫽␦i j. By taking the linear order contribution, Eqs.共3.5兲and共3.6兲are reduced to

⳵t

冉

^{(1)(1)MHˆˆi}

冊

^⫽

^冉

^{⫺共1/20 兲iki ⫺2ik0 j}

^冊 冉

^{(1)(1)MHˆˆj}

冊

^共3.10兲

in Fourier components. The eigenvalues of the coefficient matrix of Eq. 共3.10兲, which we call amplification factors, become

⌳^l⫽共0,0,⫾i

冑

^k2兲, 共3.11兲 where k²⫽k_x²⫹k_y²⫹k_z². These factors will be compared with others later, but we note that the real parts of all the ampli- fication factors are zero.

IV. ADJUSTED ADM SYSTEMS A. Adjustments

Generally, we can write the adjustment terms to Eqs.共3.1兲 and共3.2兲using Eqs.共3.3兲and共3.4兲by the following combi- nations 共using up to the first derivative of constraints兲: adjustment term of ⳵t␥i j:

⫹P_{i j}H⫹Q^k_{i j}Mk⫹p^k_{i j}共D_kH兲⫹q_{i j}^kl共D_kMl兲, 共4.1兲

adjustment term of⳵tK_{i j}:

⫹R_{i j}H⫹S^k_{i j}Mk⫹r^k_{i j}共D_kH兲⫹s^kl_{i j}共D_kMl兲, 共4.2兲 where P,Q,R,S and p,q,r,s are multipliers共do not confuse R_{i j} with the three Ricci curvature that we write as R_{i j}⁽³⁾).

Since this expression is too general, we mention some re- stricted cases below.

We remark that our starting system, Eqs. 共3.1兲and共3.2兲, is the standard ADM system for numerical relativists intro- duced by York 关12兴. This expression can be obtained from the originally formulated canonical expression by ADM关1兴, but in that process the Hamiltonian constraint equation is used to eliminate the three-dimensional Ricci scalar. There- fore the standard ADM system is already adjusted from the original ADM system. We start our comparison with this point.

B. Original ADM vs standard ADM

Frittelli’s adjustment analysis关13兴can be written in terms of Eqs.共4.1兲and共4.2兲, as

R_{i j}⫽共1/4兲共␮⫺1兲␣␥i j, 共4.3兲 where␮ is a constant and set other multipliers in Eqs.共4.1兲 and 共4.2兲 to zero. Here ␮⫽1 corresponds to the standard ADM system共no adjustment, since R_{i j}⫽0) and␮⫽0 to the original ADM system共without any adjustment to the canoni- cal formulation by the ADM system兲.

Keeping the multiplier共4.3兲in mind, we here discuss the case of nonzero R_{i j},S_{i j}^k 共all other multipliers being zero兲. The constraint propagation equations become

⳵tH⫽␤^j共⳵jH兲⫺2␣␥^{i j}共⳵iMj兲⫹2␣^KH⫹␣共⳵l␥mk兲共2␥^ml␥^{k j}⫺␥^mk␥^{l j}兲Mj⫺4␥^{i j}共⳵j␣兲Mi⫹2KRH⫺2K^{i j}R_{i j}H

⫹2K␥^{i jS}i j

kMk⫺2K^{i j}S^k_{i j}Mk, 共4.4兲

⳵tMi⫽⫺共1/2兲␣共⳵iH兲⫹␤^j共⳵jMi兲⫹␣^KMi⫺共⳵i␣兲H⫺␤^k␥^jl共⳵i␥lk兲Mj⫹共⳵i␤k兲␥^{k j}Mj⫹␥^{k j}共⳵jR_ki兲H⫺␥^jk共⳵iR_jk兲H

⫹R_i^j共⳵jH兲⫺R_jk␥^jk共⳵iH兲⫹␥^{l j}共⳵jS_li^k兲Mk⫺␥^jl共⳵iS^k_jl兲Mk⫹S_i^{k j}共⳵jMk兲⫺␥^jlSjl

k共⳵iMk兲⫹共⳵j␥^{k j}兲R_kiH

⫹⌫jk

j R_i^kH⫺⌫ji

kR_k^jH⫺共⳵i␥^jk兲R_jkH⫹共⳵j␥^{l j}兲S_li^kMk⫹⌫jl

jS_i^klMk⫺⌫ji

lS_l^{k j}Mk⫺共⳵i␥^jl兲S_jl^kMk; 共4.5兲

that is, Eqs. 共4.4兲and共4.5兲form a first-order system. The principal part can be written as

