arX

iv:g

r-qc/

9903

030v

1 8

Mar

199

9

A Conformal Hyperbolic Formulation of the Einstein Equations

Miguel Alcubierre(1), Bernd Brugmann(1), Mark Miller(2) and Wai-Mo Suen(2,3)(1)Max-Planck-Institut fur Gravitationsphysik

Albert-Einstein-Institut

Schlaatzweg 1, D-14473, Potsdam, Germany(2)McDonnell Center for the Space Sciences

Department of Physics, Washington University, St. Louis, Missouri 63130(3)Physics Department

Chinese University of Hong Kong, Hong Kong

(January 2, 2014)

We propose a re-formulation of the Einstein evolution equations that cleanly separates the con-formal degrees of freedom and the non-conformal degrees of freedom with the latter satisfying a firstorder strongly hyperbolic system. The conformal degrees of freedom are taken to be determined bythe choice of slicing and the initial data, and are regarded as given functions (along with the lapseand the shift) in the hyperbolic part of the evolution.We find that there is a two parameter family of hyperbolic systems for the non-conformal degrees

of freedom for a given set of trace free variables. The two parameters are uniquely fixed if werequire the system to be consistently trace-free, i.e., the time derivatives of the trace free variablesremains trace-free to the principal part, even in the presence of constraint violations due to numericaltruncation error. We show that by forming linear combinations of the trace free variables a conformalhyperbolic system with only physical characteristic speeds can also be constructed.

a. INTRODUCTION. With the advent of largeamounts of observational data from high-energy astron-omy and gravitational wave astronomy, general relativis-tic astrophysics astrophysics involving gravitationalfields so strong and dynamical that the full Einstein fieldequations are required for its accurate description isemerging as an exciting research area. This calls for anunderstanding of the Einstein theory in its non-linear anddynamical regime, in order to study the physics of gen-eral relativistic events in a realistic astrophysical environ-ment. This in turn calls for solving the full set of Einsteinequations numerically. However, the complicated set ofpartial differential equations present major difficulties inall of these three tightly coupled areas: the understand-ing of its mathematical structure, the derivation of itsphysical consequences, and its numerical solution. Thedifficulties have attracted a lot of recent effort, includ-ing two Grand Challenge [1,2] efforts on the numericalstudies of black holes and neutron stars, respectively.

One major obstacle in solving the Einstein equationsnumerically is that we lack a complete understanding ofthe mathematical structure of the Einstein equations.The difficulties in numerically integrating the Einsteinequations in a stable fashion have motivated intense ef-fort in rewriting the Einstein equations into a form thatis explicitly well-posed [318] (for an excellent overviewsee [19]). The main idea has been re-writing the sixspace-space components of the Einstein equations intoa first order hyperbolic system. These space-space partsof the Einstein equations are dynamical evolution equa-tions, while the time-time and space-time parts of theEinstein equations are (elliptic) constraint equations.

The central question we raise in this communication

is: In order to enable an accurate and stable numericalintegration of the full set of the Einstein equations, whatpart of the system should be taken to form a hyperbolicsystem?

Our question is motivated by two observations. First,there are many recent proposals on re-formulating the sixspace-space parts of the Einstein equations into a firstorder hyperbolic system [818]. Three of the hyperbolicformulations have been coded up for numerical treatment(to the best of our knowledge), namely, the York et. al.formulation [9,14,16] (see, e.g., [20,21]), the Bona-Massoformulation [15] (see e.g., [17]), and Friedrichs formula-tion [3,4,2226] (which is rather different from the firsttwo formulations in its use of a global conformal trans-formation of the four-metric to compactify hyperboloidalslices). However, in all these cases, the numerical integra-tion of the first order hyperbolic system consisting of thesix space-space components of the Einstein equations sofar have not lead to a substantial improvement over thoseusing the traditional ADM [27] evolution equations. Thisis despite the original hope that the well-posedness of thehyperbolic formulations leads to an immediate numericaladvantage.

The second observation is that there have been vari-ous attempts in re-writing the traditional ADM form ofthe evolution equations by separating out the conformaldegree of freedom, beginning with Nakamura et. al. [28](see references cited therein). Lately this has receivedmuch attention with [29] reporting that a variant of theapproach leads to highly stable numerical evolutions. Adetailed study of the approach using gravitational wavesystems carried out by our group [30] confirmed that theapproach has advantages over the standard ADM for-

mulation. We find that the approach yields results withaccuracy comparable to that obtained by the standardADM formulation with the K-driving technique [31] forweak to medium waves, and has better stability proper-ties especially in the case of strong fields that needs highresolution with ADM [32] (see also [33]).These two observations motivated us to study the pos-

sibility of a formulation that separates out the confor-mal degree of freedom in the 6 evolution equations, whilerequiring the remaining 5 equations governing the non-conformal degrees of freedom to form a first order hyper-bolic system.A re-cap of the various components of the Einstein

equations is in order for a clearer discussion of our ap-proach. In the standard ADM 3+1 formulation, the Ein-stein equations are broken into (a) the Hamiltonian con-straint equation (the time-time part),

