Numerical relativity: (Almost) all you need to know

Seth Hopper

Vitor Cardoso

Fig: Ian Hinder CERN - November 16, 2016 Binary black holes come in a few different varieties

396 Chapter 12 Binary black hole initial data

quasi-circular plunge & ringdown inspiral merger

4 BHPT + 3 Post-Newtonian Theory PN ) h(t)

r/M 2 Black Hole

log( Perturbation 1 Numerical Theory t Relativity

0 01234 post-Newtonian numerical perturbation log(M/µ) techniques relativity methods Figure 12.1 The different phases of compact binary inspiral and coalescence. The gravitational wave amplitude Fig: Forseth et al. h(t) is sketched schematically and the analysisFig: Baumgarte technique is identified& Shapiro for each phase.

Here ij is the reduced quadrupole moment, I 1 2 3 1 2 jk ρ x j xk δ jkr d x m A x j xk δ jkr , (12.7) I ≡ − 3 = A − 3 ! " # $ " # the bracket denotes an average over several orbital periods, and the triple dot denotes ⟨⟩ the third time derivative. For a binary at large separation we may treat the stars as point i masses and insert their stellar masses m A and Newtonian trajectories x (t) into equa- tion (12.7) to evaluate equations (12.5) and (12.6). For a binary orbit with eccentricity e the emission of gravitational radiation always leads to a decrease in the eccentricity, e˙ < 0.3 Put differently, gravitational radiation circularizes elliptical orbits.Thisresultimplies that during the late stages of compact binary inspiral, we may approximate the orbit as circular.4

Exercise 12.1 Consider a Newtonian binary consisting of two point masses m1 and m2 at a binary separation r. Write the binary’s Hamiltonian H(r, φ, Pr , Pφ), which is equal to its conserved energy E,as

1 P2 1 P2 µM E H r φ , (12.8) = = 2 µ + 2 µr 2 − r

where M m1 m2 is the total mass and µ m1m2/M is the reduced mass. Define = + = $orb φ˙ to be the orbital angular velocity, and J Pφ to be the orbital angular momentum.≡ ≡

3 Peters (1964); see also Lightman et al. (1975), Problem 18.7. 4 See also exercise 12.2,whichshowsthatoncetheybecomecircular,theorbitsremaincircularastheyshrinkinradius. This is a video simulating the second detection, GW151226

M1 = 14.2 M⊙ M2 = 7.5 M⊙ • Simulation: Simulating eXtreme Spacetime project MFinal = 20.8 M⊙ MGWs = 1 M⊙ Outline

History and formalism Initial data

Evolution of field equations Wave extraction

The tools we use There is a long history of successes and failures in numerical relativity

Darmois Shibata & Nakamura Arnowitt, Deser, & Lichnerowicz Pretorius Misner Baumgarte & Shapiro Hahn & Fourès-Bruhat Lindquist

1927 1944 1956 1962 1964 Long, long 1995 1998 2005 years pass “Grand Challenge” Here are two good resources 330 HAHN AND LINDQUIST

This time is to be compared with the time torit at which the throat collapses to Hahn & Lindquistzero circumference did the : first work in 1964

terit W ?rm M 2$.

We have used here a limiting form for Eq. (2.20), m m 7r2/2p:,TWO-BODY PROBLEM which IN GEOMETRODYNAMICS is valid 307 ANNALS OF PHYSICS: forf8, 304-331small (1964)~0 . By choosing ~0 sufficiently small, one can evidently make the ratio tl/tcrit as small as one pleases. The Two-BodyThis ratherProblem surprisingin Geometrodynamics result does not contradict the analysis of Fuller and

Wheeler, SUSANfor it G.applies HAHN only to the case of strongly interacting masses. There are good reasons for regarding the latter system as a single compositeI / particle\ , with an (a) (b) International internalBusiness Machinesstructure”; Corporation, violationsNew York, Newof causalityYork within this super-particle would not be at variance ANDwith commonly accepted physical principles. Instead one should ask whether one can send a light ray from the asymptotic region through the RICHARD W. LINDQUIST wormhole and back out to infinity without encountering a singularity in the Adelphi University, Garden City, New York metric. We believe this to be impossible, although we have not been able(C) to prove FIG. 2. Three models for the two-body problems are displayed as two-dimensional sur- The problem itof twoconclusively. interacting masses is investigated within the framework faces imbedded in flat a-space. The first, (a), a direct generalization of Fig. 1, shows a pair of Einstein-Rosen bridges whose upper sheets are joined together. In (b) their lower sheets of geometrodynamics. In It summary,is assumed that thethe space-timenumerical continuum solutionis free of of the Einsteinhave been joined fieldas well, toequations form a multiply connectedpresents space whose upper and lower sheets all real sources of mass or charge; particles are identified with multiply con- are equivalent (i.e. isometric). If the two masses represented by the two mouths are equal, the figure will also be symmetric under reflections in a vertical plane midway between them. nected regions noof empty insurmountable space. Particular attentiondifficulties. is focused onMuch an asymp- still remains These tosymmetries be done,then allow onehowever, to identify corresponding in thepoints on the two sheets. and totically flat space containing a “handle” or “wormhole.” When the two also to identify corresponding points at the two mout,hs; the space which results is shown investigation both of stable difference schemes (a inproof (c). of stability being one of “mouths” of the wormhole are well separated, they seem to appear as two cen- ters of gravitational the attractionoutstanding of equal mass.unsolved To simplify problems)the problem, it andis of coordinatemodel was firstconditions proposed by Wheeler that (1) andare later calledwell by him a “wormhole” assumed that the metric is invariant under rotations about the axis of sym- manifold (2). In order that the two mouths shall match up sufficiently smoothly metry, and symmetricsuited with torespect numerical to the time work.t = 0 of maximumThe practicalseparation impossibility(i.e., analytically),of carrying they must bethe identical; numerical thus this model describes a universe of the two mouths. Analytic initial value data for this case have been ob- containing two particles of equal mass.5 tained by Misner; solutionthese contain sufficientlytwo arbitrary parameters,far into whichthe arefuture uniquely limits the conclusionsWhile there is no compellingwhich reasoncan for bepreferring drawn one model over another, we have focused attention exclusively on the wormhole picture. As far as the purely determined when aboutthe mass theof the dynamicaltwo mouths and theirbehavior initial separation of thehave wormhole system. Nevertheless, one sees been specified. We treat a particular case in which the ratio of mass to initial 6 The present investigation has been restricted to the case of pure gravitation, for which the appropriate field equations are those given in Eq. (l.la) with the right-hand side set separation is approximatelyevidence one-half.for a Togravitational determine a unique collapsesolution of theof each mouth,equal to zero analogous(i.e., with F,, = 0). Oneto can, thathowever, alsoof investigate the solutions to the corn remaining (dynamic) field equations, the coordinate conditions go- = -& are plete set of equations (1.1) on this manifold. In this more general approach it is also possible Schwarzschild metric, together with an interactionto trap electricbetween lines of force inthe the “wormhole,”two givingof them.the appearance of two particles with imposed; then the set of second order equations is transformed into a quasi- equal mass and equal but opposite charge. The time-symmetric initial value problem for this linear first order Thesesystem andtwo the effectsdifference schemecan onlyof Friedrichs be properlyused to ob- disentangledcase has beenthrough solved by Lindquistmeasurements (13). in tain a numerical solution. Its behavior agrees qualitatively with that of the one-body problem, theand asymptoticcan be interpreted region;as a mutual attractionwith theand pinching-limited data at our disposal, such an analysis off of the two mouthshas notof the beenwormhole. possible. I. INTRODUCTION Wheeler (1, 2) has used the term “geometrodynamics” to characterizeACKNOWLEDGMENTS those solutions of the field equations for gravitation and electromagnetism’ We are grateful to Professor J. A. Wheeler and Professor C. W. Misner for many valuable 41 =discussions R,v - ?4 g&about = 2(F,,FPthe formulation- Pi gj,.F,sF=B) of the problem(l.la) and interpretation of the results, and to Professor FPu;vR. 0. = Friedrichs0 for much useful (l.lb)advice on the mathematical approach. We also

1 Throughout this paperthank Greek Professorsubscripts andM. superscriptsFerentz rangefor hisfrom assistance0 to 3 and Latin with the analysis and programming during ones from 1 to 3. Also, theunits earlierare chosen stagesso that Gof (universalthe work.gravitation constant) = c = 1. 304 RECEIVED: December 9, 1963.

REFERENCES 1. J. A. WHEELER, Phys. Rev. 97,511 (1955). 2. C. W. MISNER AND J. A. WHEELER, Ann. Phys. (N.Y.) 2, 525 (1957).

15 See, e.g., Brill and Lindquist (18). 2.2 Maxwell’s equations in Minkowski spacetime 27

We denote the 4-dimensional spacetime metric by gab, the 3-dimensional spatial metric by γij, and its conformally related metric byγ ¯ij. All of these are objects that we will encounter in later chapters. Four-dimensional objects associated with gab are denoted with (4) asuperscript2.2in Maxwell’s front of equations the symbol, in Minkowski objects spacetime associated with27 γ ¯ij carry a bar, and objects i ¯ i related to γij carry no decorations. For example, " jk is associated with γij, " jk withγ ¯ij, We denote the 4-dimensional(4) i spacetime metric by gab, the 3-dimensional spatial metric ¯ and " jk with gab.ThecovariantderivativeoperatorisdenotedwithDi and Di when by γij, and its conformallyassociated related with metric the spatial byγ ¯ij. metric All of and these the are conformally objects that related we will metric, respectively, but with encounter in later chapters. Four-dimensional objects associated with gab are denotedThe with 3+1 decompostion of (4) the nabla symbol a when associated with the 4-dimensional metric gab.Weoccasionally asuperscript in front of the symbol, objects∇flat associated withγ ¯ij carry a bar, and objects use the symbol # for thei flat scalar Laplace2 operator.¯ i related to γij carry no decorations. For example, " jk is associated with γij, " jk withγ ¯ij, (4) i We denote the symmetric and antisymmetric partsEinstein’s¯ of a tensor with brackets equations( ) and [ ] and " jk with gab.ThecovariantderivativeoperatorisdenotedwithDi and Di when associated with the spatialaround metric indices and the in conformally the usual way. related For metric, example respectively, but with the nabla symbol a when associated with the 4-dimensional metric gab.Weoccasionally ∇flat 1 1 use the symbol # for the flat scalar LaplaceT(ab operator.) (Tab Tba)andT[ab] (Tab Tba) (2.4) We denote the symmetric and antisymmetric parts≡ 2 of a tensor+ with brackets (≡ ) and2 [ ] − around indices in the usual way. For example represent the symmetrized and antisymmetrizedThe major purpose tensors constructed of this book from is toTab describe.Wewritea how to determine the dynamical evolution flat 4-dimensional1 spacetime metric1 as ηab (Minkowski spacetime) and a flat 3-dimensional T (T T )andT (T of aT physical) system(2.4) governed by Einstein’s equations of general relativity. For all but the (abspatial) metricab ba as η ; these[ab] symbolsab areba meant to apply in any coordinate system. Only ≡ 2 + ij ≡ 2 simplest− systems, analytic solutions for the evolution of such systems do not exist. Hence when specifically stated will η denote the Minkowski metric in Cartesian (inertial) represent the symmetrized and antisymmetrized tensors constructedab the task from ofTab solving.Wewritea Einstein’s equations must be performed numerically on a computer. flat 4-dimensional spacetimecoordinates metric withas ηab components(Minkowski spacetime) diag( 1, and1, 1 a, flat1). 3-dimensional Finally, we refer to a 4-dimensional line 2 −To construct algorithms to do this we first have to recast Einstein’s 4-dimensional field spatial metric as ηij; theseinterval symbols in spacetime are meant as tods applyand in aany 3-dimensionalcoordinate system. line interval Only on a spatial hypersurface 2 equations (1.32) into a form that is suitable for numerical integration. In this chapter we when specifically statedas dl will. ηab denote the Minkowski metric in Cartesian (inertial) coordinates with components diag( 1, 1, 1, 1). Finally, we referpresent to a 4-dimensional such a formulation. line − interval in spacetime as ds2 and a 3-dimensional line interval onThe a spatial problem hypersurface of evolving the gravitational field in general relativity can be posed in as dl2. 2.2 Maxwell’s equationsterms of in a traditional Minkowski initial spacetime value problem or “Cauchy” problem. This is a fundamental problem arising in the mathematical theory of partial differential equations. In classical Many of the concepts that we will encounter in this chapter are more transparent in the dynamics, the evolution of a system is uniquely determined by the initial positions and 2.2 Maxwell’ssimpler equations framework of in electromagnetism Minkowski spacetime in special relativity as described by Maxwell’s The Maxwell equations serve as a guidevelocities of its constituents. By analogy, the evolution of a general relativistic gravitational Many of the conceptsequations. that we will In encounter several places in this chapter throughout are more this transparent book we will in the return to electromagnetism to simpler framework ofillustrate electromagnetism various features in special of relativity Einstein’sfield as equations. described is determined by Maxwell’s by specifying the metric quantities gab and ∂t gab at a given (initial) equations. In several placesMaxwell’s throughout equations this book naturally we will split returninstant into to electromagnetism two of groups. time t. The In particular, to first group we can need be written to specify as the metric field components and their illustrate various features of Einstein’s equations. first time derivatives everywhere on some 3-dimensional spacelike hypersurface labeled 28 Chapter 2 The 3+1 decompostioni of Einstein’s0 equations Maxwell’s equations naturally24 split intoChapter two groups. 2 The The 3+1 firstE by decompostion groupD coordinatei E can4 beπρ writtenx of0(2.5) Einstein’s ast constant equations. The different points on this surface are distinguished C ≡ − = = = i i by their spatiali coordinates x .Nowthesemetricquantitiescanbeintegratedforwardin2 constraint 28 Chapter 2 The 3+1 decompostionETheD secondi E of Einstein’s4 groupπρ 0(2.5) of equations MaxwellB D equationsi B 0, is (2.6) equations 2 equationsC ≡ − = timeC ≡ provided= we can obtain from the Einstein field equations expressions for ∂ gab at all Maxwell’s i t i B i Di B 0, j(2.6)k The second groupwhere of MaxwellE andC equations≡B are= the is electricpoints and the∂t onE magnetici theϵ hypersurface.ijkD fieldsB and4π Thatjiρ is way the charge we can den- integrate these(2.7) expressions to compute g equations = a0 − a0 ab i i G 8πT (2.2) where E and B aresity. the Here electric and and throughout, the magneticjDik fieldsdenotesand and a∂ρ spatial,t gisab theon covariant charge a new spacelike den-j derivativek hypersurface with6 evolution respect at to some the new time t δt, and then, by repeating ∂t Ei ϵijkD B 4π ji ∂t Bi ϵijkD=E ,(2.7) (2.8) i 0 i + 1 sity. Here and throughout,coordinateDi denotesx .InflatspaceandCartesiancoordinates,itreducestoanordinarypartial a spatial,= covariant− derivative with= respect− to the equations j k the process, obtain gab for all other points x and x in the (future) spacetime. i do not∂ B furnishi ϵ anyD E of, the information required(2.8) for the dynamical evolution of the fields. coordinate x .InflatspaceandCartesiancoordinates,itreducestoanordinarypartialderivative.where tj i is theijk charge 3-current.Obtaining These the equations appropriate describe expressions how the fields for ∂2 evolveg for forward such an integration is not so trivial. Rather, they= − supply four constraints on the initial data, i.e., four relationst ab between g and 24derivative.whereChapterj i is 2 the The charge 3+1The 3-current. decompostion abovein time, equations These and are equations of involve therefore Einstein’s describe only calledWe equations spatial how require the derivatives theevolution fields 10 second evolve of equations the forward derivatives electric. To and completely and, magnetic at first determine fields sight, there the time appearab to be 10 field equations, The above equations involve only spatial derivatives of the electric and magnetic0 fields in time, and are thereforeand hold called∂evolutiont atgab each theonevolution instant the of the initial of electromagnetic equations time hypersurface independently. To completely fields at of we determinex the also priort have. the The or time to subsequent only specify truly how evolution dynamical the sources of equationsρ and j i must be ab and hold at each instant of time independently of the prior orGab subsequent8πTab evolution, that= might of furnish them. But note that the Bianchi identities bG 0 equationsevolution of thethe electromagnetic fields.providedevolve They fields according therefore by we the also to six constrain have the remaining net to specify force any= possible acting how relations the on sourcesconfigurations them.ρ Theirand motionj i of the depends fields, and on what are forces are ∇ = the fields. They therefore constrain any possible configurationsgive of the fields, and are evolve accordingcorrespondingly toEinstein’s the netacting force acting on called them, on the them. butconstraint the Their motion motion equations of depends the sources. on what is less forces relevant are for our discussion here. We do correspondingly called the constraint equations.a0 a0 acting on them, butequations the motion of the sourcesG is less8π relevantT for our discussionij here. We doija0 (2.2)ai bc a ab c note, however, that the total charge is conserved,G as8 canπ∂Tt G be. seen by∂i takingG theG spatial$ diver-G $ . (2.3) (2.1) note, however, that the total charge is conserved,= as can be seen by taking the spatial= diver- = − − bc − bc gence of equation4 second (2.7 order) and substituting the6 constraintsecond order (2.5 ) to get the continuity equation, do notgence furnish of equation any of (2.7 the) and informationIt substituting is not surprisingconstraint the required constraint equations that for (2.5Since there the) to dynamicalget nois the a term mismatch continuityevolution on evolution the equation, equations right between hand of the the side fields. required of equation number (2.1) (10) contains of second third time derivatives or 2 ∂ρ a0 Rather, they supply four constraints∂ρon the initiali data,higher, i.e., the four four relations quantitiesi betweenG cannotg and contain second time derivatives. Hence the four time derivativesDi j∂t gab0. and the availableD numberi j 0(2.9). (6) of dynamicalab field equations.(2.9) After all, ∂t 0+ = ∂t + = ∂t gab on the initial hypersurfacethere is at alwaysx at. fourfold The only ambiguity truly dynamical associated equations with the must freedom be to choose four different It is possible to bring Maxwell’sIt is possible equations= to into bring a form Maxwell’s that is closer equations to the 3 into1 form a form of that is closer to the 3 1 form of provided by the six remainingcoordinates relations to label points in spacetime. So,+ for example, we could always+ choose Gaussian Einstein’s equations that weEinstein’s will derive equations in this chapter. that we1 ToHere will do so, we derive are we assuming, introduce in this of chapter. the course, vector that To suitable do so, boundary we introduce conditions the and vector initial data are chosen so that these solutions potential Aa (%, Ai ) andnormal write B coordinatesi asa i and set g00 i 1 and g0i 0. That way we have six metric variables = potential A ij(%, A ) andij writedo indeedB exist.as G= 8πT . = − = (2.3) 2 g to evolve, six dynamicalj k equations (2.3) to provide the required quantities ∂ g ,and ij Bi ϵijk=D A . j k(2.10) t ij = Bi ϵijkD A . (2.10) four constraint equations (2.2)thatrelate= gij and ∂t gij on the23 initial hypersurface. The It is notBy surprising construction, thatBi automatically there is a mismatch satisfies the constraint between (2.6 the). required The two evolution number equa- (10) of second 2 initialBy construction, value problemBi automatically thus appears satisfies to be the solved, constraint at least (2.6 in). The principle. two evolution2 equa- time derivativestions (2.7) and∂t (g2.8ab)canberewrittenintermsofand the available numberEi and (6)A ofi : dynamical field equations. After all, there is always a fourfold ambiguitytions (2.7) and associated (2.8)canberewrittenintermsof with the freedom to chooseEi and fourAi : different ∂t Ai Ei Di % (2.11) = −Exercise− 2.1 Demonstrate that the constraint equations (2.2), if satisfied ini- coordinates to label points in spacetime.j So, forj example,∂t Ai weE couldi Di always% choose Gaussian (2.11) ∂t Ei tially,Di D A arej D automaticallyD j Ai 4π j=i . satisfied− − at later(2.12) times when the gravitational field is = − − normal coordinates and set g00 evolved1 and byg0 usingi 0. the That dynamical way wej equations havej six (2.3 metric). Equivalently, variables show that the relation = − = ∂t Ei Di D A j D D j Ai 4π ji . (2.12) g to evolve,Exercise six dynamical 2.2 Show that equations the∂ evolution(Ga0 (2.3 equations8)π toT a0 provide) (2.110willbesatisfiedattheinitialtime)and(= the2.12 required)preservethe− quantities− ∂2g x,and0 t, hence conclude that ij constraint (2.5); i.e., show that t t ij Exercise− 2.2 Show= that the evolution0 equations (2.11)and(2.12=)preservethe four constraint equations (2.2)thatrelateequation (2.2gij) willand be∂t g satisfiedij on the at x initialt hypersurface.δt,etc. The constraint∂ (2.5); i.e., show that = + Hint: UseE the0. Bianchi identities together(2.13)2 with the equations of energy-momentum initial value problem thus appears to be∂t C solved,= at least in principle. ab ab 0 conservation to evaluate b(G ∂ 8πT )atx t. With the vector potential Ai we have introduced a gauge freedom∇ into− electrodynamicsE 0. = (2.13) whichExercise is expressed 2.1 Demonstrate in the freely specifiable that the gauge constraint variable equations%. (∂2.2t C ),= if satisfied ini- The above discussion reveals that formulating the Cauchy problem in general relativity tially, are automaticallyWith satisfied the vector at later potential timesA wheni we have the introduced gravitational a gauge fieldfreedom is into electrodynamics Exercise 2.3 Show that a transformation to a new “tilded” gauge according to evolved by using the dynamicallogicallywhich is expressed involves equations in a (2.3 the decomposition). freely Equivalently, specifiable of show 4-dimensional gauge that variable the relation% spacetime. into 3-dimensional space a0 a0 ∂& 0 ∂t (G 8πT ) 0willbesatisfiedattheinitialtimeand one-dimensional%˜ % time. In thisx chaptert, hence(2.14) we conclude will explore that how this split induces a natural − = Exercise= 2.3− ∂Showt that a transformation= to a new “tilded” gauge according to equation (2.2) will be“3 satisfied1” decomposition at x0 t δt,etc. of Einstein’s equations and leads to the standard “3 1” equations A˜i =Ai +Di & (2.15) Hint: Use the Bianchi identities+ together= + with the equations of energy-momentum∂& + of general relativity. The 3 1 equations%˜ % are entirely equivalent to(2.14) the usual field equa- leaves the physical fields Ei andabBi unchanged.ab 0 conservation to evaluate b(G 8πT )atx t+. = − ∂t tions∇ (1.32−) but they focus= on the evolution of 12 purely spatial quantities closely related to A˜i Ai Di & (2.15) The initial value problem in electrodynamics can now be solved in= two steps.+ In the Thefirst above step, discussioninitial data (A reveals,gEij),and together that∂t g formulatingij withand the the sources constraints the (ρ Cauchy, j i ), are that specified problem they must that in satisfy general satisfy on relativity spatial hypersurfaces. Once these i i leaves the physical fields Ei and Bi unchanged. logicallythe constraintinvolves (2.5 a decomposition). In thespatial second step, field of these 4-dimensional quantities fields are are evolved specified spacetime according on into to some the 3-dimensional evolution initial “time slice” space (i.e., spatial hypersurface) and one-dimensional time.consistent InThe this initial chapter with value wethe problem will 3 explore1 constraint in electrodynamics how equations, this split can induces the now 3 be a solved1 natural evolution in two equations steps. In the can then be + + i “3 1” decomposition of Einstein’sfirst step, initial equations data ( andAi , E leadsi ), together to the with standard the sources “3 1” (ρ, equationsj ), are specified that satisfy + the constraint (2.5). In the second step, these fields are+ evolved according to the evolution of general relativity. The 32 1 equations are entirely equivalent to the usual field equa- +Only four of the 12 functions gij and ∂t gij represent true dynamical degrees of freedom that can be independently tions (1.32) but they focus onspecified the evolution on the initial of hypersurface. 12 purely The spatial reason quantities is as follows: closely In addition related to the four to constraint equations, one can choose g and ∂ g and the constraintsthree that arbitrary they functions must satisfy to induce on coordinate spatial transformations hypersurfaces. on the Once hypersurface these without changing its geometry. Plus ij t ij there exists the freedom to choose the initial hypersurface in the embedding spacetime, which can be accomplished by spatial field quantities are specifiedspecifying on one some other arbitrary initial function. “time The slice” remaining (i.e., 12 spatial4 3 hypersurface)1 4freelyspecifiablequantitiescanbeidentified − − − = consistent with the 3 1 constraintwith two sets equations, of the pair of the metric 3 functions1 evolution (gij,∂t gij), equations i.e., the 3-metric can and then its “velocity”. be These four functions specify + the two dynamical degrees of freedom+ characterizing a gravitational field in general relativity (e.g., the two polarization states of a gravitational wave). For further discussion, see Chapter 3 below and Wald (1984), Chapter 10.2.