⳵t

冉

^MHi

冊

^⯝

冉

^{⫺共1/2兲␣␦il⫹␤Rlil⫺␦ilRkm␥km ␤l␦ij⫹⫺Sijl2⫺␣␥␥mkjl␦ilSmkj}

冊

^⳵l

冉

^MHj

冊

^{. 共4.6兲}

The general discussion of the hyperbolicity and characteristic speed of the system 共4.6兲 is hard, so hereafter we restrict ourselves to the case

R_{i j}⫽␬1␣␥i j, S_{i j}^k⫽␬2␤^k␥i j, 共4.7兲 where we recover Eq. 共4.3兲by choosing ␬1⫽(␮⫺1)/4 and

␬2⫽0. The eigenvalues of Eq.共4.6兲then become

␭^l⫽„␤^l,␤^l,共1⫺␬2兲␤^l

⫾

冑

␣²␥^ll共1⫹4␬1兲⫹共␬2␤^l兲²… 共no sum over l兲 共4.8兲 and the hyperbolicity of Eq. 共4.6兲 can be classified as 共i兲 symmetric hyperbolic when␬1⫽3/2 and␬2⫽0, 共ii兲strongly hyperbolic when ␣²␥^ll(1⫹4␬1)⫹␬2

2(␤^l)2⬎0 where ␬1⫽

⫺1/4, and 共iii兲 weakly hyperbolic when ␣²␥^ll(1⫹4␬1)

⫹␬2

2(␤^l)2⭓0.

For the case of Eq. 共4.7兲 on a Minkowskii background metric, the linear order terms of the constraint propagation equations become

⳵l

冉

^{(1)(1)MHˆˆi}

冊

^⫽

^冉

^{⫺共1/2兲共10⫹4␬1兲iki ⫺2ik0 j}

^冊 冉

^{(1)(1)MHˆˆj}

冊

共4.9兲 whose Fourier transform gives the eigenvalues

⌳^l⫽„0,0,⫾

冑

⫺k²共1⫹4␬1兲…. 共4.10兲 That is 共two zeros, two purely imaginary兲 for the standard ADM system and共four zeros兲for the original ADM system.

Therefore, according to our conjecture, the standard ADM system is expected to have better stability than the original ADM system.

C. Detweiler’s system

1. Detweiler’s system and its constraint amplification Detweiler’s modification to the ADM system关14兴can be realized through one choice of the multipliers in Eqs. 共4.1兲 and 共4.2兲. He found that with a particular combination the

evolution of the energy norm of the constraints, H²⫹M², can be negative definite when we apply the maximal slicing condition K⫽0. 共We will comment more on his criteria in Sec. IV C 2.兲His adjustment can be written in our notation in Eqs.共4.1兲and共4.2兲as

P_{i j}⫽⫺L␣³␥i j, 共4.11兲 R_{i j}⫽L␣³共K_{i j}⫺共1/3兲K␥i j兲, 共4.12兲

S_{i j}^k⫽L␣²关3共⳵(i␣兲␦j)

k⫺共⳵l␣兲␥i j␥^kl兴, 共4.13兲

s_{i j}^kl⫽L␣³关␦(i k␦j)

l ⫺共1/3兲␥i j␥^kl兴, 共4.14兲

everything else zero, where L is a constant. Detweiler’s ad- justment, Eqs. 共4.12兲–共4.14兲, does not put the constraint propagation equation in first-order form, so we cannot dis- cuss hyperbolicity or the characteristic speed of the con- straints.

For the Minkowskii background spacetime, the adjusted constraint propagation equations with above choice of mul- tiplier become

⳵l

冉

^{(1)(1)HMˆˆ i}

冊

^⫽

^冉

^{⫺⫺共1/22Lk兲ik2 i ⫺共L/2兲k⫺2␦2ikij⫺共jL/6兲kikj}

^冊

⫻

冉

^{(1)(1)MHˆˆj}

冊

^{. 共4.15兲}

The eigenvalues of the Fourier transform are

⌳^l⫽„⫺共L/2兲k²,⫺共L/2兲k²,⫺共4L/3兲k²

⫾

冑

^k2关⫺1⫹共4/9兲L²k²兴…. 共4.16兲 This indicates negative real eigenvalues if we choose small positive L.