H = (3)R+K2 KijKij 16

ADM= 0, (1)

(b) the 3 momentum constraint equations (the time-space part)

Hi = jKij ijjK 8j

i = 0, (2)

where ADM

, ji, Sij , S = gijSij are the components of the

stress energy tensor projected onto the 3-space, and (c)the 6 evolution equations (the space-space part) given as12 first order equations

tgij = 2Kij, (3)

tKij = ij+ (Rij +K Kij 2KimKmj

8(Sij 1

2gijS) 4ADM gij), (4)

where i denotes a covariant derivative with respect tothe 3-metric gij , t stands for tL with L being theLie derivative with respect to i, and Rij is the Ricci cur-vature of the 3-metric. In the ADM formulation, Eqs. (3,4) are used to evolve the 12 variables Kij , gij forward intime for given lapse and shift vector i. The constraintequations are automatically satisfied if {Kij , gij} satisfythem on the initial time slice. However, in numerical evo-lutions the constraints will be violated due to truncationerror. One major difficulty in numerical relativity is thatthe constraint violations often drive the development ofinstabilities, at least in the case of numerical evolutionusing the standard ADM equations (3, 4).In the hyperbolic re-formulations of the evolution equa-

tions [912,1417], one makes use of the constraint equa-tions (1),(2), and introduces additional variables (e.g.,dijk = gij,k or its linear combinations) to cast Eqs. (3 ,4) into a first order strongly hyperbolic system (often thesymmetric hyperbolic subclass). (More variables wouldhave to be introduced for formulations involving higherderivatives [9,11].) However, we note that hyperbolicityis often shown only under the assumption that some of

the variables involved in the evolution equations, in par-ticular the lapse and the shift j , are considered asgiven functions of space and time. In actual numericalevolutions with no pre-determined choice of spacetimecoordinates, and j have to be given in terms of thevariables {Kij, gij , dijk} (e.g., ,

j determined in a setof elliptic equations involving {Kij, gij , dijk}). In theBona-Masso formulation [15], the lapse can be part of thehyperbolic system for some choice of slicings (while theinclusion of the shift into the hyperbolic system severelyrestricts the class of applicable shifts). In [9,11], in addi-tion to the lapse and the shift, the trace of the extrinsiccurvature, K = gijKij , is also regarded as a given func-tion (K is used to specify the slicing, e.g., K = 0 formaximal slicing). The point we want to bring out hereis that in all of the existing hyperbolic re-formulationsof the Einstein evolution equations, part of the quanti-ties {Kij , gij , dijk, ,

j} are considered to be given, whileothers are evolved using hyperbolic equations.

In the following, we present a formulation in which thenon-conformal degrees of freedom are separated out forhyperbolic evolution.

b. FORMULATION. For the evolution of the three-geometry, the conformal degree of freedom is representedby g (the determinant of the spatial 3-metric gij), itsspatial derivative g,i and its time derivative K (K =1/(2g)tg). For the non-conformal degrees of free-dom, we define

gij = gij/g1/3, (5)

Aij = (Kij 1

3gijK)/g

1/3, (6)

Dijk = gij,k. (7)

gij has unit determinant, and Aij is the rescaled trace-free part of Kij . All indices of tilde quantities are raisedand lowered with the conformal 3-metric gij . We note

that Dijk is trace-free with respect to the indices (i, j).We take {gij , Dijk, A

ij}, or their covariant componentcounterparts, to represent the non-conformal degrees offreedom.

In the following we develop a first order hyperbolic sys-tem for the non-conformal degrees of freedom, under thesimplifying assumption that the 5 conformal degrees offreedom {g, g,i,K} and the gauge choice functions {,

i}can be regarded as given functions of space and time.Note that these variables cannot be specified indepen-dently of each other. A concrete example is that of max-imal slicing, K = 0, and vanishing shift, i = 0, in whichcase both g and g,i are part of the initial data (time in-dependent), and are therefore truely given functions inthe numerical evolution. In other cases, with K given tospecify the slicing, it involves a non-trivial time integra-tion to determine g (from the definition of K in terms ofthe time derivative of g).