2 Only four of the 12 functions gij and ∂t gij represent true dynamical degrees of freedom that can be independently specified on the initial hypersurface. The reason is as follows: In addition to the four constraint equations, one can choose three arbitrary functions to induce coordinate transformations on the hypersurface without changing its geometry. Plus there exists the freedom to choose the initial hypersurface in the embedding spacetime, which can be accomplished by specifying one other arbitrary function. The remaining 12 4 3 1 4freelyspecifiablequantitiescanbeidentified − − − = with two sets of the pair of metric functions (gij,∂t gij), i.e., the 3-metric and its “velocity”. These four functions specify the two dynamical degrees of freedom characterizing a gravitational field in general relativity (e.g., the two polarization states of a gravitational wave). For further discussion, see Chapter 3 below and Wald (1984), Chapter 10.2. 1 1

R = 1 R = 1

R =2M R =2M

R =0 R =0

a nae(i) =0 a 378 Chapter 11 Recasting the378 evolutionChapter equations 11 Recastingn theae evolution(i) =0 equations

↵⌦ = n = t however,378 exponentiallyChapter 11 growing Recastinghowever, modes the exponentially evolution are still very equations growing bad, and modesa cana are easilyr stilla terminate very bad, a and can easily terminate a simulation after only a shortsimulation time. In fact, after many only of a short the symmetric time. In fact, or strongly many of hyperbolic the symmetric or strongly hyperbolic ↵⌦a = na = at systemshowever, that have exponentially been introducedsystems growing over that the have years modes been lead introduced are to such still exponentially over veryr the bad, years growing and lead to can modessuch easily exponentially terminate growing a modes a 9 9 in numericalsimulation implementations. after onlyin a numerical Unless short thesetime. implementations. modes In fact, can many be controlled, Unless ofn the theseab symmetricthese=0 modes systems can or be are controlled, strongly not hyperbolicthese systems are not very useful for numerical simulations.very useful From for numerical a numerical simulations. perspective, From then, a numerical well-posedness perspective, then, well-posedness systems that have been introduced over the yearsa lead to such exponentially growing modes is a necessary but not a sufficientis a necessary condition. but not a sufficientn ab condition.=0 in numerical implementations. Unless these modes can be controlled,9 these systems are not In the following we will discussIn the severalfollowing different we will approaches discuss↵ several that leadi different to different approaches that lead to different reformulationsvery useful of for the numerical evolutionreformulations equations simulations. that of the are From evolution both a strongly numerical equations hyperbolic perspective, that are and both that then,strongly have well-posedness hyperbolic and that have beenis used a necessary successfully but in not numericalbeen a sufficient used simulations. successfully condition. In in the numerical↵ remainderi simulations. of this chapter In the we remainder of this chapter we will pursueIn the378 a more followingChapter heuristic 11 wewill Recasting approach will pursue discuss the a than more evolution the several heuristic one equations sketched different approach in this approaches than section the one and that sketched will lead in to this different section and will Ain marshal mathematical intuitionmarshal and exploit mathematical analogies intuition to motivate and exploit the differenti analogies approaches. to motivate the different approaches. reformulationshowever,378 Chapter exponentially of the 11 evolution Recasting growing the equations evolution modes equations that are are still both very strongly bad, and hyperbolic can easily and terminate that have a In fact, some of the systemsIn discussed fact, some below of the were systems developed discussedi in exactly below this were way developed and were in exactly this way and were been used successfully in numerical simulations.A ni In the remainder of this chapter we discoveredsimulation tohowever, have desirable exponentiallyafterdiscovered only numerical a short growing to properties time. have modes desirable In fact,empirically are many numerical still very of before the bad,properties symmetric they and were can empirically analyzed or easily strongly terminate before hyperbolic they a were analyzed i mathematicallywill pursuesystemssimulation and a that revealed more after havemathematically heuristic to only been be stronglya introduced short approach time. hyperbolic. and over In revealed fact, than the To many years to illustratethe be of lead strongly one the to some symmetric sketchedn such hyperbolic. of exponentially these or in heuristic strongly To this illustrate section growing hyperbolic some andmodes of these will heuristic 9 argumentsmarshalin forsystems numericalmathematical a simple that implementations. haveandarguments intuition familiar been introduced example for and Unless a exploit simple over we these the will and analogies years modes begin familiar lead can by to to such example discussingbe motivate controlled, exponentially we theMaxwell’s willthese different growing begin systems by modes approaches. discussing are not Maxwell’s i equationsIn fact, invery some Sectionin numerical useful of11.2 the for. implementations. systemsnumericalequations discussed simulations. in Section Unless below11.2 these From. modes were a numericaln can developed be controlled, perspective, in exactly9 these then, systems this well-posedness way are not and were isvery a necessary useful for but numerical not a sufficient simulations. condition. From a numericali perspective, then, well-posedness discovered to have desirable numerical properties empiricallyA ij before they were analyzed In the following we will discuss several different() approaches that lead to different mathematicallyis aThe necessary Maxwell and revealed but not aequations sufficient to be strongly condition. can hyperbolic. be written To in illustrate a 3+1 someform of these heuristic reformulationsIn the11.2 following Recastingof the we evolution will Maxwell’s discuss equations11.2 several Recasting that equationsi different are both approaches Maxwell’s strongly hyperbolic that lead equations to and different that have A ij argumentsreformulations for a simple of the and evolution familiar equations example that() are we both will strongly begin hyperbolic by discussing and that have Maxwell’s In Chapterbeen2.2 we used have successfully seenIn Chapter that in we numerical2.2 canwe bring have simulations. Maxwell’s seen that evolutioni we In the can remainder bring equations Maxwell’s of in this a evolution chapter we equations in a equations in Section 11.2. E ij Minkowskiwill spacetimebeen pursue used into successfully a more theMinkowski form heuristic in spacetime numerical approach into simulations. than the form the In one the() sketched remainder in of this this section chapter and we will marshalwill pursue mathematical a more heuristic intuition approach and exploitMaxwell than analogies the one tosketched motivate in the this different section and approaches. will Einstein Ei marshal mathematical∂t Ai intuitionEi Di and" exploit∂t analogiesAi E toiij motivateDi " the different(11.4) approaches. (11.4) In fact, some of the= systems− − discussed below were()= − developed− in exactly this way and were InPrimary fact, some field of11.2 the systems Recastingj discussed below Maxwell’sj Ai were developedEij equationsij in exactlyKj ij this way and were discovered to∂ havet Ei desirableD D j numericalAi Di D propertiesA∂tjEi 4π jiD empiricallyD j Ai D beforei D A j(11.5) they4π wereji analyzed (11.5) discovered to have= desirable− numerical+ properties−= − empirically+ before they− were analyzed In Chaptermathematically2.2 we and have revealed seen thatto be strongly we can hyperbolic. bring Maxwell’s To illustrate evolution some of these equations heuristic in a (see equationsmathematically2.11 and 2.12(see and). Recall revealed equations that to2.11 beA stronglyisand the2.12 three-vector hyperbolic.). Recall Topotential, that illustrateAi is the the some magnetic three-vector of these heuristic potential, the magnetic Secondary field i Ai Ei ij Kij arguments forj ak simple and familiar examplej k we will begin by discussing Maxwell’s fieldMinkowski satisfiesargumentsBi spacetimeϵijkD forAfield aand into simple satisfies therefore the and formB familiari is automaticallyϵijk exampleD A and divergence-free, we therefore will begin is automatically byφ discussingis a gauge divergence-free, Maxwell’s φ is a gauge equations= in Section 11.2. = potential, andequations the electric in Section fieldpotential,Ei11.2has. and to satisfy the electric the constraint field Ei equationhas to satisfy (2.5), the constraint equation (2.5), Gauge variables∂t Ai Ei Di " (11.4) = i− − i Di E 4πρj e. jDi E 4πρe. (11.6) (11.6) 11.211.2∂t Ei Recasting=D D j A Maxwell’si Maxwell’sDi D A equationsj equations=4π ji (11.5) In the above equations ρ isIn the the electric above= charge equations− densityρ is and+ thej electricthe current charge− density. density and j the current density. In ChapterMatter sources2.2e we have seen that wee canand bringi Maxwell’s evolutioni equations in a In(see Chapter equationsIn2.7 Chapterwe have2.112.2 discussedandweIn2.12 Chapter have some). seen Recall2.7 of that thewe similarities havethat we can discussedAi bringis of the Maxwell’s Maxwell’s some three-vector of the equations evolution similarities potential, in equations the of Maxwell’s the in magnetic a equations in the MinkowskiMinkowski spacetime spacetimej into intok the form form abovefield form satisfies with theB ADMi ϵ evolutionijkaboveD formA equationsand with therefore the (2.134 ADM) is evolution and automatically (2.135 equations), namely divergence-free, (2.134) and (2.135φ),is namely a gauge = potential, andEvolution the electric field∂∂t EAii hasE toEi i satisfyDDi "i " the constraint equation (2.5), (11.4)(11.4) ∂t γij 2αKij Di β j D∂t γj βiji 2αKij Di β j (11.7)D j βi ? (11.7) ==−− −− equations = − + j j + = j−j + + ∂∂t Eii DD DDij Aj Ai i DDi Di DA jA j 4π4jiπ ji (11.5)(11.5) and and ==−−Di E +4+πρe. − − (11.6) = (see(see equations equations2.112.11k andand 2.122.12).). Recall Recall that thatAkAi iisis the the1 three-vector three-vector potential, potential, the themagnetic1 magnetic ∂t Kij α(Rij 2KikK j ∂tKKKijijj ) kα(DRiijD j α2Kik8παK j(SijKKijγ)ij(SDi Dρ))j α 8πα(Sij 2 γij(S ρ)) In the=field abovefield satisfies satisfies equations− BBi +ρϵeϵijkisDD thej=A−k electricandand therefore therefore− charge− is is automatically+ density automatically− 2 and− − divergence-free,ji divergence-free,the− current(11.8)φ density.−isφ ais gauge a gauge− (11.8) k i = ijkk kk k k potential,β ∂k Kij andK theik=∂ electricj β K fieldkj∂iββE ∂.hask Kij to satisfyKik∂ j theβ constraintKkj∂i β equation. (2.5), In Chapterpotential,+ 2.7 and+we the have electric discussed+ field+ Eii somehas to+ of satisfy the similarities the+ constraint of equation Maxwell’s (2.5 equations), in the above formConstraint with the ADM evolution equationsi (2.134) and (2.135), namely Di E i 4πρe. (11.6) 9 9 Di E = 4πρe. (11.6) See, e.g., Scheel et al.equations(1998); KidderSee,et al. e.g.,(2001Scheel)forexamples.et al. (1998); Kidder=et al. (2001)forexamples. In the above equations ρ is the electric charge density and j the current density. In the above equations ρ∂e tisγij the electric2αK chargeij D densityi β j andD jiβji the current density. (11.7) In Chapter 2.7 we havee discussed= − some of+ the similarities+ of Maxwell’si equations in the In Chapter 2.7 we have discussed some of the similarities of Maxwell’s equations in the and above form with the ADM evolution equations (2.134) and (2.135), namely above form with the ADM evolution equations (2.134) and (2.135), namely k 1 ∂t Kij α(Rij 2KikK ∂tKKγij ij) 2αKDiji D jDαi β j 8παD j β(Si ij γij(S ρ)) (11.7) = − j +∂ γ = −−2αK + D−β+ D β − 2 − (11.7)(11.8) k t kij kij i j j i and β ∂k Kij Kik∂ j β =K−kj∂i β .+ + + + + and k 1 ∂t Kij α(Rij 2KikK j KKij) Di D j α 8πα(Sij 2 γij(S ρ)) = − k + − − − 1 − (11.8) ∂t Kij α(Rijk 2KikK KKk ij) Dki D j α 8πα(Sij γij(S ρ)) 9 See, e.g., Scheel et= al. (1998β );∂−kKidderKij etKik al.j∂+(j2001β )forexamples.Kkj−∂i β . − − 2 − (11.8) + k + k+ k β ∂k Kij Kik∂ j β Kkj∂i β . + + + 9 See, e.g., Scheel et al. (1998); Kidder et al. (2001)forexamples. 9 See, e.g., Scheel et al. (1998); Kidder et al. (2001)forexamples. 30 Chapter 2 The 3+1 decompostion of Einstein’s equations