We confirmed numerically, using perturbation on Minkowskii spacetime, that Detweiler’s system presents bet- ter accuracy than the standard ADM system, but only for small positive L. See the Appendix.

2. Differences with Detweiler’s requirement

We comment here on the differences between Detweiler’s criteria for stable evolution and ours.

Detweiler calculated the L2 norm of the constraints C_␳ over the three-hypersurface and imposed the negative defi- niteness of its evolution:

Detweiler

⬘

^{s criteria⇔}⳵t

冕

^{C␳C␳dV⬍0, ᭙ nonzero C␳,}

共4.17兲 where C_␳C^␳⫽:G^␳␴C_␳C_␴ and G_␳␴⫽diag关1,␥i j兴 for the pair of C_␳⫽(H,Mi).

Assuming the constraint propagation to be ⳵tCˆ_␳⫽A_␳^␴Cˆ_␴ in the Fourier components, the time derivative of the L2 norm can be written as

⳵t共Cˆ

␳Cˆ␳兲⫽共A^␳␴⫹A¯␴␳⫹⳵tG¯␳␴兲Cˆ

␳C¯ˆ

␴. 共4.18兲 Together with the fact that the L2 norm is preserved by Fourier transform, we can say, for the case of a static back- ground metric,

Detweiler

⬘

^{s criteria}⇔eigenvalues

of共A⫹A^†兲are all negative᭙ k. 共4.19兲 On the other hand,

our criteria⇔eigenvalues of Aare all negative᭙ k.

共4.20兲 Therefore for the case of a static background, Detweiler’s criterion is stronger than ours. For example, the matrix

A⫽

冉

^{⫺01 ⫺a1}

冊

where a is constant, 共4.21兲 for the evolution system (Cˆ

1,Cˆ

2), satisfies our criterion but not Detweiler’s when兩a兩⭓

冑

2. This matrix, however, gives asymptotical decay for (Cˆ

1,Cˆ

2). Therefore we may say that Detweiler requires monotonic decay of the constraints, while we assume only asymptotical decay.

We remark that Detweiler’s truncations on higher-order terms in the C norm corresponds to our perturbational analy- sis; both are based on the idea that the deviations from the constraint surface 共the errors expressed as a nonzero con- straint value兲are initially small.

D. Another possible adjustment 1. Simplified Detweiler system

Similar to Detweiler’s equation 共4.11兲, we next consider only the adjustment

P_{i j}⫽␬0␣␥i j, 共4.22兲 all other multipliers being zero in Eqs. 共4.1兲and共4.2兲.

On the Minkowskii background, the Fourier components of the constraint propagation equation can be written as

⳵t

冉

^{(1)(1)HMˆˆ i}

冊

^⫽

^冉

^{⫺共21/2␬0k兲2iki ⫺2ik0 j}

^冊 冉

^{(1)(1)MHˆˆj}

冊

^,

共4.23兲 and the eigenvalues of the coefficient matrix are

⌳^l⫽„0,0,␬0k²⫾

冑

^k2共⫺1⫹␬0

2k²兲…. 共4.24兲 That is, the amplification factors become共0, 0, two negative real values兲for the choice of relatively small negative␬0.

We also confirmed that this system works as desired. We give a numerical example in the Appendix.

2. Adjusting the Hamiltonian constraint system Our final example is a combination of the one in Sec.

IV B and that above, that is,

P_{i j}⫽␬0␣␥i j, 共4.25兲 R_{i j}⫽␬1␣␥i j, 共4.26兲 all other multipliers being zero in Eqs.共4.1兲and共4.2兲. Simi- lar to the previous one, the Fourier-transformed constraint propagation equation is

⳵t

冉

^{(1)(1)MHˆˆi}

冊

^⫽

^冉

^{⫺共1/22兲ik␬0i⫺k22␬1iki ⫺2ik0 i}

^冊 冉

^{(1)(1)MHˆˆj}

冊

^,

共4.27兲 which gives the eigenvalues

⌳^l⫽„0,0,␬0k²⫾

冑

^k2共⫺1⫹␬0k²⫺4␬1兲…. 共4.28兲 We can expect a similar asymptotical stable evolution by choosing␬0and␬1, so as to make the eigenvalues共0, 0, two negative real values兲.