We now discuss hyperbolicity of the evolution of thenon-conformal variables, {gij, D

ijk, A

ij}, by examiningthe principal part of the evolution equations, which is thepart that decides about strong hyperbolicity of the sys-tem [34]. To obtain the principal part we drop all termsthat can be expressed by (1) the variables {gij , Dijk, A

ij}themselves, and (2) spacetime functions that are re-garded as given, i.e. {, i, g, g,i,K} and their space andtime derivatives. We have

tgij 0, (8)

tDijk 2A

ij,k, (9)

tAij g1/3(Rij

1

3gijR), (10)

where represents equal up to principal part, andwhere for the evolution equation of Dijk we have usedthat spatial derivatives i and the time derivative t com-mute.To evaluate Rij and R in (10), we use

Rij g2/3Rij (11)

1

2g2/3(gklDijk,l g

ilDjkl,k gjlDikl,k), (12)

where the relation

gklgkl,i = g,i/g 0, (13)

and the spatial derivatives of it have been used. We ob-tain

tAij

1

2g1/3(gklDijk,l g

ilDjkl,k gjlDikl,k

+2

3gijDklk,l). (14)

To make the non-conformal system strongly hyper-bolic, one can add a combination of the momentum con-straint to (9). To principal part the momentum con-straint (2) is Hi g1/3Aij ,j. We obtain

tDijk 2A

ij,k 2g

1/3(gikHj + gjkH

i) (15)

2(Aij ,k gikA

jl,l g

jkA

il,l). (16)

An energy norm can be constructed for the system:

E =

gij gij + A

ijAij +1

4g1/3DijkDij

k. (17)

It is straightforward to demonstrate using (8), (14), and(16) that tE is a total derivative up to terms that canbe expressed by the variables {gij, Aij , Dijk} themselves.One can also show directly that the characteristic metricof the system (8), (14), and (16) has a complete set ofeigenvectors with real eigen values. The system is similarto but not contained in the one parameter family of thehyperbolic systems constructed in [10].

Next we go one step beyond hyperbolicity. We makethe following observations:(i) Since Aij and Dijk are trace-free, one can add aterm 1g

1/3gijH to (14), and a term 2gijHk to (16)

without affecting the hyperbolicity. We have therefore atwo parameter family of hyperbolic evolution equations(without making a variable change).(ii) With these two terms added respectively to (14) and(16), the trace of the principle parts of the RHSs of theequations are 31D

ksk,s (proportional to the principal

part of the Hamiltonian constraint), and (32 4)Akl,l

(proportional to the principal part of the momentum con-straint), respectively. On the other hand, the LHS of theequations, tA

ij and tDijk are trace-free to the princi-

pal order. This means that truncation error in the nu-merical evolution which leads to a violation of the con-straints will drive Aij and Dijk to evolve away from beingtrace-free, even up to the principal order.(iii) We therefore propose to fix the freedom in the pa-rameters 1 and 2 by requiring the system to be consis-tently trace-free, i.e., 1 = 0 and 2 = 4/3, so that theequations are trace-free to principal order consistently.Hence (14) for Aij is left unchanged, but

tDijk

2Aij ,k 2g1/3(gikH

j + gjkHi) +

4

3gijHk (18)

2(Aij ,k gikA

jl,l g

jkA

il,l +

2

3gij gkmA

ml,l). (19)

The system {(8),(14),(19)} forms a strongly hyperbolicsystem with the same energy norm (17).(iv)The remaining freedom in constructing conformal-hyperbolic systems that are consistently trace-free isthrough forming linear combinations of the variables.There are clearly infinite choices. Here we show for ex-ample a linear combination that leads to a system withonly physical characteristic speeds, a property advocatedby York et. al., see e.g., [14]. (14) can be written as

tAij (g1/3)gkllU

ijk g

kllUijk, (20)

where

U ijk =1

2(Dijk g

ikD

ill g

jkD

jll +

2

3gij gkmD

mll). (21)

We can take U ijk to be our basic non-conformal variables(note gijU

ijk = 0). Taking the time derivative of U

ijk

and commuting time and space derivatives leads to

tUijk (A

ij,k g

ikA

jl,l g

jkA

il,l +

2

3gij gkmA

ml,l).

(22)

To make the system strongly hyperbolic, we follow thestep leading to (15) and add the combination of momen-tum constraints g1/3(gikH

j+ gjkHi)2gijHk/3 to (22)

to arrive at

tUijk A

ij,k. (23)

(20) and (23) form a conformal hyperbolic system for{U ijk, A

ij} with only physical characteristic speeds. Thesystem can be symmetrized by contracting (23) with gkl.

c. DISCUSSION AND CONCLUSION. We raise thequestion of what part of the variables in the Einstein the-ory should be evolved in a hyperbolic fashion in numericalrelativity. We propose a re-formulation of the Einsteinevolution equations that cleanly separate the conformaldegrees of freedom {g, g,i,K} and the non-conformal de-

grees of freedom {gij, Dijk, Aij} (or their linear combi-

nations), with the latter satisfying a first order stronglyhyperbolic system. The conformal degrees of freedom aretaken to be determined by the choice of slicings and theinitial data, and are regarded as given functions in thehyperbolic part of the evolution equations, along withthe lapse and the shift.