na M

2.7 Choosing basis vectors: the ADMt4 equations 43

∑4 This is the evolution equation for the extrinsic curvature. Note that all differentialt3 operators

and the Ricci tensor Rab are associated∑3 with the spatial metric γab. t2 Exercise 2.22 Show that raising an index in equation (2.106) yields ∑ 2 t a a a a a 1 a 1 t K b D Dbα α(R b KK b) 8πα(S b γ b(S ρ)) L = − ∑1+ + − − 2 − a β K . (2.107) + L b Figure 2.1 AfoliationofthespacetimeM.Thehypersurfaces! are level surfaces of the coordinate time t, The evolution equation for the spatial metric γ , the last missing piece, can be found a ab "a a t.Thenormalvectorn is orthogonal to these t constant spatial hypersurfaces. directly= ∇ from equation (2.53), again using equation (2.98= ),

tγab 2αKab β γab. (2.108)2 The 4-metric gab allows usL to compute= − the+ normL of "a,whichwecall α− , The coupled evolution equations (2.106) and (2.108) determine the evolution− of the 1 1 gravitational field data (γab, Kab). Together2 withab the constraint equations (2.90) and (2.96) " g at bt . (2.21) they are completely equivalent to∥ Einstein’s∥ = equations∇ ∇ (2.83≡−). Noteα2 we have succeeded Asin recasting we will Einstein’s see more equations, clearly which later, areα measures second order how in time much in their proper original time form, elapses between neighboringas a coupled set time of partialslices along differential the normal equationsR = vector that are"a nowto the first slice, order and in time. is therefore As in called the 44 electrodynamics,Chapter 2 The 3+1the evolution decompostion equations of Einstein’s conserve1 equations the constraint equations,i.e.,ifthefield1 lapse function. We assume that α > 0, so that "a is timelike and the hypersurface ! is data (γab, Kab)satisfytheconstraintsatsometimet and are evolved with the evolution spacelikeequations, then everywhere. the data will also satisfy the constraint equations at all later times (see R =2M exercises 2.1 and 2.2). 2.6 The constraint and evolution equations 41 R = Exercise 2.5 Show that1 the normalized 1-form The Hamiltonian constraint (2.90) and the momentum constraint (2.96)arethedirect1 2.7 Choosing basis vectors:ωa theα" ADMa equations (2.22) 1 equivalent of the constraints (2.5) andR (2.6=0)≡ in electrodynamics. They involve only the R =2M spatialSo far, metric, weis rotation-free have the expressed extrinsic our curvature, equations and in a covariant,their spatial coordinate-independent derivatives. They aremanner, the conditions i.e., the basis vectors ea have been completely arbitrary and have no particular relationship (bR) = Figurethat 2.3 allowSpatial a 3-dimensional basis vectors ea are slice Lie dragged! with from data one spacelike (aγab, K sliceab) to to the be next embedded along the coordinate in a 4-dimensional to the 1-form &a or to the(i) congruence1 definedωa by[a t b.Itisquiteintuitive,though,thatthingsωc] 0. 2.3 Foliations of spacetime (2.23)31 a n e ∇=0 = R = We foliatecongruencemanifoldwill spacetime simplifyt . Consequently,M with if weinto data adopt these spacelike ( basisg aabR coordinate). vectors=0 Field connectslices data systema points ((γi)ab that with, K reflectsab the) same that our spatial are 3 being coordinates1 split imposed1 of on spacetime on a timeslice neighboringWe slices can (e.g., now points defineA, B theand C uniton slice normalt have to the the same slices spatial coordinates as + as points A , B and C , a ! inhaveThus a natural toγ satisfyab way.is a Weprojection the will two see constraint tensor that the that Lie equations. projects derivative out We in all the will geometric evolution discuss objects equations strategies′ lying′ (2.106 alongfor′ ) solvingn . the respectively,andThis (2.108 on metric slice)thenreducestoapartialderivativewithrespecttocoordinatetimeand,asant allowsdt). R us=2 toM compute distances within a slice ".Toseethatγ is purely constraint equations+ and finding initialn dataa thatgab representω . a snapshot ofab the gravitational(2.24) a ⌦a = at b a R =2M additionalspatial, benefit, i.e., resides we will entirelyn beae( ablei) =0 in to ignore" with all no timelike≡− piece components along n , of we spatial contract tensors. it with the fieldsExercise at a certain 2.23 Show instant that of a time spatial in vector Chapterea thatr3. is Lie dragged along ta remains Tonormal do so,na we, first introduce a basis of three(i) spatial vectorsa ea (the subscript i 1, 2, 3 HereThespatial, evolution the i.e., negative show equations that sign has that been evolve chosen the so data that (nγ points, K(i) ) in forward the direction in time= of can increasing be foundt, distinguishes the vectors,R not=0 the components; we again referab theab reader to Section 2.1 for from (2.53), which can bena consideredγ na ga a asn thean definitionnab n n of the140. extrinsic curvature,(2.28) and the • Define surfacesa summary of ofconstant our notation) t: ⌦a that=ab resideat t(n! inaabnωe aa particular=) 0↵.a⌦gab timeω ωb sliceb'1., (2.111) (2.25) r=L (ai+) = =a b− = a R =0 Ricci equation (2.82). However, the Lie= derivative− along= n , n, is not a natural time a a a As theIntuitively, fourth basisγab calculates vectora we the pick spacetimee&ae(i) distancet .Recallthatwewanttosetthevector0. with gab and thenL kills off(2.109) the timelike derivativeBy construction, since na isn notisa normalized dual to the(0) andsurface timelike,= 1-form # ,i.e.,theirdotproductisnotunity • 3 spacelike basis vectors: nae(i) =0 = a contributiona (normal to the spatial surface) with nanb. The inversea spatial metric can be congruenceWe extendt to our be spatial tangent vectors to the to coordinate other slices linea ' congruenceby Lie dragging and along thereforet , connect points but rather na = ↵⌦a a n abab =0ab with thefound same by spatial raising coordinates the indices on of neighboringγabnwithna g g, timeω slices.aωb The1 duality, n e conditiona =0 (2.99) (2.26) • ea 0, a (i) (2.110) Lie drag them to get thema on next2.7 Choosing slice:a 15 basist (i) vectors:= ab the ADM= equations− 1 45 then implies that e(0) has the componentsn ab#a Lac αbd=g at abbt a αb− . (2.97) and may therefore be thoughtγ of asg theg 4-velocityγcd g ofn an “normal”. observer whose(2.29) worldline as illustrated in Figure 2.3. a == − ∇= ∇ + = n aaba=0 a i •andFourth fromInstead, the basisis normalization always vector consider normal defined condition the to vectorby then t:a spatialn t 1wefind slicese(0) ↵(1!,.0, 0, 0). (2.112) Next we break up 4-dimensional==− = tensors by decomposing↵⌦ them= n into= a purelyt spatial 14 In the language of differential forms, the spatial vectors ea do not pierce the spatiala hypersurfacea ': &a, e With the normaln vectora ( weα, 0, can0, 0)a now. construct(i) the spatial(2.117) metric rγab that(i) is induced by This meanspart,a which that the lies Lie in derivative the hypersurfaces along t "areduces, and aa timelike to a partiala part, derivative which is normal with respect⟨ to the⟩ tospatial= • &a e(i) 0. = − Time vector has= normal part and “shift”:i t αn β , (2.98) From thet : definitiongtabsurface.on∂t of(see the the To 3-dimensional equation spatial do so, metric weA.10 need↵). (2.27 hypersurfaces two)wehave projection= ! operators., + The first one, which projects a 4- L = i ! Fromdimensional equation (2.109 tensor)wenowfind into a spatial slice, canA beni founda by raising only one index of the spatial which is dual to #a for any spatial shift vector β , a • Define spatial metric tensor: γij gij, γab gab nanb. n(2.118)ab =0 (2.27) metric γab = 1 = + 1 aa a a a meaning that the metric on # is just the spatial!i a parte of the 4-metric.nae Since0, zeroth components (2.113) A nti (ai#) a a αn a#(i)a βa #a a 1. (2.99) • Define spatial projection operator: γ =g=−α n inb a=+0 δ n =nb. (2.30) of spatial contravariant tensors have to vanish,b we= alsob + haven γ = b0.+ The inverse metric which, since the ea span $, impliesa that the covariant= spatial components of the normal can thereforeIt will be expressed prove useful( asi) to choose t toa beb the congruencea along which we propagate the spatial Exercise 2.6 Show that γ av ,wherev is an arbitrary spacetime vector, is purely • vector have to vanish, i b a Definecoordinate normal projection grid from operator: one timen2 slicen nb to2 thei next slice. In other words, t will connect points ab spatial.ab a b α− α− β g γ n n − i . (2.119) a with the= same− spatial= coordinatesα 2β j γ ij n onαA neighboring20β.i β jij time slices. Then the(2.114) shift vector β ! − −i − () " • Decomposewill measure tensorsToprojecthigherranktensorsintothespatialsurface,eachfreeindexhastobecontracted the into amount timelike by and which spacelike the= spatial parts coordinates are shifted within a slice with Exercise 2.24 Show that a i Since spatialwith a tensors projection vanish operator. whenAi t contracted It is↵ sometimes with the convenient normal vector,to denote this this also projection means that with a respect to the normal vector,() as illustratedij in Figure 2.4.Aswehavenotedbefore,thelapse all components of spatial tensorsγ ikγ withδ ai contravariant. index equal(2.120) to zero must vanish. For symbol ,e.g., kj j Ei function α measures⊥ how much= proper timea elapsesij0 between neighboring time slices along the shift vector, for example,ij this implies naβ ()n0β 0andhence Equation (2.120) implies that γ and γij are 3-dimensional inverses,c d and can hence be the normal vector.2 The lapseji and thei shiftTab = thereforeγa γb =Tcd determine. how the coordinates(2.31) evolve used to raise and lower spatial@t Ai indices+ DjDE ofA spatiali Hij( tensors.t,a ⊥ x )= ForD=ii@ example,t 4⇡j thei covariant form () aβ (0, β ). (2.115) of the shiftin vector time.Similarly, is The choice we may of defineα and the normalβ is= quite projection arbitrary, operator and as we will postpone a discussion of a Ai Ei ij Kij Solvingsome equation common (2.98 choices)forn tothen Chapter yieldsj 4 the.Thefreedomtochoosethesefourgaugefunctions contravariant components a βi γijβ a. a a a (2.121) α and β completelyAi arbitrarilyEi = ijN embodiesb Kij n nb theδ four-foldb γ b, coordinate degrees(2.32) of freedom naS1,S(2α≡−1, α 1β=i ), − (2.116) We can now invert (2.119) and find the components11 − of the 4-dimensional− metric inherent in general relativity. =Specifically,− the lapse function reflects thea freedom to even though in most cases it is just as easy to write out the normal vectors n nb.We 2 l 15 chooseStrictly, equation the sequence (2.112) is not of a tensor timeα equation, slices,βl β but pushing ratherβi a specification them forward of components by in different the adopted basis.amounts In the of proper can now use theseg two projection operators. to decompose any tensor(2.122) into its spatial and abstract index notation of Waldab (1984−), the+ index a appearing on the left hand side would be denoted with a Greek timetimelike at different parts. spatial For= example,! pointsβ wej onJ can a write sliceγij " an and arbitrary thus exploiting vector va as “the many-fingered nature letter. Since the meaning is clear from the context, we do not make this distinction in the few places that it arises in 12 Equivalently,ofthis time”. book. the line elementThe shift may be vector decomposeda reflectsa b as thea freedoma b toa relabela spatialb coordinates on each v δ bv (γ b N b)v v n nbv . (2.33) slice in an2 arbitrary2 2 way.= Observersi =i who+j arej “at=⊥ rest”− relative to the slices follow the ds α dt γij(dx β dt)(dx β dt), (2.123) = − a+ M+1/M2 + normal congruenceExercise 2.7n Showand are that calledfor the second either rank normal tensor orTab Eulerianwe have observers, while observers which is often refered to as the metric in 3 1 form. We may interpret this line element as + c 2 c c d the Pythagorean theorem for a 4-dimensionalTab Tab spacetime,nan Tcbds nbn (properTac na timenbn betweenn Tcd . (2.34) =⊥ − ⊥ − = −⊥ + 11 a 2 a 2 neighboringRecall spatial that hypersurfaces)β is spatial and therefore(proper subject distancerad to the within constraint the thatspatialn βa hypersurface)0, hence only. three of its components may Exercise 2.7 illustrates+ that thehab symbol has to be used with= some care, since it applies This equationbe thus freely determines specified. the invariant interval⊥ between neighboring points A and B, as illustrated12 See, inonly Figure e.g., toMisner the2.4. freeet al. indices(1973), of p. the 527. tensor that it operates on. To avoid confusion, we will usually write out the projection operators explicitly. Exercise 2.25 Show that the determinant g det(gradab) of the spacetime metric gab gab = ⌘ab + hab can be written as =

√ g α√γ , (2.124) − = 4 where γ det(γij) is the determinant ofij the= spatial ⌘ij metric γij. = 1 Hint: Recall that for any square matrix Aij the following is true: (A )ij cofactor − = of A ji/ det A. Exercise 2.26 Use equation (2.123) directlyR = to determine the proper time dτ measured by a clock carried by a normal observer1 na in a coordinate time interval dt.