V. CONFORMAL-TRACELESS ADM SYSTEMS The so-called ‘‘conformally decoupled traceless ADM formulation’’ was first developed by the Kyoto group 关15兴. After the rediscovery that this formulation is more stable than the standard ADM formula by Baumgarte and Shapiro 关16兴, several groups began to use the CT-ADM formulation for their numerical codes and reported an advantage in sta- bility 关17,18兴. Along with this conformal decomposition, several hyperbolic formulations have also been proposed 关19–21兴, but they have not yet been applied to numerical simulations.

However, it is not yet clear why the CT-ADM formula- tion gives better stability than the ADM formulation. The Potsdam group 关22兴 found that the eigenvalues of the CT- ADM evolution equations have fewer ‘‘zero eigenvalues’’

than those of the ADM system, and they conjectured that instability can be caused by ‘‘zero eigenvalues’’ that violate the ‘‘gauge mode.’’ Miller关23兴applied von Neumann’s sta- bility analysis to plane wave propagation and reported that the CT-ADM formulation has a wider range of parameters that give us stable evolution. These studies provide support for the CT-ADM formulation in some sense, but on the

other hand, it is also shown that an example of an ill-posed solution in the CT-ADM formulation exists 共as well in the ADM formulation兲 关24兴.

Here, we apply our constraint propagation analysis to this CT-ADM system.

A. CT-ADM equations

Since one reported feature of the CT-ADM formulation is the use of the momentum constraint on the RHS of the evo- lution equations 关22兴, we here present the set of CT-ADM equations carefully for such a replacement of the constraint terms.

The widely used notation 关15,16兴is to use the variables (␾^,˜␥i j,K,A˜

i j,⌫˜ⁱ) instead of the standard ADM variables (␥i j,K_{i j}), where

␥

˜i j⫽e^⫺4␾␥i j, 共5.1兲 A˜

i j⫽e^⫺4␾共K_{i j}⫺共1/3兲␥i jK兲, 共5.2兲

⌫

˜ⁱ⫽⌫˜

jk

i ^˜␥^jk, 共5.3兲

and we impose det␥^˜i j⫽1 during the evolutions. The set of evolution equations becomes

共⳵t⫺L_␤兲␾⫽共⫺1/6兲␣^K, 共5.4兲

共⳵t⫺L_␤兲^˜␥i j⫽⫺2␣^A˜i j, 共5.5兲 共⳵t⫺L_␤兲K⫽␣共1⫺␬1兲R⁽³⁾⫹␣共1⫺␬1兲K²⫹␣␬1A˜

i jA˜^{i j}

⫹共1/3兲␣␬1K²⫺␥^{i j}共ⵜiⵜj␣兲, 共5.6兲 共⳵t⫺L_␤兲A˜

i j⫽⫺e^⫺4␾共ⵜiⵜj␣兲^TF⫹e^⫺4␾␣^Ri j (3)