We find a two parameter family of non-conformal hy-perbolic system for {gij , Dijk, A

ij}. The two parametersare uniquely fixed if we require the system to be consis-tently trace-free, i.e., the time derivative of the trace-free variables {gij , Dijk, A

ij} remains trace-free to prin-cipal part, even in the presence of constraint violationscaused by numerical truncation error. We also show thatcertain linear combinations of the Dijk lead to a confor-mal hyperbolic system with physical characteristic speed.

This formulation merges two recent trends in re-writing the Einstein evolution equations for numericalrelativity: first order hyperbolicity and the separatingout of the conformal degrees of freedom. We believe itwill lead to many interesting investigations: Given thecoordinate conditions, e.g., maximal slicing and an ap-propriate shift condition, can the combined elliptic hy-perbolic system be shown to be well-posed analytically[9,35]? When posted as initial boundary value problem,what are the suitable boundary conditions for stabilityin numerical evolutions? How will the constraints prop-agate under this system of conformal-hyperbolic equa-tions? One particularly interesting issue that will be re-ported on in a follow up paper is the stability of thisformulation in numerical evolution, and how the stabil-ity is related to the slicing conditions (K) one chooses.

This research is supported by the NSF grant Phy96-00507, and NASA HPCC Grand Challenge AwardNCCS5-153. We thank Carles Bona, Helmut Friedrich,Alan Rendall, and Ed Seidel for discussions.

[1] For the Binary Black Hole Grand Challenge, seewww.npac.syr.edu/projects/bh/.

[2] For the NASA Neutron Star Grand Challenge Project,see, e.g., http://wugrav.wustl.edu/Relativ/nsgc.html.

[3] H. Friedrich, Proc. Roy. Soc. London A 375, 169 (1981).[4] H. Friedrich, Proc. Roy. Soc. London A 378, 401 (1981).[5] Y. Choquet-Bruhat and T. Ruggeri, Comm. Math. Phys

89, 269 (1983).[6] H. Friedrich, Comm. Math. Phys. 100, 525 (1985).[7] C. Bona and J. Masso, Phys. Rev. Lett. 68, 1097 (1992).[8] C. Bona, J. Masso, E. Seidel, and J. Stela, Phys. Rev.

Lett. 75, 600 (1995).[9] Y. Choquet-Bruhat and J. York, C. R. Acad. Sc. Paris

321, 1089 (1995).[10] S. Fritelli and O. Reula, Phys. Rev. Lett. 76, 4667 (1996).[11] A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.

York, Phys. Rev. Lett. 75, 3377 (1995).[12] H. Friedrich, Class. Quant. Grav. 13, 1451 (1996).[13] M. H. van Putten and D. Eardley, Phys. Rev. D 53, 3056

(1996).[14] A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.

York, C.R. Acad. Sci. Paris 835 (1996).[15] C. Bona, J. Masso, E. Seidel, and J. Stela, Phys. Rev. D

56, 3405 (1997).[16] A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.

York, Class.Quant.Grav A9 (1997).[17] C. Bona, J. Masso, E. Seidel, and P. Walker, (1998), gr-

qc/9804065. Submitted to Physical Review D.[18] A. Anderson and J. York, gr-qc/9901021 (1999).[19] O. Reula, Living Reviews in Relativity 1, (1998).[20] M. Scheel et al., Phys. Rev. D 56, 6320 (1997).[21] M. A. Scheel et al., Phys. Rev. D .[22] P. Hubner, Ph.D. thesis, Ludwig Maximilians Universitat

Munchen, 1993.[23] P. Hubner, Phys. Rev. D 53, 701 (1996).[24] P. Hubner, gr-qc/9804065 (1998).[25] J. Frauendiener, Phys. Rev. D 58, 064002 (1998).[26] J. Frauendiener, Phys. Rev. D 58, 064003 (1998).[27] R. Arnowitt, S. Deser, and C. W. Misner, in Gravita-

tion: An Introduction to Current Research, edited by L.Witten (John Wiley, New York, 1962), pp. 227265.

[28] M. Shibata and T. Nakamura, Phys. Rev. D 52, 5428(1995).

[29] T. Baumgarte and S. Shapiro, Physical Review D 59,024007 (1999).

[30] M. Alcubierre and et. al., (1999), to be submitted.[31] J. Balakrishna et al., Class. Quant. Grav. 13, L135

(1996).[32] M. Alcubierre and et. al., (1999), in preparation for Phys.

Rev. D.[33] A. Arbona, C. Bona, J. Masso, and J. Stela, gr-

qc/9902053 (1999).[34] B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time depen-

dent problems and difference methods (Wiley, New York,1995).

[35] L. Andersson and V. Moncrief, (1998), in preparation.