R =2M

R =0

a nae(i) =0

⌦ = t a ra

n = ↵⌦ a a

a n ab =0 2.7 Choosing basis vectors: the ADM equations 45

a and from the normalization condition nan 1wefind = − na ( α, 0, 0, 0). (2.117) = − 2.6 The constraint and evolution equations 41 From the definition of the spatial metric (2.27)wehave

γij Thegij, Hamiltonian constraint (2.90)(2.118) and the momentum constraint (2.96)arethedirect =equivalent of the constraints (2.5) and (2.6) in electrodynamics. They involve only the meaning that the metric on # is just the spatial part of the 4-metric. Since zeroth components spatial metric, the extrinsic curvature, and their spatial derivatives. They are the conditions of spatial contravariantThe tensors lapse have and to the vanish, shift we can also be have definedγ a0 with0. The great inverse metric that allow a 3-dimensional= slice ! with data (γab, Kab) to be embedded in a 4-dimensional can therefore be expressed as freedom manifold M with data (gab). Field data (γab, Kab) that are being imposed on a timeslice 2 2 i ab ab a b α−! haveα to− satisfyβ the two constraint equations. We will discuss strategies for solving the g γ n n − 2 j ij 2 i j . (2.119) = − = α− βconstraintγ α equations− β β and finding initial data that represent a snapshot of the gravitational ! − " fields at a certain instant of time in Chapter 3. Exercise 2.24 Show that 46 Chapter 2 The 3+1 decompostion of Einstein’s equations The evolution equations that evolve the data (γ , K ) forward in time can be found ik i ab ab γ γkj δ j . (2.120) + from= (2.53), which can be considered as thei definitioni ofB the extrinsict dt curvature, and the dx +ß dt xi +dx i ij i a Equation (2.120) implies that γ and γijRicciare 3-dimensional equation (2.82 inverses,). However, and can the hence Ließidt derivative be xi dx along n , n, is not a natural time a L used to raise and lower spatial indices of spatialderivative tensors. since Forn example,is not dual the to covariant the surfacea form 1-form #a,i.e.,theirdotproductisnotunity a t dt dxa of the shift vector is but rather an dt t j a ab 1 βi γijβ . n #a xi (2.121)αg at bt α− . (2.97) = A= − ∇ ∇ = We can now invert (2.119) and find the components of the 4-dimensional metric Instead, considerFigure 2.4 thePythagorean vector theorem in 3 1 dimensional spacetime. The normal vector αna and the time vector ta + 2 l connect points on two neighboring spatial slices. The shift vector βi resides in the slice and measures their 1 α βl β βi a a a gab − + difference.. The infinitesimal displacementt (2.122) vectorαndxa connectsβ , two nearby, but otherwise arbitrary, points on (2.98) = ! β j γijneighboring" slices (e.g., the point A at xi on= slice t and+ the point B at xi dxi on slice t dt). The total + + displacement vector dxa ta dt dxi ,wheredxi is thea spatial vector drawn in the figure, may1 be decomposed which is dual to #a for any spatial= + shift vector β , Equivalently, the line element may be decomposed as alternatively into two vectors that form the legs of a right-triangle, dxa (αna dt) (dxi βi dt), as shown. a i 2 a = + + 2 2 2 Thei vectori t Using connectsj ↵ this decompositionj points towith evaluatea the the same invarianta spatial intervalads dx dxa ,commonlyexpandedasin1 ds α dt γij(dx β dt)(dx β dt), t #a (2.123)αn #a β #a= 1. (2.99) = − + coordinates+ onequation neighboring+ (1.1), yields the time Pythagorean slices.= theorem, equation+ (2.123). = a which is often refered to as the metric in 3 It1 will form.a prove We useful mayi The interpret entire to choose content thist of lineto any be element spatial the congruence tensor as is available along from which their spatial we propagate components. the This spatial +Lapse:t ↵ Measures2 proper time along normal vector a the Pythagorean theorem for a 4-dimensionalcoordinate spacetime, gridis obviouslyds from one true(proper time for contravariant slice time to between the components, next slice. In since other their words, zeroth componentt will connect vanishes, points 2 a ji = −i neighboring spatial hypersurfaces)2 @t A(properti +Shift:withD↵ distancej D the A sameHowi but withinH coordinates spatial also(t, the holds x )= coordinates spatial covariantD “move”i@ hypersurface)t components. onon4⇡ neighboringtheji slice2. Therefore, as time time the evolves entire slices. content Then of the the shift decomposed vector βa + This equation thus determines the invariantwill interval measure betweenEinstein the neighboring amount equations by is which points contained theA inand spatial theirB, spatial coordinates components are alone, shifted and we within can rewrite a slice the with @2A + D DjA H(t, xHamiltoniani)=D @ constraint 4⇡j (2.90), as illustrated in Figure 2.4. t i j respecti to the normali vector,t as illustratedi in Figure 2.4.Aswehavenotedbefore,thelapse 2 j i R K 2 K K ij 16πρ, (2.125) @t Ai + DjD Ai functionH(t, x )=α measuresDS1i@,St2 how4⇡ muchji proper time elapsesij between neighboring time slices along Exercise 2.25 Show that the determinant g det(gab) of the spacetime metric gab + − = can be written as the normal= vector.the momentum The lapse constraint and the (2.96 shift), therefore determine how the coordinates evolve a in time.S1,S The2 choice of α and β is quite arbitrary,ij ij and wei will postpone a discussion of √ g α√γ , (2.124)D j (K γ K ) 8π S , (2.126) − some= common choices to Chapter 4.Thefreedomtochoosethesefourgaugefunctions− = S1,S2 a the evolutionJ equation for the extrinsic curvature (2.106), where γ det(γij) is the determinant ofα theand spatialβ completely metric γij. arbitrarily embodies the four-fold coordinate degrees of freedom = 1 Hint: Recall that for any square matrix Aij the following is true: (A− )ij 11cofactor k 1 inherent in general∂t Kij relativity.Di D j α α(Specifically,Rij 2KikK theKK lapseij) 8 functionπα(Sij reflectsγij(S ρ)) the freedom to J = j 2 (2.127) of A ji/ det A. = − k + − k + k − − − β Dk Kij Kik D j β Kkj Di β , choose the sequence+ of time slices,+ pushing+ them forward by different amounts of proper J M /M Exercise 2.26 Use equation (2.123)time directly at to different determineand1 the spatial evolution2 the proper points equation time on for ad sliceτ the spatial and metric thus exploiting(2.108), “the many-fingered nature measured by a clock carried by a normalof observer time”.12naThein a shiftcoordinate vector time reflects interval the freedom to relabel spatial coordinates on each M /M ∂t γij 2αKij Di β j D j βi . (2.128) dt. slice in1 an2 arbitrary way. Observers who= − are “at+ rest”+ relative to the slices follow the Equations (a2.125)–(2.128) are equivalent to Einstein’s equations (2.83) and comprise the Mnormal1/M2 congruencerad n and are called either normal or Eulerian observers, while observers “standard”hab 3 1 equations. Sometimes they are referred to as the “ADM” equations after + Arnowitt, Deser and Misner,16 even though these equations had been derived earlier,17 and rad 11 hab a a 18 Recall that β evenis spatial though andArnowitt therefore subjectet al. (1962 to the) constraint derived the that equationsn βa 0, in hence a different only three form. of its components may = radbe freely specified. 12hab rad See, e.g.,gabMisner=16 Arnowitt⌘etab al.+(et1973h al.ab(),1962 p.). 527. 17 E.g., Darmois (1927); Lichnerowicz (1944); Foures-Bruhat` (1956). 18 See,rad e.g., Anderson and York, Jr. (1998)foradiscussion. gab = ⌘ab + hab rad gab = ⌘ab + hab 4 ij = ⌘ij 4 ij = ⌘ij = 4⌘ ij ij R = 1 R = 1 R = 1 R =2M R =2M R =2M R =0 R =0 R =0 a nae(i) =0 a nae(i) =0 a nae(i) =0 ⌦ = t a ra ⌦ = t a ra ⌦ = t a ra n = ↵⌦ a a n = ↵⌦ a a na = ↵⌦a a n ab =0 a n ab =0 a n ab =0 ↵i ↵i ↵i 100 Chapter 4 Choosing coordinates: the lapse and shift

slicing or a time coordinate therefore amounts to making a choice for the lapse function. Letting the lapse vary with position across the spatial slice takes advantage of the freedom that proper time can advance at different rates at different points on a given slice (“the many-fingered nature of time”). The shift βi ,ontheotherhand,determineshowspatial points at rest with respect to a normal observer na are relabeled on neighboring slices. The I amspatial forced gauge to ask: is therefore imposed by a choice for the shift vector. In the rest of this chapter we will discuss a few different gauge choices that are commonly used in numerical simulations. We will focus here on some of the simpler conditions that lend themselves to straightforward geometric interpretation and provide us with valuable intuition. In later chapters we will observe these and other choices in action when we study actual simulations that construct numerical spacetimes. There we will see how well different gauge conditions perform in different physical situations. A brief summary of a few common gaugeWhy choices is providedare inyou Box 4.1 for themaking reader eager and willing to order from a limited,things but representative, so menu. complicated? 4.1 Geodesic slicing Since the lapse α and the shift βi can be chosen freely, let us first consider the simplest possible choice, Geodesic slicing: α 1, βi 0. (4.1) = = In the context of numerical relativity this gauge choice is often called geodesic slicing; the resulting coordinates are also known as Gaussian-normal coordinates.4 Recall that coordinate observers move with 4-velocities ua ta ea (i.e., spatial = = (0) velocities ui 0). Thus with βi 0, coordinate observers coincide withBoom! normal observers = = (ua na). With α 1, the proper time intervals that they measure agree with coordinate = = time intervals. Their acceleration is given by equation (2.51),

ab Db ln α 0. (4.2) = = Evidently, since their acceleration vanishes, normal observers are freely-falling and there- fore follow geodesics, hence the name of this slicing condition. Clearly, the evolution equations (2.134) and (2.135) simplify significantly when geodesic slicing is adopted.

Exercise 4.1 Consider the Robertson–Walker metric for a flat Friedmann cosmol- ogy,

2 2 2 i j ds dt a (t) ηijdx dx , (4.3) = − + where the expansion factor a(t) is a function of time only. We can immediately read off the lapse and shift to be α 1andβi 0, showing that this spacetime is geodesically sliced. = = (a) Identify the spatial metric γij and find the extrinsic curvature Kij.

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Recall2.7 of thewe+ similarities havethatabove discussedAiaboveis− form of the Maxwell’s formk some with three-vector with the of− theADM the equations similarities ADM evolutionabove potential,k evolution− in form the of equations withMaxwell’s theequations−k the magnetic ( ADM2.134 equations (2.134) evolution and=)(11.8) and (2.135 in ( the2.135 equations), namely), namely (2.134) and (2.135), namely k k equationk (1.1), yields the Pythagorean theorem, equation (2.1232 ∂).t γija = 2αK+ij D+i β j D j βi (11.7) Minkowski spacetimeβ ∂k intoKij theK formik∂ j β Kkj∂Usingiβ ∂ this.k K decompositionij Kik∂ toj β evaluateK thekj invariant∂i β . interval ds dx dxa ,commonlyexpandedasin Minkowski spacetimej intok the form = = − + + abovefield form satisfies with theB ADMi ϵ+ evolutionijkaboveD formA equationsand+ with thereforeExercise the (2.134 ADM+) 3.7 is evolution and automaticallyConsider (+2.135equation equations), (1.1 namely the), yields+ weak-field divergence-free, (2.134 the Pythagorean) and+ limit (2.135 theorem, ofφ equation), equationis namely a gauge (2.123 (3.37). )underthesame and∂t γij∂t γij 2αK2ijαKijDi βDj i β∂Dj γj βDi j βi 2αK D β (11.7)D β(11.7) (11.7) = The entire content of any= − spatial= − tensor+ + is available+ t+ij from their spatialij components.i j j i This potential, andEvolution the electric field∂∂t EAii hasassumptionsE toEi i satisfyDDi "i " as the in Exercise constraint2.28 equation.ThencomparewiththePoissonequationforthe (2.5), (11.4)(11.4) = − + + ∂t γij 2αKij==−−Di β−−j D∂t γj βiji is obviously2αKij trueDi β forj contravariant(11.7)D j βi components, sincek (11.7) their zeroth component vanishes,1 Newtonian9 See, e.g.,andScheel gravitationaland etThe al. (1998 entire potential); Kidder content&etand ofto al. any show∂(2001t K spatialij)forexamples. thatα tensor(Rij is2 availableKikK j fromKK theirij) spatialDi D j α components.8πα(Sij Thisγij(S ρ)) 9 See,equations e.g., Scheel et al. (=1998−∂);EKidder+etD al.j jD(2001+A )forexamples.D =D j−Aj 4π+j + = (11.5)− + − − − 2 − (11.8) ∂t Eii D Dij j Ai i isDi i obviouslybutD jA alsoj holds true4iπ ji forcovariant contravariant components. components,k Therefore,(11.5) since the entire theirk zeroth contentk component of the decomposed vanishes, ==−−D E +4+πρ . − − k k β ∂k Kij(11.6)Kkik∂ j β 1 Kkj1∂i β . 1 and and i ∂t Kij∂teKEinsteinijα(Rijα( equationsRij2Kik2KK isikj K contained∂ KKK ij1KK) α inij(RD their) i DD spatialj αi2DKj α8K componentsπα8(παSijKK(Sij alone,γ)ij(SγDij and(DSρ)) weα ρ can))8πα rewrite(S the γ (S ρ)) = but= also= holds− covariant− ψ +t1 components.j +ij &−+ −ij Therefore,−+ik −j the entire−+ij2 − content2 (3.39)−i ofj− the decomposed(11.8)ij(11.8)2 ij (see(see equations equations2.112.11andand 2.122.12).). Recall Recall that thatAAi isis the the three-vector three-vectork k potential, potential,k the=k themagnetic magnetick − k + − − − − (11.8) k k i Hamiltonian1 β ∂kβKij∂k constraintKijKik=∂Kj βik (2.90∂−j β2),Kkj1∂Ki βkjk ∂.i β . k k ∂t Kij α(Rij 2KikK j ∂tKKKijijj ) kα(DRiijD j α2Kik8παK j(SijEinsteinKKijγ)ij( equationsSDi Dρ))j α is contained8πα(Sij in theirβγij∂(k spatialSKij ρ)) componentsKik∂ j β K alone,kj∂i β and. we can rewrite the In the abovefield satisfies equationsBi ρeϵijkisD thej Ak electricand therefore charge is automatically density2+ and+ divergence-free,ji the+ + current+φ density.is+2 a gauge =field satisfies− Bi +ϵijkD =A−and therefore− − is+ automatically− − − divergence-free,−9 See,(11.8) e.g., Scheel−+φetis al.2 a(1998 gauge− );+Kidderij et al.(11.8)(2001+ )forexamples. βk∂ K K ==∂ βk K ∂ββinkk∂. theK weak-fieldK ∂ βHamiltonian limit,k K assuming∂ β constraintk. suitable (2.90 boundary), R K conditionsKijK for16ψπρand, &. (2.125) In Chapterpotential,potential,k2.7 andij andwe the the haveik electric electricj discussed field fieldkj i EEi somehaskhasij to to ofsatisfy satisfy theik j the similarities the constraint constraintkj i equation of equation Maxwell’s (2.5 (),2.5 equations), + − in the= + + + + i + 9 + 9 See, e.g.,See,Scheel e.g.,theScheelet momentum al. (et1998 al. ();1998Kidder constraint);9 Kidderet al. (et2001 (2.96 al.R()forexamples.2001), K)forexamples.2 K K ij 16πρ, (2.125) above formConstraint with the ADM evolutionIn addition equations toi the (2.134 spatial) and metric (2.135 and),See, extrinsic namely e.g., Scheel et curvature, al. (1998ij); Kidder itet may al. (2001 also)forexamples. be necessary Di E i 4πρe. + (11.6)− = 9 Di E 4πρe. i ij (11.6)ij i 9 See, e.g., Scheel et al.equations(1998); KidderSee,et al. e.g.,(2001toScheel)forexamples. transformet al. (1998); theKidder= matteret al.the(2001 sourcesmomentum)forexamples.ρ constraintand S in (2.96 (3.37), D j)(K and (γ3.38K )) to8π insureS , uniqueness of (2.126) = − = In the above equations ρ∂e tisγij the electric152αK chargeij densityDi β j andDjjiβthei current density. (11.7) In the above equations ρsolutions.is the electricWe charge will largely densitythe ignore evolution and j this equationthe issue current for in the this density. extrinsic chapter,ij curvature butij it is (2.106 neverthelessi), instructive e = − + + i D j (K γ K ) 8π S , (2.126) In Chapter 2.7 we haveto discussed discuss its some origin of the in similarities passing. of Maxwell’s equations in the− = In Chapter 2.7 we have discussed some of the similarities of Maxwell’s equationsk in the 1 and above form with the ADM evolution equations (2.134the evolution∂)t andKij (2.135 equationDi D),j namelyα forα the(Rij extrinsic2KikK curvatureKKij ()2.1068πα), (Sij γij(S ρ)) We start by considering the linear= − equation+ − j + − − 2 − (2.127) above form with the ADM evolution equations (2.134) and (2.135βk D K), namelyK D βk K D βk, k 1k ij ik j kj i ∂ K α(R 2K K ∂ KKγ ) 2αKD D Dα β∂ K8παD β(SD+D α γα((+RS ρ2K)) (11.7)+K k KK ) 8πα(S 1 γ (S ρ)) t ij ij ik j t ij ij iji j i tj ij j i iji j 2 ij2u ij fu ik j ij ij 2 ij (3.40) = − +∂t γij = −−2αKij+ D−iandβ+j the=D− evolutionj βki − equation+ − for− thek spatial+(11.7) metric(11.8)k (2.108− ), − − (2.127) k k k β Dk K∇ij K=ik D j β Kkj Di β , and β ∂k Kij Kik∂ j β =K−kj∂i β .+ + + + + and + + on some+ domain '.Here f is some given function,∂t γij and2αKij weD willi β j assumeD j βi . u 0onthe(2.128) k and the evolution1 equation for the spatial= − metric+ (2.108+), = ∂t Kij α(Rij 2KboundaryikK j KK∂'ij).IfDifDisj α nonnegative8πα(Sij everywhere,2 γij(S ρ)) we can apply the maximum principle to = − k + − −Equations (−2.1251)–(2.128− ) are equivalent(11.8) to Einstein’s equations (2.83) and comprise the ∂t Kij α(Rijk 2KikK j KKk ij) Dki D j α 8πα(Sij γij(S ∂t γρij)) 2αKij Di β j D j βi . (2.128) 9 See, e.g., Scheel et= al. (1998β );∂−kKidderKij showetKik al.∂+ that(j2001β u)forexamples.Kkj−0∂i β everywhere.. “standard”− The 3 point−12 equations. is that− Sometimes if=u−were(11.8) they nonzero+ are referred+ somewhere to as the “ADM” in ' equations,say after + k + k+ = k + 16 17 β ∂k Kij positive,Kik∂ j β thenK itkj∂ musti β . Equations haveArnowitt, a maximum (2.125 Deser)–( and2.128 somewhere. Misner,) are equivalenteven Atthough the to these Einstein’s maximum equations equations the had been left (2.83 derived hand) and side earlier, compriseand the + + + 18 “standard”even though 3 Arnowitt1 equations.et al. Sometimes(1962) derived they the are equations referred to in as a different the “ADM” form. equations after 9 See, e.g., Scheel et al. (1998);ofKidder (3.40et al.)mustbenegative,buttherighthandsideisnonnegativeif(2001)forexamples. + f 0, which is Arnowitt, Deser and Misner,16 even though these equations had≥ been derived earlier,17 and 9 a contradiction. Clearly,16 the argument works the same way if u is negative somewhere, See, e.g., Scheel et al. (1998); Kidder et al. (2001)forexamples.even thoughArnowitt etArnowitt al. (1962). et al. (1962) derived the equations in a different form.18 implying that u 0everywhereif17 E.g., Darmois (f1927); 0.Lichnerowicz (1944); Foures-Bruhat` (1956). = 18 See, e.g., Anderson≥ and York, Jr. (1998)foradiscussion. Now consider the nonlinear16 Arnowitt et equation al. (1962). 17 E.g., Darmois (1927); Lichnerowicz (1944); Foures-Bruhat` (1956). 2 n 18 See, e.g., Anderson and York,u Jr. (1998fu)foradiscussion., (3.41) ∇ = and assume there exist two positive solutions u1 and u2 u1 that are identical, u1 u2, ≥ = on the boundary ∂'.Thedifference)u u2 u1 must then satisfy an equation = − 2 n 1 )u nfu˜ − )u, (3.42) ∇ = where u˜ is some positive function satisfying u1 u˜ u2. Applying the above argument to ≤ ≤ )u, we see that the maximum principle implies )u 0 and hence uniqueness of solutions =