⫺e^⫺4␾␣共1/3兲␥i j共1⫺␬3兲R⁽³⁾

⫹␣共KA˜

i j⫺2A˜

ikA˜

j

k兲⫹e^⫺4␾␣共1/3兲␥i j␬3

⫻关⫺A˜_klA˜^kl⫹共2/3兲K²兴, 共5.7兲

⳵t⌫˜ⁱ⫽⫺2共⳵j␣兲A˜^{i j}⫺共4/3兲␬2␣共⳵jK兲␥^{˜i j}

⫹12␬2␣^A˜ji共⳵j␾兲⫺2␣^A˜k j共⳵j␥^˜ik兲

⫺2␬2␣⌫˜

l j kA˜

k

j␥^˜il⫺2共1⫺␬2兲␣共⳵jA˜_kl兲␥^˜ik␥^˜jl

⫹2␣共1⫺␬2兲A˜

j

i⌫˜^j⫺⳵j„␤^k⳵k^˜␥^{i j}⫺␥^{˜k j}共⳵k␤ⁱ兲

⫺␥^˜ki共⳵k␤^j兲⫹共2/3兲␥^{˜i j}共⳵k␤^k兲…, 共5.8兲

whereL_␤ is the Lie derivative along the shift vector␤^{i, and} R⁽³⁾is the three-metric scalar curvature. Here we introduced parameters ␬ which show where we replace the terms with constraints. For example (␬1,␬2,␬3)⫽(0,0,0) is the case of no replacement 关the standard ADM equations expressed us- ing Eqs.共5.1兲–共5.3兲兴, while Baumgarte and Shapiro关16兴use (␬1,␬2,␬3)⫽(1,1,0).

The constraint equations in the CT-ADM system can be expressed as

H⫽e^⫺4␾R˜⁽³⁾⫺8e^⫺4␾␥^{˜i j}共⳵i⳵j␾兲⫺8e^⫺4␾˜␥^{i j}共⳵i␾兲共⳵j␾兲

⫹8e^⫺4␾共⳵i␾兲⌫˜ⁱ⫹共2/3兲K²⫺A˜_{i j}A˜^{i j}, 共5.9兲

Mi⫽共⳵jA˜

ki兲␥^{˜k j}⫺共2/3兲共⳵iK兲⫺A˜

ji⌫˜^j⫹6共⳵j␾兲A˜

i j⫺⌫˜

ji kA˜

k j, 共5.10兲

Gⁱ⫽⌫˜ⁱ⫹⳵j␥^˜ji. 共5.11兲

Here H,M are the Hamiltonian and momentum constraints and the third one, G, is a consistency relation due to the algebraic definition of Eq.共5.3兲.

B. Constraint propagation equations of the CT-ADM formulation

Similar to the ADM cases, we here show the propagation equations for Eqs.共5.9兲–共5.11兲. The expressions are given using Eqs. 共3.5兲and 共3.6兲, but we have to be careful to keep using the new variable ⌫i wherever it appears. Following关16兴, we express R˜

i j (3)as R˜

i j

(3)⫽⫺共1/2兲^˜␥^lm共⳵l⳵m␥^˜i j兲⫹共1/2兲^˜␥ki⳵j⌫˜^k⫹共1/2兲^˜␥k j⳵i⌫˜^k⫹共1/2兲⌫˜^k⌫˜_{(i j)k}⫹␥^˜lm⌫˜

li

k⌫˜_jkm⫹^˜␥^lm⌫˜

l j

k⌫˜_ikm⫹^˜␥^lm⌫˜

im k ⌫˜_{kl j}.

共5.12兲

The constraint propagation equations, then, are obtained by straightforward calculations as

⳵tH⫽␤^j共⳵jH兲⫺2␣^e⫺4␾␥^{˜i j}共⳵iMj兲⫹2␣^KH⫺2␣^e⫺4␾共⳵i␥^{˜i j}兲Mj⫺4␣^e⫺4␾共⳵i␾兲␥^{˜i j}Mj⫺4e^⫺4␾˜␥^{i j}共⳵j␣兲Mi

⫹2␬2e^⫺4␾共⳵i␣兲␥^{˜i j}Mj⫹2␬2e^⫺4␾␣共⳵i␥^{˜i j}兲Mj⫹2␬2e^⫺4␾␣␥^{˜i j}共⳵iMj兲⫹16␬2␣^e⫺4␾共⳵i␾兲␥^{˜i j}Mj⫺共4/3兲␬1␣^KH, 共5.13兲

⳵tMi⫽⫺共1/2兲␣共⳵iH兲⫹␤^j共⳵jMi兲⫹␣^KMi⫺共⳵i␣兲H⫺4␤^j共⳵i␾兲Mj⫹␤^k␥^˜jl共⳵i␥^˜lk兲Mj⫹共⳵i␤k兲e^⫺4␾␥^{˜k j}Mj