15 See, e.g., Yo r k , J r. (1979). 1 1

R = 1 R = 1

R =2M R =2M

R =0 R =0

378 Chapter 11 Recasting the evolution equations a nae =0 a (i) nae =0 however, exponentially growing modes are still(i) very bad, and can easily terminate a simulation after only a short time. In fact, many of the symmetric or strongly hyperbolic systems that have been introduced over the years lead to such exponentially growing modes ↵⌦a = na = at in numerical implementations. Unless these↵⌦a modes= na = can bea controlled,t 9 these systems are not r r very useful for numerical simulations. From a numerical perspective, then, well-posedness is a necessary but not a sufficient condition. In the followingna we will=0 discuss severala different approaches that lead to different ab n ab =0 reformulations of the evolution equations that are both strongly hyperbolic and that have been used successfully in numerical simulations. In the remainder of this chapter we will pursue a more heuristic approach than the one sketched in this section and will ↵i i marshal mathematical intuition and exploit analogies↵ to motivate the different approaches. In fact, some of the systems discussed below were developed in exactly this way and were discovered to have desirable numerical properties empirically before they were analyzed mathematically and revealedi to be strongly hyperbolic.i To illustrate some of these heuristic A ni A ni arguments for a simple and familiar example we will begin by discussing Maxwell’s equations in Section 11.2.

i Curvature can be intrinsicn or extrinsic ni 11.2 Recasting Maxwell’s equations In Chapter 2.2 we have seen that we can bring Maxwell’s evolution equations in a i i Minkowski spacetimeA into theij form A () () ij ∂t Ai Ei Di " (11.4) = − − ∂ E D j D A D D j A 4π j (11.5) i t i j ii i j i E ij = − E + − () () ij (see equations 2.11 and 2.12). Recall that Ai is the three-vector potential, the magnetic j k field satisfies Bi ϵijkD A and therefore is automatically divergence-free, φ is a gauge Intrinsic= curvature: Extrinsic curvature: potential, and the electric field Ei has to satisfy the constraint equation (2.5), Ai Ei ij KAiij Ei ij Kij i Di E 4πρe. (11.6) =

In the above equations ρe is the electric charge density and ji the current density. In Chapter 2.7 we have discussed some of the similarities of Maxwell’s equations in the above form with the ADM evolutionRelation: equations (2.134) and (2.135), namely

∂t γij 2αKij Di β j D j βi (11.7) = − + + and k 1 ∂t Kij α(Rij 2KikK KKij) Di D j α 8πα(Sij γij(S ρ)) = − j + − − − 2 − (11.8) k k k β ∂k Kij Kik∂ j β Kkj∂i β . + + +

9 See, e.g., Scheel et al. (1998); Kidder et al. (2001)forexamples. Outline

History and formalism Initial data

Evolution of field equations Wave extraction

The tools we use 46 Chapter 2 The 3+1 decompostion of Einstein’s equations

i i B t+dt dx +ß dt xi +dx i i 1 1 46 Chapter 2 The 3+1 decompostion of Einstein’s equations ßidt xi dx

a a t dt dxa i i B an dtt+dt dx +ß dt xi +dx i i t ßidt xi dx R = R = 1 1xi a a t dt a A an dt dx Figure 2.4 Pythagorean theoremt in 3 1 dimensional spacetime. The normal vector αna and the time vector ta + connect points on two neighboring spatial slices. The shift vector βi resides in the slice and measures their xi R =2MR =2M A difference. The infinitesimal displacement vector dxa connects two nearby, but otherwise arbitrary, points on neighboring slices (e.g., the point A at xi on slice t and the point B at xi dxi on slice t dt). The total + + Figure 2.4 Pythagorean theorem in 3 1 dimensionaldisplacement spacetime. vector Thedx normala t vectora dt αdxnai ,whereand thedx timei is vector the spatialta vector drawn in the figure, may be decomposed + i = + connect points on two neighboring spatial slices. Thealternatively shift vector intoβ tworesides vectors in the that slice form and the measures legs of their a right-triangle, dxa (αna dt) (dxi βi dt), as shown. Constructing initiala data = + + difference. The infinitesimal displacement vector dx connects two nearby,R =0 butR otherwise=0 arbitrary, points on 2 a ConstructingUsing initial this decomposition to data evaluate the invariant interval ds dx dxa ,commonlyexpandedasin 3 neighboring slices (e.g., the point A at xi on slice t and the point B at xi dxi on slice t dt). The total = Initial data equation (1.1), yields the+ Pythagorean+ theorem, equation (2.123). 3displacement vector dxa ta dt dxi ,wheredxi is the spatial vector drawn in the figure, may be decomposed = + a a i i alternatively into two vectors that form the legs of a right-triangle,The entiredx content(αn ofdt) any(dx spatialβ dt tensor), as shown. is available from their spatial components. This 2 a = + + Using this decomposition to evaluate the invariant interval ds dx dxaa ,commonlyexpandedasina is obviously= n trueae forn=0 contravariante =0 components, since their zeroth component vanishes, equation (1.1), yields the Pythagorean theorem, equation (2.123). (i) a (i) but also holds covariant components. Therefore, the entire content of the decomposed The entire content of any spatial tensorEinstein is available equations from their is contained spatial components. in their spatial This components alone, and we can rewrite the is obviously true for contravariant components,Hamiltonian since constraint their zeroth (2.90 component), vanishes, ↵⌦a =↵n⌦aa == na =at at As we have seenAs in we Chapterbut have also seen2 holds, the in Chapter covariantspatial2 metric, components. the spatialγij, metricthespatial Therefore, extrinsicγ metric, the the extrinsiccurvature entire contentr curvatureKij ofand theK decomposed any2andextrinsic any ij curvature 12 initial degrees of freedom ij rR Kij KijK 16πρ, (2.125) matter fields havematter to satisfyEinstein fields the have equations Hamiltonian to satisfy is contained the Hamiltonian constraint in their spatial( constraint2.132 components), (2.132), alone, and we+ can rewrite− the = Hamiltonian constraint (2.90), the momentum constraint (2.96), 2 ij 2 ij a R K KRijK K 16K2 πρijK, ij16πρ, a (3.1)ij (3.1)ij i + R −K Kij=K 16n πρab, n=0ab =0D j (K (2.125)γ K ) 8π S , (2.126) - 4 constraint equations+ − =+ − = − = and the momentum constraint (2.133), and the momentum constraintthe momentum (2.133 constraint), (2.96), the evolution equation for the extrinsic curvature (2.106), ij ij i ij ijD j (K ijγ iK )ij 8π S i (3.2) D j (K γ K )D j (−K8π S∂γ K=K ) 8Dπ SD, α iα(R 2K(3.2)K k (2.126)KK ) 8πα(S 1 γ (S ρ)) − t ij = ↵i j ij i ik j ij ij 2 ij - 4 coordinate conditions −(lapse and= shift) = − k +↵− k + k − − − (2.127) on every spacelike hypersurface $. Before we can evolveβ theDk fieldsKij toKik obtainD j β a spacetimeKkj Di β , on every spacelike hypersurfacethe evolution$ equation. Before for we the can extrinsic evolve curvature the fields (2.106+ to), obtain+ a spacetime+ that satisfies Einstein’s equations, we have to specify gravitational fields (γij, Kij)onsome that satisfies Einstein’s equations, we have to specify gravitationalkand the evolution fields equation (γ ,1 K for)onsome the spatial metric (2.108), ∂t Kij Di D j α α(Rij 2KikK KKij) 8πα(Sij ij γijij(S ρ)) initial spatial slice= − $ that+ are compatible− withj + the constraint− equations.− 2 These− fields(2.127) can initial spatial slice $ that are compatibleβk D K withK D theβk constraintK D βk, equations. These fields can then be used as “startingk ij values”ik forj a dynamicalkj i evolution obtained by∂t γij integrating2αKij the Di β j D j βi . (2.128) + + + = − + + then be used as “startingevolution equations values” ( for2.135 a) dynamical and (2.134). evolution1 obtained by integrating the 4 remainingand degrees the evolution of equationfreedom1 for the spatialEquations metric2 dynamical (2.108 (2.125), )–(2.128 GW) are polarizations equivalent to Einstein’s equations (2.83) and comprise the evolution equationsClearly, (2.135 the) and four (2.134 constraint). equations (3.1) and (3.2) cannot determine all of the gravita- ∂ γ 2α“standard”K D β 3 D1 equations.β . Sometimes they(2.128) are referred to as the “ADM” equations after Clearly, the four constraint equations (3.1) andt ij (3.2) cannotij determinei j + j i all of the16 gravita- 17 tional fields (γij, Kij). Since both γij and= −KijArnowitt,are+ symmetric, Deser+ and 3-dimensional Misner, even tensors, though they these equations had been derived earlier, and tional fields (γij,togetherKij).Equations Since have both12 ( independent2.125γij)–(and2.128K components.)ij areare equivalent symmetric, Theeven to four Einstein’s though 3-dimensional constraintArnowitt equations equationset (al. tensors,2.83(1962) can and) only theyderived comprise deter- the the equations in a different form.18 together have 12mine independent four“standard” of these, components. 3 leaving1 equations. eight The undetermined. Sometimes four constraint they Four are of referred equations these eight to as the undetermined can “ADM” only deter- equations functions after + 16 17 mine four of these,are leaving relatedArnowitt, toeight coordinate Deser undetermined. and choices: Misner, three Foureven specify though of these16 Arnowitt thesethe eight spatial equationset al. undetermined(1962 coordinates). had been within derived functions the earlier, slice $and, evenHow though doArnowitt youet al.specify(1962) derived those17 E.g., theDarmois equations degrees(1927 in); Lichnerowicz a different of freedom?( form.1944);18Foures-Bruhat` (1956). are related to coordinateand these choices: coordinates three can specify be chosen the arbitrarily spatial18 See, coordinates without e.g., Anderson changing and within York, the Jr. the physical(1998 slice)foradiscussion. properties$, of the slice. One function, associated with the time coordinate, can be used to specify the and these coordinates can16 Arnowitt be chosenet al. (1962). arbitrarily without changing the physical properties of the slice. Onechoice function,17 ofE.g., the associatedDarmois hypersurface(1927); withLichnerowicz$ the. For time(1944 any); given coordinate,Foures-Bruhat` spacetime(1956 can). solution be used to to Einstein’s specify equations, the 18 See, e.g., Anderson and York, Jr. (1998)foradiscussion. choice of the hypersurfacechoosing different$. For hypersurfaces any given spacetime on which to solution obtain initial to Einstein’s data, and equations, then propagating choosing differentthese hypersurfaces data, regenerates on physically which to equivalent obtain initial spacetimes. data, Thisand leavesthen propagating four undetermined functions that represent the two dynamical degrees of freedom characterizing a gravita- these data, regenerates physically equivalent spacetimes. This leaves four undetermined tional field in general relativity, e.g., two independent sets of values for the conjugate pair functions that represent the two dynamical degrees of freedom characterizing a gravita- (γij, Kij). These two dynamical degrees of freedom correspond to the two polarization tional field in generalmodes relativity, of a gravitational e.g., two wave independent in general relativity. sets of2 values for the conjugate pair (γij, Kij). These twoIt is dynamical quite intuitive degrees that the ofstate freedom of a dynamical correspond field, like to a the gravitational two polarization wave, cannot be 2 modes of a gravitationaldetermined wave from in constraint general relativity.equations. Waves satisfy hyperbolic equations, and their state It is quite intuitive that the state of a dynamical field, like a gravitational wave, cannot be 1 If there is any matter present, then the matter fields must be specified on the initial hypersurface as well, and the determined from constraint equations. Wavesab satisfy hyperbolic equations, and their state matter evolution equations, b T 0, must be integrated simultaneously with the field evolution equations to build ∇ = aspacetime.SometypicalmattersourcesandtheirevolutionequationsarediscussedinChapter5. 1 If there is any matter2 This present, is the then same the number matter of degrees fields must of freedom be specified in the linearized on the theory initial of hypersurface general relativity as well, appropriate and the for a weak ab matter evolution equations,gravitationalb T field;0, the must field equations be integrated in this simultaneously limit reduce to propagation with the equations field evolution for a spin-2 equations linear field to buildin Minkowski ∇ = aspacetime.SometypicalmattersourcesandtheirevolutionequationsarediscussedinChapterspacetime. 5. 2 This is the same number of degrees of freedom in the linearized theory of general relativity appropriate for a weak gravitational field; the field equations in this limit reduce to propagation equations for a spin-2 linear field in Minkowski spacetime. 54

54 3.1 Conformal transformations 59

Schwarzschild solution in isotropic coordinates, which we have seen before in Chapter 2. Inspection of the metric (2.147),

4 4 2 i j M i j M 2 2 2 2 2 dl γijdx dx 1 ηijdx dx 1 dr r (dθ sin θdφ ) , = = + 2r = + 2r + + ! " ! " # $ (3.19) shows that it is explicitly written in the form (3.5), where the conformally related metric is the flat metric in spherical polar coordinates. This solution forms the basis of the so- called puncture methods for black holes, which we will discuss in much greater detail in Chapters 12.2.2 and 13.1.3. While the formalism of conformal decomposition may appear unnecessarily technical and perhaps confusing initially, this example demonstrates that it provides an extremely powerful tool for constructing solutions to Einstein’sequations. In fact, once the formalism has been developed, it is much easier to derive the Schwarzschild solution by going through the above steps then deriving it from Einstein’s equations directly. Perhaps even more impressively, we will see below that we can trivially generalize this method to construct multiple black hole initial data. Before we do that, though, it is useful to discuss the single black hole solution in more detail. The solution (3.19)issingularatr 0. However, we can show that this singularity is = only a coordinate singularity by considering the coordinate transformation

2 1 r M , (3.20) = 2 rˆ ! " under which the isotropic Schwarzschild metric (3.19)becomes

4 dl2 1 M drˆ2 rˆ2(dθ 2 sin2 θdφ2) . (3.21) = + 2rˆ + + ! " # $ The geometry described by metric (3.21)evaluatedataradiusrˆ a is identical to that = of the metric (3.19)evaluatedatr a. The mapping (3.20)thereforemapsthemetric = into itself, and is hence an isometry. In particular, this demonstrates that the origin r 0 = is isomorphic to spatial infinity, which is perfectly regular. The same conclusion can be reached by considering a coordinate transformation between the isotropic radius r to a SchwarzschildSchwarzschild areal radius Rin. Thisisotropic demonstrates coordinates that the isotropicis good radiusfor r covers only the black holeinitial exterior, data and that avoiding each Schwarzschild singularitiesR corresponds to two values of the isotropic radius r.