⫹共1/3兲共2␬1⫹␬3兲共⳵i␣兲H⫹共1/3兲共2␬1⫹␬3兲␣共⳵iH兲⫺2␬2␣^A˜i

jMj⫺共1/3兲␬3␣G^j␥^˜jiH⫹2␬3␣共⳵i␾兲H, 共5.14兲

⳵tGⁱ⫽2A˜

j

iG^j⫹2␬2␣␥^{˜i j}Mj. 共5.15兲

These form a first-order system, and the characteristic part can be extracted as

⳵t

冉

^HMGii

冊

^⬵

冉

^{关共2/3兲␬1⫹共1/3␤0兲l␬3⫺共1/2兲兴␣␦il 2共⫺1␤⫹0l␬␦i2j兲␣␥l j 000}

冊

^⳵l

冉

^HMGjj

冊

^{, 共5.16兲}

whose characteristic speeds are

␭^l⫽„0,0,0,␤^l,␤^l,␤^l⫾␣

冑

␥^ll共1⫺␬2兲关1⫺共4/3兲␬1⫺共2/3兲␬3兴… 共no sum over l兲. 共5.17兲 By analyzing the reality of the eigenvalues, the diagonalizability of the characteristic matrix, and the possibility of a symmetric characteristic matrix, we can classify the hyperbolicity of the system共5.16兲as

weakly hyperbolic⇔共1⫺␬2兲关1⫺共4/3兲␬1⫺共2/3兲␬3兴⭓0, 共5.18兲 strongly hyperbolic⇔共1⫺␬2兲⫽关1⫺共4/3兲␬1⫺共2/3兲␬3兴⫽0,

or 共1⫺␬2兲关1⫺共4/3兲␬1⫺共2/3兲␬3兴⬎0, 共5.19兲 symmetric hyperbolic⇔共⫺1⫹␬2兲⫽关1⫺共4/3兲␬1⫺共2/3兲␬3兴. 共5.20兲 That is, for the nonadjusted system (␬1,␬2,␬3)⫽(0,0,0), constraint propagation forms a strongly hyperbolic system, while the Baumgarte-Shapiro form gives only weak hyperbolicity. 共We note that the first-order version of the CT-ADM system by Frittelli and Reula 关20兴has also well-posed constraint propagation equations.兲

C. Amplification factors on a Minkowskii background For a Minkowskii background, the constraint propagation equations at linear order become

⳵t

冉

^{(1)(1)(1)HMGˆˆˆi i}

冊

^⫽

^冉

^{关共2/3兲␬1⫹共1/300兲␬3⫺共1/2兲兴iki 2共␬22␬⫺02␦1i j兲ikj 000}

^冊 冉

^{(1)(1)(1)HMGˆˆˆj j}

冊

^{. 共5.21兲}

The constraint amplification factor becomes

⌳^l⫽„0,0,0,0,0,⫾

冑

⫺k²共1⫺␬2兲关1⫺共4/3兲␬1⫺共2/3兲␬3兴…. 共5.22兲

That is,⌳^lare either zero, purely imaginary, or⫾ real num- bers. For the nonadjusted system they are zero and purely imaginary 关that is, the same as Eq. 共3.11兲兴, while the Baumgarte-Shapiro form gives us all zero eigenvalues.

Therefore, from our point of view, these two are not very different in their characterization of constraint propagation.

VI. CONCLUDING REMARKS

We have reviewed ADM systems from the point of view of the adjustment of the dynamical equations by constraint terms. We have shown that characteristic speeds and ampli- fication factors of the constraint propagation change due to their adjustments. We compared the equations for the ADM,

adjusted ADM, and conformal traceless ADM 共CT-ADM兲 systems, and tried to find a system that is robust for violation of the constraints, which we can call an ‘‘asymptotically constrained’’ system.

We conjectured that if the amplification factors共eigenval- ues of the coefficient matrix of the Fourier-transformed con- straint propagation equations兲are negative or purely imagi- nary, then the system has better asymptotically constrained features than a system where they are not. According to our conjecture, the standard ADM system is expected to have better stability than the original ADM system 共no growing mode in amplification factors兲. Detweiler’s modified ADM system, which is one particular choice of adjustment, defi- nitely has good properties in that there are no growing modes

in the amplification factors. We also showed that this can be obtained by a simpler choice of adjustment multipliers.

We also studied the CT-ADM system which is popular with numerical relativists nowadays. However, from our point of view, we do not see any particular advantages for the CT-ADM system over the standard ADM system.