1 Exercise 3.4 Find the coordinate transformation between the isotropic radius r and 1 1 Schwarzschild radius R that brings the isotropic Schwarzschild metric (3.19)into Schwarzschild coordinates: R = the Schwarzschild form 1 R = 1 1 R = 2M − 1 2 2 2 2 2 2 R =2M dl 1 dR R (dθ sin θdφ ). (3.22) R =2M = − R + 3.1 Conformal+ transformations 59 3.1 Conformal transformations 59 ! " R =2M R =0 Schwarzschild solution in isotropic coordinates,Schwarzschild which solution we have seen in isotropic before in coordinates, Chapter 2. which we have seenR =0 before in Chapter 2. Inspection of theIsotropic metric (2.147 coordinates:), Inspection of the metric (2.147), R =0 4 4 4 4 2 i j M i 2j iM j 2 M2 2 2i j 2 M 2 2 2 2 2 dl γijdx dx 1 ηijdx dldx γij1dx dx dr1 r (dθ ηijsindx dxθdφ ) , 1 dr r (dθ sin θdφ ) , = = + 2r == + 2r = ++ 2r + = + 2r + + ! " ! " # ! " $! " # $ (3.19) (3.19) shows that it is explicitly written in the form (3.5), where the conformally related metric Conformal factorshows that it is explicitly written in the form (3.5), where the conformally related metric is the flat metric in spherical polar coordinates.is the flat This metric solution in sphericalFlat forms space the polar basis coordinates. of the so- This solution forms the basis of the so- called puncture methods for black holes,called whichpuncture we willmethods discuss in for much black greater holes, detail which in we will discuss in much greater detail in Chapters 12.2.2 and 13.1.3. Chapters 12.2.2 and 13.1.3. While the formalism of conformal decomposition may appear unnecessarily technical • r = 0 is a coord. singularity,While corresponding the formalism of conformal decomposition may appear unnecessarily technical and perhaps confusing initially, this example demonstrates that it provides an extremely to infinity in the “otherand universe” perhaps confusing initially, this example demonstrates that it provides an extremely powerful tool• forOutside constructing the horizon solutions this to Einstein’sequations. is standard Schwarzschild In fact, once the formalism has been developed, it is much easier topowerful derive the tool Schwarzschild for constructing solution solutions by going to through Einstein’sequations. In fact, once the formalism the above steps then deriving it fromhas Einstein’s been developed, equations it directly. is much Perhaps easier to even derive more the Schwarzschild solution by going through impressively, we will see below that wethe can above trivially steps generalize then deriving this method it from to construct Einstein’s equations directly. Perhaps even more multiple black hole initial data. Beforeimpressively, we do that, though, we will it is see useful below to discuss that we the can single trivially generalize this method to construct black hole solution in more detail. multiple black hole initial data. Before we do that, though, it is useful to discuss the single The solution (3.19)issingularatr black0. However, hole solution we can in show more that detail. this singularity is = only a coordinate singularity by consideringThe the solution coordinate (3.19 transformation)issingularatr 0. However, we can show that this singularity is = only a coordinate singularity by considering the coordinate transformation 2 1 r M , (3.20) = 2 rˆ 2 1 ! " r M , (3.20) = 2 rˆ under which the isotropic Schwarzschild metric (3.19)becomes ! " 4 under which the isotropic Schwarzschild metric (3.19)becomes dl2 1 M drˆ2 rˆ2(dθ 2 sin2 θdφ2) . (3.21) = + 2rˆ + + 4 ! " # dl2 $1 M drˆ2 rˆ2(dθ 2 sin2 θdφ2) . (3.21) The geometry described by metric (3.21)evaluatedataradius=rˆ a +is identical2rˆ to that+ + =! " of the metric (3.19)evaluatedatr a. The mapping (3.20)thereforemapsthemetric# $ = The geometry described by metric (3.21)evaluatedataradiusrˆ a is identical to that into itself, and is hence an isometry. In particular, this demonstrates that the origin r 0 = of the metric (3.19)evaluatedatr a. The= mapping (3.20)thereforemapsthemetric is isomorphic to spatial infinity, which is perfectly regular. The same conclusion= can be into itself, and is hence an isometry. In particular, this demonstrates that the origin r 0 reached by considering a coordinate transformation between the isotropic radius r to a = Schwarzschild areal radius R. This demonstratesis isomorphic that to the spatial isotropic infinity, radius whichr covers is perfectly only regular. The same conclusion can be the black hole exterior, and that each Schwarzschildreached by consideringR corresponds a coordinate to two values transformation of the between the isotropic radius r to a isotropic radius r. Schwarzschild areal radius R. This demonstrates that the isotropic radius r covers only the black hole exterior, and that each Schwarzschild R corresponds to two values of the Exercise 3.4 Find the coordinate transformationisotropic radius betweenr. the isotropic radius r and Schwarzschild radius R that brings the isotropic Schwarzschild metric (3.19)into the Schwarzschild form Exercise 3.4 Find the coordinate transformation between the isotropic radius r and 1 2M − Schwarzschild radius R that brings the isotropic Schwarzschild metric (3.19)into dl2 1 dR2 R2(dθ 2 sin2 θdφ2). (3.22) = − R +the Schwarzschild+ form ! " 1 2M − dl2 1 dR2 R2(dθ 2 sin2 θdφ2). (3.22) = − R + + ! " 12.2 The conformal transverse-traceless approach 403 3.1 Conformal transformations 57 located at Exercise 3.1 Verify the transformation law (3.7)from(2.141). 1/5 rISCO (24λ) R 2.05R, (12.36) Exercise 3.2 Show that the covariant derivative= associated≈ with the connection (3.7) is compatibleimplying with that such the conformally a binary would related reach metric, the ISCO just before the two stars touch. We have plotted this result in Figure 12.2 for identical stars of compaction m/R 0.2, where D¯ i γ¯jk 0. (3.9)= m M/2 is the mass of the individual= stars; we also show the combined effects of = For the Riccirelativistic tensor corrections we similarly and find tidal interactions in determining the ISCO. For compressible stars λ is smaller than 3/2, so that in Newtonian gravitation these stars would merge before lm theyRij couldR¯ encounterij 2 D¯ anyi D¯ ISCO.j ln ψ However,γ¯ijγ¯ D in¯ l generalD¯ m ln ψ relativity, for identical irrotational = − + companions constructed with a moderately stiff equation of state, the entire equilibrium ! lm " 4 (D¯ i ln ψ)(D¯ j ln ψ) γ¯ijγ¯ (D¯ l ln ψ)(D¯ m ln ψ) , (3.10) 58 Chapter 3sequence Constructing actually+ initial terminates data at a finite− separation prior to merger, at which point the stellar density profiles! form a cusp (see Chapter 15.3). Orbital decay beyond" this point will then and for the scalar curvature Let us further choosetrigger the some conformally tidal disruption related prior metric to eventual to be flat, merger (see Chapter 16). After reaching the ISCO,R theψ binary4 R¯ 8 companionsψ 5 D¯ 2ψ. plunge and merge on an orbital(3.11) time γ¯ η− . − (3.14) scale. Depending on the natureij= andij masses− of the compact companions, their coalescence 2 ij = Here D¯46 γChapter¯ D¯ i D 2¯ Thej is 3+1 the decompostion covariant of Laplace Einstein’s operator equations associated withγ ¯ij. Whenever this= isleads the case, to the we formation call the of physical a merged spatial object, metric or remnant,γij conformally that may flat either. 3.1 be Conformal a rotating transformations neutron 61 We again emphasizestar or a black that we hole. only During consider the final conformalringdown transformationsphase of the binary of the evolution spatial this remnant Exercise 3.3 Verify equations (3.10)and(3.11). t+dt settles down into an equilibriumi i state. In theB case of a hypermassive neutron star, secular metric in this text. Accordingly, “conformaldx +ß flatness”dt refers, forxi +dx ouri purposes, to the spatial i metricInserting and not theeffects the spacetime scalar (viscosity curvature metric. or magnetic Inß ( fouridt3.11 or) fields)x into anyi dx higher the can Hamiltonian lead dimensions, later to delayed constraint we can collapse evaluate (3.1 to) the yieldsa black hole (see

Weyl tensor to examineChapters whether14 and any16.) givena metric is conformally flat. This is a consequence ¯ 2 nadt t¯ dt 5 2dxa 5 ij 5 of the fact that theDuring Weyl8 tensor theD aψ inspiral (1.26ψ)R isphase, invariantψ upK to under reasonablyψ K conformalijK small transformations binary16πψ separations,ρ, of the the binary(3.12) can − − + = −t spacetime metricbe – modeled this explains very accurately why it is by often post-Newtonian called the conformal expansions tensor (see.7 AppendixSince it D). The ring- which, for a given choice ofi the conformally related metricγ ¯ ,wemayinterpretas vanishes for a flatdown metric, phase it must can be also describedx vanish for very all well geometries by strong-field that are perturbative conformallyij techniques. related Numerical an equation for the conformalA factor ψ. The extrinsic curvature K has to satisfy the to the flat metric.relativity In three simulations dimensions, are however, needed tothe connect Weyl tensor these vanishes two regimes, identically,ij beginning so with the late Figure 2.4 Pythagorean theorem in 3 1 dimensional spacetime. The normal vector αna and the time vector ta momentuminspiral constraint phase, (3.2 and), continuing and+ it will through be useful the dynamical to rescale plungeKij conformally and merger phase. as well. Numerical We that it no longerconnect providespoints on two a neighboring useful diagnostic. spatial slices. The For shift a spatialvector βi resides metric, in the we slice may and measuresinstead their evaluate 3.1 Conformal transformations 61 thewillBach doordifference. thatCotton–Yorkrelativity in The Section infinitesimal istensor also3.1.3 displacement required, after vector to discussing treatdxa connects delayed in two collapse. Section nearby, but otherwise Not3.1.2 surprisingly, arbitrary,some points elementary considerable on solutions effort in to (3.12neighboring) fornumerical which slices (e.g.,K relativity the point0.A at hasxi on focused slice t and on the point compactB at xi binariesdxi on slice int closedt). The quasicircular total orbits. In the ij 1 + + displacement vector dxa ij=ta dt Figure1dx/3i ,where 3.2 Schematicdxi is the spatialj embedding vectorj diagram drawn in of the the figure, geometry may be described decomposed by metric (3.23)fortwoblackholesata rest of thisB chapter= γ+ we[ willikl]D constructk Rl initialδl R data, for binary black holes(3.15) in such orbits. Thisalternatively generalizes into two vectors to= that momentmultiple form the of legs time of black symmetry.a right-triangle, −holes This4 dx isa the three-sheeted(αna dt) (dx topology,i βi dt), which as shown. does not satisfy an isometry across each ! 2 a =" + + Using this decomposition to evaluatethroat. the invariant interval ds dx dxa ,commonlyexpandedasin = 8 which vanishesequation if (1.1 and), yields only the if Pythagorean3.1.2 the spatial Elementary theorem, geometry equation ( is2.123 black conformally). hole solutions flat. In equation (3.15), 1 [ikl] is the completely12.2 antisymmetric, The conformal or permutation, transverse-traceless symbol, with [123] approach:1. Bowen–York we obtain the solution simply by adding the= individual contribution of each black hole AsAt an this aside, pointThe we entire it note is content instructive that any of anyspherically to spatial consider tensor symmetric is some available spatial simple, from metric their but spatial physically is always components. conformally interesting, This solutions Assume: 12.2.1 Solving the momentum constraint to the constraintis obviously true equation. for contravariantaccording Consider components, to vacuum sincesolutions their zeroth for4 which component the mattervanishes, source terms flat, meaning that we can always write such a metric4 as γij ψ ηij.Foranyspherically but• Conformal also holds covarianti flatness: components. Therefore,ij = ⌘ij the entire= content of the decomposed symmetricvanish space, (ρ In0 we the may transverse-tracelessS , hence etc.) and assume focus conformal approach on a “moment we flatness assume without of both time conformal loss symmetry”. of generality. flatness,α At so a moment thatγ ¯ij ofηij, Einstein= equations= is contained in their spatial components alone,ψ and1 we canM rewrite. the = (3.23) • Maximaland maximal slicing: slicing, so that K 0. The momentum constraint (12.4)thenreducesto Assumingtime symmetry, conformal all flatness time derivatives dramatically of simplifiesγij are zero all calculations, and the= 4-dimensional since+ α D¯ 2i rreducesα line to interval has Hamiltonian constraint (2.90), = ! theto flat be covariant invariant• Time derivative reversal under (and timeinvariance: in particular reversal, tot partialt derivatives. Theij latter in Cartesian condition coordinates). implies that the shift 2 i i ij D¯ A¯ 0, (12.37)i HereRrα K x →−KijCKα is the16j πρ (coordinate), separation from(2.125) the center Cα of the αth black Moreover,must satisfyFigure the Ricci 3.2βSchematici tensor0and,hence,byequation( embedding and scalar diagram curvature+ of=| the− geometryR−= associated| described=2.134 by with metric=), the ( the3.23 extrinsic)fortwoblackholesata conformally curvature related also has to hole. The total mass1 of the spacetime is the sum of the coefficients α.However,since momentthe momentumwhere of time= symmetry.D¯ constraint¯is now This¯ is ( a2.96 the flat-space three-sheeted), covariant topology, which derivative. does not6 satisfy In an Cartesian isometry across coordinates each thisM operator metricvanish mustThen: everywhere now vanish, jR onij theRthe slice, total0. Under i.e., massK this willij assumption, also0 includeK . the contributionsOn Hamiltonian such a from time constraint the slice black the hole momen- interactions, can throat. = = = = α becomes the remarkablyreduces to simple the partial Laplace derivative equationij ij∂ j . Our assumptionsi have simplified the momentum M tum constraints• Momentum (3.2 constraint) are satisfiedbe identified D j ( K trivially. with γ K the ) The mass 8 π S Hamiltonian of, satisfied the αth blacktrivially constraint hole only(2.126) in (3.12 the limit)reduces of large separations. − = ¯ constraint to the pointParticularly where2 theinterestingR =2 decompositionM astrophysically of Aij andinto for a transverse the generation and of a gravitational longitudinal waves is the to wethe• Hamiltonian obtain evolution the equation solution constraint: for simply the extrinsic byD¯ addingψ curvature the0, individual (2.106), contribution of each black(3.16) hole accordingpart to (see equationcase3.48 of) binary is somewhat= black holes, pointless, in which since case both (3.23 of)reducesto them now have vanishing ¯ 2 k 2 1 1 where D is∂t nowKij theD flati D j α Laplaceα(Rij operator.2KikK D We¯ ψKK willij) beψ8 interestedπαR¯.(Sij γ inij(Sasymptoticallyρ)) flat (3.13) j 2 1 2 = − k + − k + k= 8−α − − (2.127) solutions thatThis satisfy equationβ D kisK ijlinear!Kik D j β ψ Kkj1RDi =0β , M . ψ 1 M M(3.23). (3.24) + + + = + α 2rα = + 2r1 + 2r2 ! and the evolution equationψ for1 the spatial(r 1)for metric (2.108r ), (3.17) 6 i i This simple− solution to the constraint equationsi for two black holes instantaneously at rest A4-geometryissaidtobetime-symmetricifthereexistssuchatimeslice.Here rα x Cα is→ the (coordinate)+ O separation from→∞ the center Cα of the αth black =| − | at a moment of time symmetry can be used as initial data for head-on collisions of black where r ishole. the coordinate The total mass radius. of the Spherically spacetime∂t γij 2 isα symmetric theKija sumD ofi β j the solutions coefficientsD j βi . are α.However,since(2.128) = − nae(i)+=0 + M the total mass will also includeholes contributions (see Chapter from13.2). the black hole interactions, α can Equations (2.125)–(2.128) are equivalent to Einstein’s equations (2.83) and compriseM the be identified with the mass ofWe the canαth now black define hole mappings only in the equivalent limit of large to (3.20 separations.), which represent reflections through “standard” 3 1 equations. Sometimesψ 1 theyM are. referred to as the “ADM” equations(3.18) after Particularly interesting+ astrophysicallythe αth throat.= and+ In for2 general,r the generation the existence of gravitational of other black waves holes is the destroys the symmetry that we Arnowitt, Deser and Misner,16 even though these equations had been derived earlier,17 and case of binary black holes, in which case (3.23)reducesto We will show in exercise 3.20 belowfound that, for a in⌦singlea this= black particularat hole. Each case, Einstein–Rosen the constant18 bridgeis in therefore connects to its own even though Arnowitt et al. (1962) derived ther equations in a different form. M fact the black hole mass M. It shouldn’tasymptotically come flat as Universe.Misner, a great Drawing1960 surprise an that embedding this is just diagram the for such a geometry yields ψ 1 1 2 . (3.24) 16 several differentM “sheets”,M where each sheet corresponds to one Universe. A geometry Arnowitt et al. (1962). = + 2r1 + 2r2 7 See, e.g., page17 E.g., 130Darmois in Carroll(1927(2004); Lichnerowicz). containing(1944); FourNes-Bruhat` black( holes1956). may contain up to N 1 different asymptotically flat universes 8 This18 See, simple e.g., Anderson solution and York,to the Jr. constraint(1998)foradiscussion. equationsna = ↵ for⌦a two black holes instantaneously+ at rest See Eisenhart (1926); Yo r k , J r. (1971); see(see also Figure Problem3.2 21.22). in Misner et al. (1973). Note that the Bach tensor is a tensor densityat arather moment than a of tensor; time see symmetry AppendixIf candesired,A.3. be used however, as initial the data isometry for head-on across collisions the throats of black can be restored as follows. Recall holes (see Chapter 13.2). that equation (3.16) is equivalent to the Laplace equation in electrostatics, so that we We can now define mappings equivalent to (3.20), which represent reflections through na =0 the αth throat. In general, the existence of otherab black holes destroys the symmetry that we found for a single black hole. Each Einstein–Rosen bridge therefore connects to its own asymptotically flat Universe. Drawing an embedding diagram for such a geometry yields several different “sheets”, where each sheet↵ correspondsi to one Universe. A geometry containing N black holes may contain up to N 1 different asymptotically flat universes + (see Figure 3.2). If desired, however, the isometry across the throats can be restored as follows. Recall i that equation (3.16) is equivalent to theA Laplaceni equation in electrostatics, so that we

ni

Ai () ij

Ei () ij

Ai Ei ij Kij 12.2 The conformal transverse-traceless approach 403

located at

1/5 rISCO (24λ) R 2.05R, (12.36) = ≈ implying that such a binary would reach the12.2 ISCO The just conformal before transverse-traceless the two stars touch. approach We 403 have plotted this result in Figure 12.2 for identical stars of compaction m/R 0.2, where located at = m M/2 is the mass of the individual stars; we also show the combined effects of = 1/5 relativistic corrections and tidal interactionsrISCO in determining(24λ) R 2 the.05 ISCO.R, For compressible(12.36) = ≈ stars λ is smaller than 3/2, so that in Newtonian gravitation these stars would merge before implying that such a binary would reach the ISCO just before the two stars touch. We they could encounter any ISCO. However, in general relativity, for identical irrotational have plotted this result in Figure 12.2 for identical stars of compaction m/R 0.2, where = companions constructedm M/2 is with the mass a moderately of the individual stiff equation stars; weof state, also show the entire the combined equilibrium effects of = sequence actuallyrelativistic terminates corrections at a finite and separation tidal interactions prior to in merger, determining at which the ISCO. point Forthe stellar compressible density profilesstars formλ is smaller a cusp (see than 3Chapter/2, so that15.3 in). Newtonian Orbital decay gravitation beyond these this stars point would will merge then before trigger somethey tidal could disruption encounter prior any to eventual ISCO. However, merger in (see general Chapter relativity,16). for identical irrotational After reachingcompanions the ISCO, constructed the binary with companions a moderately plunge stiff equation and merge of state, on an the orbital entire equilibriumtime scale. Dependingsequence on the actually nature terminates and masses at a offinite the12.2 separation compact The conformal prior companions, to merger, transverse-traceless their at which coalescence point approach the stellar 405 leads to the formationdensity profiles of a merged form a object, cusp (see or Chapter remnant,15.3 that). Orbital may either decay be beyond a rotating this neutron point will then star or a blacktrigger hole.Given some During that tidal the disruption momentum final ringdown prior constraint tophase eventual ( of12.37 merger the)islinear,wecanconstructa binary (see evolution Chapter 16 this). remnantbinary black settles down intoholeAfter an solution equilibrium reaching by the superposition state. ISCO, In the the binary of case single companions of a solutions hypermassive plunge and neutron merge star, on an secular orbital time scale. Depending on the nature and masses of the compact companions, their coalescence effects (viscosity or magnetic fields) can lead laterij to delayedij collapseij toij a black hole (see leads to the formation of a mergedA¯ij object,A¯ orA remnant,¯ A¯ that mayA either¯ . be a rotating neutron(12.44) Chapters 14 and 16.) = C1P1 + C1S1 + C2P2 + C2S2 star or a black hole. During the final ringdown phase of the binary evolution this remnant During thesettles inspiralThis down completes phase, into an up an equilibrium to analytic reasonably solution state. small In of the the binary case momentum of separations, a hypermassive constraint the neutron binary describing star, can secular two black be modeled veryeffectsholes accurately (viscosity with arbitrary by or magnetic post-Newtonian momenta fields) and can spins. expansions lead later to (see delayed Appendix collapseD). to The a black ring- hole (see down phase can be described very well by strong-field perturbative techniques. Numerical Chapters Exercise14 and 16 12.6.) Show that9 relativity simulationsDuring are the needed inspiral tophase, connect up to these reasonably two regimes, small binary beginning separations, with the the binarylate can be modeled very accurately by post-NewtonianP P1 expansionsP2 (see Appendix D). The(12.45) ring- inspiral phase, and continuing through the dynamical plunge= + and merger phase. Numerical relativity is alsodown required phaseis the can to total treat be described linear delayed momentum very collapse. well of bythe Not strong-field solution surprisingly, (12.44 perturbative)and considerable techniques. effort Numerical in numerical relativityrelativity has simulations focused on are compact needed to binaries connect in these close two quasicircular regimes, beginning orbits. In with the the late J C1 P1 C2 P2 S1 S2 (12.46) rest of this chapterinspiral we phase, will and construct continuing initial through= data for× the binarydynamical+ × black plunge+ holes+ and in merger such orbits. phase. Numerical Bowen - York initialrelativity dataits is also total for required angular binary momentum to black treat delayed holes about collapse. the is origin Not of the surprisingly, coordinate system. considerable effort in (relatively) simplenumerical relativity has focused on compact binaries in close quasicircular orbits. In the1 12.2 Therest conformal ofFor this a binary chapter transverse-traceless system, we willS construct1 and S2 initialcan be data associated approach: for binary with black the Bowen–York holes spin of in such the individual orbits. black holes only in the limit of infinite binary separation, but we nevertheless take the liberty to Assume: define12.2.1 the orbital Solving angular the momentum momentumL as constraint 12.2 The conformal4 transverse-traceless approach: Bowen–York • ConformalIn the transverse-traceless flatness: approachij = we⌘ij assume both conformal flatness, so thatγ ¯ η , L J S1 S2. ij ij (12.47) 12.2.1 Solving the≡ momentum− − constraint = • Maximaland maximal slicing: slicing, so that K 0. The momentum constraint12.2 ( The12.4 conformal)thenreducesto transverse-traceless approach 405 = In the transverse-traceless approachij we assume both conformal flatness, so thatγ ¯ij ηij, 12.2.2D¯ j A¯ Solving0, the Hamiltonian constraint (12.37) = Then: and maximal slicing,GivenR = so that that theK momentum0.= The momentum constraint constraint (12.37)islinear,wecanconstructa (12.4)thenreducesto binary black 1 = ¯ With a solutionhole solution to the momentum by superposition constraintij of single at hand solutions we can now proceed to solve the • Momentumwhere D j isconstraint now a flat-space with analytic covariant solutions: derivative. D¯ j InA¯ Cartesian0, coordinates this operator(12.37) = reduces to theHamiltonian partial derivative constraint∂ . (12.3 Our). assumptions Under the assumptions haveij simplifiedij of conformal theij momentum flatnessij and maximal j A¯ij A¯ A¯ A¯ A¯ . (12.44) • Use linearity forwhere binaryD¯ j isblack now holes: a flat-space covariant derivative.C1P1 In CartesianC1S1 coordinatesC2P2 C this2S2 operator constraint to theslicing, point where this equation theR =2 decomposition reducesM to of A¯ij=into a transverse+ + and a longitudinal+ reduces to the partial derivative ∂ . Our assumptions have simplified the momentum part (see equation 3.48) is somewhat pointless,j since both of them now have vanishing This completes an analytic2 solution1 ¯7 of theij momentum constraint describing two black One equationconstraint to tosolve the pointnumerically: where the decompositionD¯ ψ ofψ−AijA¯intoij A¯ a. transverse and a longitudinal(12.48) part (see equationholes with3.48) arbitrary is somewhat momenta pointless,= and−8 spins. since both of them now have vanishing

We have reducedR the=0 construction of binary9 black hole initial data to solving a single SExercise1 12.6S2Show that ij nonlinear elliptic equation. On the rightP hand2 side, the term A¯ij A¯ can be computed analytically from (12.44). Unfortunately this termP divergesP1 P at2 the black holes’ centers C(12.45). = + i Dealing with this singularitya requires some extra care. naeis the=0 total linear momentum of the solution (12.44)and Two different approaches(i) have been adopted in the literature to solve this problem; they differ in the topology of the resulting solution.J C1 RecallP1 ourC2 discussionP2 S1 inS2 Chapter 3.1,which(12.46) P1 = × + × + + demonstrated that initial data sets representing multiple black holes are not unique. As a ⌦ its= totalt angular momentum about the origin of the coordinate system. starting point, wea canr looka for generalizations of the time-symmetric solution For a binary system, S1 and S2 can be associated with the spin of the individual black 1 2 holes only in the limit ofψ infinite1 MbinaryM separation,, but we nevertheless(12.49) take the liberty to = + rC + rC definen = the orbital↵⌦ angular momentum1 L as 2 a a 9 i i To represent the momenta P and spins S of these solutionsL it is convenientJ S1 to useS2 bold-face. notation P and S instead (12.47) of index notation. ≡ − − a n ab =0 12.2.2 Solving the Hamiltonian constraint With a solution to the momentum constraint at hand we can now proceed to solve the i Hamiltonian↵ constraint (12.3). Under the assumptions of conformal flatness and maximal slicing, this equation reduces to 1 i ¯ 2 7 ij A n D ψ ψ− A¯ij A¯ . (12.48) i = −8 We have reduced the construction of binary black hole initial data to solving a single ij nonlinear elliptic equation. On the right hand side, the term A¯ij A¯ can be computed ni analytically from (12.44). Unfortunately this term diverges at the black holes’ centers Ci . Dealing with this singularity requires some extra care. Two different approaches have been adopted in the literature to solve this problem; they i differA in the topologyij of the resulting solution. Recall our discussion in Chapter 3.1,which () demonstrated that initial data sets representing multiple black holes are not unique. As a starting point, we can look for generalizations of the time-symmetric solution i E ij 1 2 () ψ 1 M M , (12.49) = + rC1 + rC2

Ai Ei ij Kij 9 To represent the momenta Pi and spins Si of these solutions it is convenient to use bold-face notation P and S instead of index notation. The initial data is mathematically correct, but 1 astrophysically wrong

4 Does this assumption make sense? ij = ⌘ij No! But we have bigger problems R…= 1

R =2M

R =0

a nae(i) =0

⌦ = t a ra Fig: inside.hlrs.de/ Fig: spaceplace.nasa.gov/ n = ↵⌦ a a

a n ab =0

↵i

i A ni

ni

Ai () ij

Ei () ij

Ai Ei ij Kij Outline

History and formalism Initial data

Evolution of field equations Wave extraction

The tools we use Here is a simulation of the first detection, GW150914

• Simulation: Simulating eXtreme Spacetime project • Movie: Roland Haas, Max Planck Institute • Real time: ~ 1 s 376 Chapter 11 Recasting the evolution equations

implement these equations, at least in three spatial dimensions, turn out to be unstable.1 The failure of these equations can be understood in terms of their mathematical properties, as we will briefly discuss in Section 11.1.Followingthissummary,wewilldiscussreformulations 376 Chapter 11 Recasting the evolution equations of the evolution equations that avoid these problems and have proven more robust and implementsuccessful these numerically. equations, But at least first, in in three order spatial to model dimensions, the shortcomings turn out to be unstable. of the standard1 The failureevolution of these equations equations of can Chapter be understood2 in a simple in terms way of and their to mathematical obtain some properties, guidance on as we how willto briefly fix them, discuss we in shall Section return11.1 to.Followingthissummary,wewilldiscussreformulations Maxwell’s equations in Section 11.2. Fortified by the ofinsight the evolution provided equations by this example, that avoid we these will problems then present and several have proven different more reformulations robust and of the 3 1 evolution equations that have proven very successful in many numerical 1 successful+ numerically. But first, in order to model the shortcomings of the standard evolutionsimulations. equations of Chapter 2 in a simple way and to obtain some guidance on how to fix them, we shall return to Maxwell’s equations in Section 11.2. Fortified by the 1 R = insight provided by this example,11.1 Notions we will then of hyperbolicity present several different1 reformulations of the 3 1 evolution equations that have proven very successful in many numerical 1 + R = simulations.Recall from Chapter 6 that we can cast1 a hyperbolic partialR differential=2M equation – for example the simple wave equation (6.3) – in the first-order form

RR=2= M ∂t u A1 ∂x u S, (11.1) 11.1 Notions+ of· hyperbolicity= R =0 Recallwhere fromu is Chapter a solution6 vector,that weS canS cast(u)isasourcevector,andwherewehavecalledthe a hyperbolic partial differential equation – for = RR=2=0M matrix A the velocity matrix (see equation 6.7). Here we havea assumed that the solution example the simple wave equation (6.3) – in the first-order formnae(i) =0 depends on t and x only; in more than one spatial dimension we may generalize equation ∂t u A ∂x u S, (11.1) (11.1)toread Ra=0 n+ae(i) =0· = ↵⌦a = na = at where u is a solution vector, S S(u)isasourcevector,andwherewehavecalledthei r We’re gonna need a stronger= set∂ oft u equationsA ∂i u S. (11.2) matrix A the velocity matrix (see equation+ 6.7·). Here= we have assumed that the solution n ea =0 ↵⌦a =a n(ai) = at i dependsIf the solution on t and vectorx only;u has in moren components, than one spatial thenr each dimension matrix weA a is may an generalizen n matrix. equation For the n ab =0 × (11.1purposes)toread of this section we may ignore the source term and set S 0. = We call a problem well-posed if↵⌦ wea =a canna = defineat some norm ... so that the norm of the Hyperbolic equation n ab i=0r ∂t u A ∂i u S. ∥ ∥ i (11.2) solutionin first vector order satisfies form: 2 + · = ↵ i If the solution vector u has n components,ai then eachαt matrixi A is an n n matrix. For the Well-posedness: u(t, xn↵)ab =0ike u(0, x ) × (11.3) purposes of this section we may ignore∥ the∥≤ source∥ term and∥ set S i 0. A=ni forWe all call times a problemt 0.well-posed Here k andifα weare can two define constants some that norm are independent... so that of the the norm initial of the data i ≥ 2 i i ∥ ∥ solutionu(0, xStrongly). vector Stated hyperbolic: satisfies differently, solutionsThe matrix of↵ A a well-posedn i has real eigenvalues problem cannot and a increase complete more rapidly than exponentially. Clearly,set this of is eigenvectors a very desirable for all property, unit vectors but itni is not guaranteed for all u(t, xi ) keαt u(0, xi ) (11.3) hyperbolic systems. ∥ ∥≤i ∥ ∥ Weakly hyperbolic: The matrix A n in i has real eigenvalues but an incomplete for allAs times it turnst out,0. Here therek existand α differentare two types constants of hyperbolicity, that are independent and only of for the some initial of data these ≥ set of eigenvectors u(0types, xi ). are Stated the equations differently, well-posed. solutions of To a analyze well-posed these problem hyperbolicity cannot properties,increase more we rapidly consider i ni i thanan arbitrary exponentially. unit vector Clearly,n thisand constructis a very desirable the matrix property,P A n buti ,whichissometimesreferred it is not guaranteed for all = hyperbolicKey systems.point 1: Strongly hyperbolic systems are well-posed 1AsThe it high turns degree out, of spatial there symmetry exist different in spherical types symmetry of hyperbolicity, and even axisymmetry and only allows for for a some number of of these different typesstrategies are the to obtain equations stable evolutions, well-posed. including To analyze fully or partially these constrainedhyperbolicity evolution, properties, the use of wespecial consider coordinate systems,Key or point the introduction 2: ofi newThe variables. ADM equations are only iweakly hyperbolic an2 arbitrary unit vector n and construct the matrix P A ni ,whichissometimesreferred Much of this section follows the arguments of Alcubierre (2008a);= see also Kreiss and Lorenz (1989)andAlcubierre (2008b). 1 The high degree of spatial symmetry in spherical symmetry and even axisymmetry allows for a number of different strategies to obtain stable evolutions, including fully or partially constrained evolution, the use of special coordinate systems, or the introduction of new variables. 2 Much of this section follows the arguments of Alcubierre (2008a); see also Kreiss and Lorenz (1989)andAlcubierre (2008b). 11.2 Recasting Maxwell’s equations 379

If we identify the vector potential Ai with the spatial metric γij and the electric field Ei with the extrinsic curvature Kij, we see that the right-hand sides of both equations (11.4) and (11.7) contain a field variable and a spatial derivative of a gauge variable, while the right-hand sides of both equations (11.5) and (11.8)involvemattersourcesaswellas second spatial derivatives of the second field variable. In equation (11.8)thesesecond derivatives are hidden in the Ricci tensor Rij,whichwecanwrite,forexample,as