The reader might ask why we can break the time-reversal invariant feature of the evolution equations by a particular

choice of adjusting multipliers against the fact that the ‘‘Ein- stein equations’’ are time-reversal invariant. This question can be answered by the following. If we take a time-reversal transformation (⳵t→⫺⳵t), the Hamiltonian constraint and the evolution equations of K_{i j} keep their signatures, while the momentum constraints and the evolution equations of␥i j

change their signatures. Therefore if we adjust the␥i j equa- tions using the Hamiltonian constraint and/or K_{i j} equations using the momentum constraints 共supposing the multiplier has⫹ parity兲, then we can break the time-reversal invariant feature of the ‘‘ADM equations.’’ In fact, the examples we obtained all obey this rule. The CT-ADM formulation keeps its signature against the adjustments we made, so that we cannot find any additional advantage from this analysis.

Considering the constraint propagation equations is a kind of substitutional approach for numerical integrations of the dynamical equations. However, this might be one of the main directions for our future research, as Friedrich and Nagy关25兴impose a zero speed of the constraint propagation as the first principle when they considered the initial bound- ary value problem of the Einstein equations 关26兴.

We are now applying our discussion to more general spacetimes and trying to find guidelines for choosing appro- priate gauge conditions from analysis of the constraint propagation equations. These efforts will be reported else- where关27兴.

ACKNOWLEDGMENTS

H.S. appreciates helpful comments by Pablo Laguna, Jorge Pullin, Manuel Tiglio, and the hospitality of the CGPG group. We also thank Steven Detweiler for communications.

We thank Bernard Kelly for a careful reading of the manu- script. This work was supported in part by NSF grant PHY00-90091 and the Everly research funds of Penn State.

H.S. was supported by the Japan Society for the Promotion of Science.

APPENDIX: NUMERICAL DEMONSTRATIONS OF ADJUSTED ADM SYSTEMS

We here show two numerical demonstrations of the ad- justed ADM systems that were discussed in Sec. IV C共Det- weiler’s modified ADM system兲and Sec. IV D 1共simplified version兲.

Detweiler’s adjustment, Eqs. 共4.12兲–共4.14兲, can be pa- rametrized by a constant L, and our prediction from the am- plification factor on the Minkowskii background is that this system will be asymptotically constrained for small positive L. Figure 1 is a demonstration of this system. We evolved Minkowskii spacetime numerically in a plane-symmetric spacetime and added artificial error in the middle of the evo- lution. Our numerical integration uses the Brailovskaya scheme, which was described in detail in our previous paper 关6兴. The code passes convergence tests and the plots are for 401 gridpoints in the range x⫽关0,10兴, and we fix the time grid ⌬t⫽0.2⌬x. The error was introduced as a pinpoint kick, in the form of⌬g_{y y}⫽10^⫺3 at x⫽5.0 and t⫽0.25. We monitor how the L2 norm of the constraints (H²⫹Mx

2) FIG. 1. Demonstration of the Detweiler’s modified ADM sys-

tem on Minkowskii background spacetime 共the system of Sec.

IV C兲. The L2 norm of the constraints is plotted as a function of time. Artificial error was added at t⫽0.25. Here L is the parameter used in Eqs.共4.12兲–共4.14兲. We see that the evolution is asymptoti- cally constrained for small L⬎0.

FIG. 2. Demonstration of the simplified Detweiler’s modified ADM system on Minkowskii background spacetime共the system of Sec. IV D 1兲. For comparison with Fig. 1, we set L⫽⫺␬0, where

␬0 is the parameter used in Eq. 共4.22兲. We see the evolution is asymptotically constrained for small L⬎0.

behaves. From Fig. 1, we see that a small positive L reduces the L2 norm in time, which is the asymptotically constrained feature we expected. The case of slightly larger L will make the system unstable. This is the same feature we have seen in the numerical demonstration of the ␭ system or adjusted Maxwell system and Ashtekar system关11兴; for that case the upper bound of the multiplier can be explained by viola-

tion of the Courant-Friedrich-Lewy condition, while in this system we cannot calculate the exact characteristics since the system is not first order.

Similarly, we plotted in Fig. 2 the case of a simplified version共the system of Sec. IV D 1兲. We see the desired fea- ture again by changing the parameter␬0 that appears in Eq.

共4.22兲.

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关27兴H. Shinkai and G. Yoneda共in preparation兲.