1 kl kl m m Rij γ ∂l ∂i γkj ∂ j ∂kγil ∂ j ∂i γkl ∂l ∂kγij γ # #mkj # #mkl . (11.9) = 2 + − − + il − ij We can identify! the troublesome terms " ! " We can now exploit these similarities by focusing on the simpler Maxwell system of equations to identify some of the computational shortcomings of these forms of the evolution equations. We first note that equations (11.4) and (11.5) almost can be combined to yield a wave equation, which would make the system symmetric hyperbolic. To see this, take a time derivative of equation (11.4) and insert equation (11.5) to form a single equation for the vector potential Ai , 11.2 Recasting Maxwell’s equations 379 2 j j Maxwell: ∂t Ai D D j Ai Di D A j Di ∂t $ 4π ji . (11.10) If we− identify+ the vector potential− A =with the spatial− metric γ and the electric field E 11.2 Recasting Maxwell’si equations 379 ij i On the left-handwith side, the extrinsic the second curvature time derivativeK , we see combines that the right-hand with the Laplace sides of both operator equations (11.4) d’Alembertian ij D j D A to formand a wave (11.7) operator contain (d’Alembertian). a field variable and Equations a spatial (11.4 derivative) and ( of11.5 a gauge) would variable, then while the If we identify thej vectori potential Ai with the spatial metric γij and the electric field Ei right-hand sides of both equationsStrongly (11.5) and hyperbolic (11.8)involvemattersourcesaswellas without this with the extrinsicconstitute curvature a waveKij equation, we see that for the the right-hand components sidesAi ofif both it weren’t equations for the(11.4 mixed) derivative j and (11.7) containterm D ai D fieldA j variable. second and spatial a spatial derivatives derivative of the of second a gauge field variable, variable. while In the equation (11.8)thesesecond right-hand sidesIn of general both equations relativityderivatives the (11.5 are situation) hidden and (11.8 is in very the)involvemattersourcesaswellas Ricci similar. tensor TheR Ricciij,whichwecanwrite,forexample,as tensor Rij on the right hand second spatialside derivatives of equation of ( the11.8 second) contains1 field three variable. mixed In derivative equation terms (11.8)thesesecond in addition to the term with a kl kl kl m m Laplace-like operatorRij actingγ on∂l ∂iγγkj,i.e.,∂ jγ∂kγ∂il ∂ γ∂ j ∂.i Withoutγkl ∂l ∂ thesekγij mixedγ # derivative#mkj terms# #mkl . (11.9) derivatives areEinstein: hidden in the Ricci= tensor2 Rij,whichwecanwrite,forexample,asij + l −k ij − + il − ij the standard ADM equations! could be written as a set of wave equations" for!Laplacian the components " 1 kl We can now exploit these similaritieskl m by focusingm on the simpler Maxwell system of Rij γ of∂ thel ∂i γ spatialkj ∂ j ∂ metric,kγil ∂ whichj ∂i γkl would∂l ∂kγ makeij themγ # symmetric#mkj # hyperbolic.#mkl . (11.9) As we discussed in = 2 + equations− to identify− some+ of theil computational− ij shortcomings of these forms of the Section! 11.1 we would like to evolve a" system! that is well-posed," but the presence of these evolution equations. We can nowadditional exploit these terms similarities may spoil by this focusing property. on the simpler Maxwell system of We first note that equations (11.4) and (11.5) almost can be combined to yield a wave equations to identifyThese considerations some of the computational suggest that it would shortcomings be desirable of these to eliminate forms of the the mixed derivative equation, which would make the system symmetric hyperbolic. To see this, take a time evolution equations.terms. In electrodynamics, three different approaches can be taken to eliminate the D D j A derivative of equation (11.4) and insert equation (11.5) to form a singlei equationj for the We first noteterm: that one equations can make (11.4 a special) and (gauge11.5) almost choice;can one be can combined bring Maxwell’s to yield a equations wave into a first- vector potential A , equation, whichorder wouldsymmetric make the hyperbolic system symmetricform;i or, hyperbolic. one can introduce To see this, an auxiliary take a time variable.Inthe derivative of equation (11.4) and insert equation (11.52 ) to formj a single equationj for the remainder of this section we will discuss∂t Ai brieflyD D eachj Ai oneD ofi D theseA j threeDi ∂t strategies.$ 4π ji . (11.10) vector potential Ai , − + − = −

2 On thej left-hand side,j the second time derivative combines with the Laplace operator ∂t Ai jD D j Ai Di D A j Di ∂t $ 4π ji . (11.10) − +D D j Ai to11.2.1 form− a Generalized wave= operator Coulomb (d’Alembertian).− gauge Equations (11.4) and (11.5) would then constitute a wave equation for the components A if it weren’t for the mixed derivative On the left-handThe most side, straightforward the second time approach derivative to combines eliminating with the the undesirable Laplacei operator terms is to choose a j term D D j A . D D j Ai to formgauge a waveso that operator the term (d’Alembertian).i D D j A disappears. Equations Define (11.4 the) and quantity (11.5) would then In generali relativityj the situation is very similar. The Ricci tensor R on the right hand constitute a wave equation for the components Ai if it weren’t for the mixed derivative ij j side of equation (11.8) contains threei mixed derivative terms in addition to the term with a term Di D A j . # D Ai . (11.11) Laplace-like operator acting≡ on γ ,i.e.,γ kl∂ ∂ γ . Without these mixed derivative terms In general relativity the situation is very similar. The Ricciij tensor Rij onl k theij right hand side of equation (11.8) containsthe standard three mixed ADM derivative equations terms could in be addition written to as the a set term of wave with aequations for the components of the spatial metric,kl which would make them symmetric hyperbolic. As we discussed in Laplace-like operator acting on γij,i.e.,γ ∂l ∂kγij. Without these mixed derivative terms the standard ADM equationsSection could be11.1 writtenwe would as a set like of wave to evolve equations a system for the that components is well-posed, but the presence of these of the spatial metric, whichadditional would make terms them may symmetric spoil this hyperbolic. property. As we discussed in Section 11.1 we would like to evolveThese considerations a system that is suggest well-posed, that it but would the presence be desirable of these to eliminate the mixed derivative j additional terms may spoil thisterms. property. In electrodynamics, three different approaches can be taken to eliminate the Di D A j These considerations suggestterm: that one it can would make be desirable a special togauge eliminate choice the; one mixed can derivativebring Maxwell’s equations into a first- order symmetric hyperbolic form; or, one can introducej an auxiliary variable.Inthe terms. In electrodynamics, three different approaches can be taken to eliminate the Di D A j term: one can make a specialremaindergauge choice of this; one section can bring we will Maxwell’s discuss equations briefly each into one a first- of these three strategies. order symmetric hyperbolic form; or, one can introduce an auxiliary variable.Inthe remainder of this section we will discuss briefly each one of these three strategies. 11.2.1 Generalized Coulomb gauge The most straightforward approach to eliminating the undesirable terms is to choose a 11.2.1 Generalized Coulombj gauge gauge so that the term Di D A j disappears. Define the quantity

The most straightforward approach to eliminating the undesirable termsi is to choose a j # D Ai . (11.11) gauge so that the term Di D A j disappears. Define the quantity ≡

i # D Ai . (11.11) ≡ 11.2 Recasting Maxwell’s equations 379

If we identify the vector potential Ai with the spatial metric γij and the electric field Ei with the extrinsic curvature Kij, we see that the right-hand sides of both equations (11.4) and (11.7) contain a field variable and a spatial derivative of a gauge variable, while the right-hand sides of both equations (11.5) and (11.8)involvemattersourcesaswellas second spatial derivatives of the second field variable. In11.2 equation Recasting (11.8 Maxwell’s)thesesecond equations 379 derivatives are hidden in the Ricci tensor Rij,whichwecanwrite,forexample,as If we identify the vector potential Ai with the spatial metric γij and the electric field Ei

1withkl the extrinsic curvature Kij, we see that thekl right-handm sidesm of both equations (11.4) Rij γ ∂l ∂i γkj ∂ j ∂kγil ∂ j ∂i γkl ∂l ∂kγij γ # #mkj # #mkl . (11.9) = 2and (11.7) contain+ a field− variable− and a spatial+ derivativeil − of aij gauge variable, while the ! " ! " We can nowright-hand exploit sides these of similarities both equations by focusing (11.5) and on the (11.8 simpler)involvemattersourcesaswellas Maxwell system of equationssecond to identify spatial some derivatives of the computational of the second field shortcomings variable. Inof equation these forms (11.8 of)thesesecond the evolutionderivatives equations. are hidden in the Ricci tensor Rij,whichwecanwrite,forexample,as We first note that equations (11.4) and (11.5) almost can be combined to yield a wave 1 kl kl m m equation, whichRij wouldγ make∂l ∂i γkj the∂ systemj ∂kγil symmetric∂ j ∂i γkl hyperbolic.∂l ∂kγij γ To see# this,#mkj take# a# timemkl . (11.9) = 2 + − − + il − ij derivative of equation (!11.4) and insert equation (11.5) to form" a single! equation for the" We can now exploit these similarities by focusing on the simpler Maxwell system of vector potential Ai , equations to identify some of the computational shortcomings of these forms of the 2 j j evolution equations.∂ Ai D D j Ai Di D A j Di ∂t $ 4π ji . (11.10) − t + − = − We first note that equations (11.4) and (11.5) almost can be combined to yield a wave On the left-hand side, the second time derivative combines with the Laplace operator equation, which would make the system symmetric hyperbolic.11.2 To Recasting see this, Maxwell’s take a time equations 379 D j D A to form a wave operator (d’Alembertian). Equations (11.4) and (11.5) would then j i derivative of equation (11.4) and insert equation (11.5) to form a single equation for the constitute a wave equation for the components A if it weren’t for the mixed derivative vector potentialIfA we, identify the vector potentiali Ai with the spatial metric γij and the electric field Ei term D D j A . i i j with the extrinsic curvature Kij, we see that the right-hand sides11.2 of Recasting both equations11.2 Maxwell’s Recasting (11.4 equations) Maxwell’s equations379 379 In general relativity the situation2 is veryj similar. The Riccij tensor Rij on the right hand and (11.7∂)A containi D aD fieldj Ai variableDi D A andj aD spatiali ∂t $ derivative4π ji . of a gauge(11.10) variable, while the − t + − = − side of equation (11.8) containsright-hand three sidesIf mixed we of identify both derivativeIf equations we the identify terms vector (11.5 in potential the addition) vector and (A11.8 to potentiali thewith)involvemattersourcesaswellas term theA with spatiali with a metric the spatialγij and metric theγ electricij and the field electricEi field Ei kl Laplace-likeOn operator the left-hand actingsecond side, on spatialγij the,i.e.,with second derivativesγ the∂l extrinsic time∂kwithγij of. derivative Without the the curvature extrinsic second these combinesK field curvature mixedij, we variable. with see derivativeK that theij, In we the Laplace equation terms see right-hand that operator ( the11.8 sides right-hand)thesesecond of both sides equations of both (11.4 equations) (11.4) j the standardD ADMD j Ai equationsto formderivatives a wave could operator be areand written hidden (11.7 (d’Alembertian). as in) acontainand the set Ricci( of11.7 wave a) field tensor contain Equationsequations variableRij a,whichwecanwrite,forexample,as field for ( and11.4 the variable a) components and spatial (11.5 and derivative) a would spatial then of derivative a gauge of variable, a gauge while variable, the while the of380 the spatialChapterconstitute metric, 11 Recasting a which wave equationwould the evolution makeright-hand for equations them the components symmetricright-hand sides of both hyperbolic.A sidesi if equations it of weren’t both As ( equationswe11.5 for discussed) the and mixed ( (11.511.8 in derivative))involvemattersourcesaswellas and (11.8)involvemattersourcesaswellas Section 11.1 we wouldj like to evolve1 kl a system that is well-posed, but the presencekl of thesem m term Di D A j . Rij γ second∂l ∂i γkj spatialsecond∂ j ∂k derivativesγil spatial∂ j ∂i γkl derivatives of the∂l ∂k secondγij of theγ field second# variable.#mkj field In# variable. equation#mkl . In(11.9) (11.8 equation)thesesecond (11.8)thesesecond To eliminate the offending term,= 2 we may then+ set − − + il − ij additional termsIn general may spoil relativity this property. the situationderivatives! isderivatives veryare hidden similar. inare The the hidden Ricci Ricci in tensor tensor the" RicciRRijij,whichwecanwrite,forexample,as!on tensor the rightRij,whichwecanwrite,forexample,as hand " These considerationsside of equationWe suggest (11.8 can) that now contains it exploit would! three be these0. mixed desirable similarities derivative to eliminate by terms focusing the in additionmixed on(11.12) derivativethe to simpler the term Maxwell with a system of “Generalized Coulomb gauge“ =provides1 kl kl 1 insightkl for j kl m kl mm m terms. In electrodynamics,Laplace-like operatorequations three actingdifferent to identifyR onij approachesγ ,i.e., someγ Rγij∂l of can∂∂i γ∂ thekj beγ taken computational.∂ Without∂j ∂l ∂k toγi γilkj eliminate these∂∂j ∂ji∂ shortcomingsγkkl mixedγ theil D∂l∂i∂ derivativejDk∂γiijγAklj ofγ∂ thesel terms∂kγ#ij forms#mkjγ of## the##mklmkj . #(11.9)#mkl . (11.9) This choice is the familiar Coulomb gauge=condition.ij2 =l 2k +ij −+ −− +− il + − ijil − ij gravityterm: one gaugethe can standard make choices a special ADMevolution equationsgauge equations. choice could; one be written can! bring as a Maxwell’s set! of wave equations equations into for a the first- components" ! " ! " " order symmetricExerciseof the spatial 11.1 hyperbolicShow metric,We thatform; first which in note the or,We would Coulomb that one can equations can make now gaugeWe introduce exploit them can the (11.4 gauge now symmetric these) an and exploit potentialauxiliary similarities (11.5 hyperbolic. these)"almostsatisfies variable similarities bycan As focusing.Inthe be we combined discussed by on focusing the to in simpler yield on thea Maxwell wave simpler system Maxwell of system of remainderPoisson’sSection of this elliptic section11.1 we equationequation, we would will like discuss whichequations to evolve would briefly a to makeequations eachsystem identify theone that system of to some is these identify well-posed, symmetric of three the some strategies. computational but hyperbolic. of the the presence computational To shortcomings of see these this, shortcomings take of a these time forms of these of the forms of the i derivative ofevolutionD equationDi " ( equations.11.44evolutionπρe). and insert equations. equation (11.5(11.13)) to form a single equation for the additional terms may spoil this property.= − Standard E&M:vector potentialWeAi , first noteWe that first equations note that (11.4 equations) and ( (11.511.4))almost and (11.5can) almost be combinedcan be to combined yield a wave to yield a wave In generalThese relativity, considerations11.2.1 an analogous Generalized suggest approach that it can Coulomb would be taken be desirable gauge by choosing to eliminate harmonic the coor- mixed derivative equation, whichequation, would which make would the10 system make the symmetric system symmetric hyperbolic.j hyperbolic. To see this, To take see a this, time take a time dinates,380terms. whichChapter In bring electrodynamics, 11 the Recasting equations the into three evolution the different form∂2 equationsA of approaches a waveD j D equation.A canD beD taken(Thej A to reader eliminateD ∂ $ may the4π jD.i D A j (11.10) Maxwell’s equations:derivative oftderivative equationi ofj (11.4 equationi ) andi insert(11.4j ) equation andi t insert (11.5 equationi) to form (11.5 a) single to form equation a single for equation the for the Thewish most toterm: review straightforward one Chapter can make4.3 approach,and,especially,exercise a special togauge eliminating− choice+;4.10 the one undesirable.) can We bring will− return Maxwell’s terms to this= is to equations strategy choose− intoa a first- j gaugein SectionToso thatorder eliminate11.3 thesymmetricbelow. term theD offendingOni D hyperbolic theA j left-handdisappears. term,vectorform; we side, potential may Define or, thevector then one second theA set cani potential quantity, introduce timeA derivativei , an auxiliary combines variable with the.Inthe Laplace operator j This approachremainder has of disadvantages, thisD D sectionj Ai to we form too. will As a wave discuss exercisei operator briefly11.1 demonstrates, (d’Alembertian). each2 one ofj the these2 choice Equations three (11.12j strategies. (j11.4) ) and (11.5j ) would then Coulomb gauge: # D A!i . 0. ∂t Ai D ∂Dt jAAii DDi DD(11.11)j AAij (11.12)DDi iD∂t $A j 4πDjii∂.t $ 4π ji . (11.10) (11.10) forces us to satisfy a constraintconstitute that a may wave be≡ equation inconvenient= for the to− solve. components In+ general− Ai relativity,if+− it weren’t the for−= the mixed−= derivative− analogousThis choice harmonic is the gaugeterm familiar condition,D DCoulombj A . (4)!a gauge0, similarlycondition. imposes a coordinate choice – i Onj the= left-handOn the side, left-hand the second side, the time second derivative time combines derivative with combines the Laplace with the operator Laplace operator the lapse and shift now haveIn general to satisfy11.2.1 relativity equationsj Generalized the (4.44 situationj ) and Coulomb (4.45 is very). These similar. gauge coordinates The Ricci may tensor Rij on the right hand Exercise 11.1 Show thatD inD thej Ai CoulombtoD formD j aA gauge wavei to form operator the agauge wave (d’Alembertian). potential operator" (d’Alembertian).satisfies Equations ( Equations11.4) and ( (11.511.4)) would and (11.5 then) would then not be well-suited to theside problem of equation at hand, (11.8 and) contains may lead three to coordinate mixed derivative singularities. terms One in addition to the term with a TheAlternatively:Poisson’s most straightforward elliptic equationconstitute approach to aconstitute wave eliminating equation a wave the for undesirable equation the components for terms the components isAi toif choose it weren’tA ai if for it weren’t the mixed for derivative the mixed derivative way to regain coordinate freedom is not to set ! 0asinequation(kl11.12), but instead to Laplace-like operatorj actingj on γijj,i.e.,γ ∂l ∂kγij. Without these mixed derivative terms gauge so that the term D DtermA Ddisappears.i Di term=A j . D Definei D A j the. quantityi set ! equal to some yet-to-be-chosenthe standardi ADMgaugej equationsD sourceDi " function could4πρ bee H. written(t, x ), as a set of wave equations(11.13) for the components Define gauge In general relativityIn= general− the relativity situation the is situation very similar. is very The similar. Ricci tensor The RicciRij on tensor the rightRij on hand the right hand of the spatial metric, whichi wouldi make them symmetric hyperbolic. As we discussed in Insource general function: relativity, an analogousside! ofH equation( approacht, xside#). of (11.8D can equationA)i be. contains taken (11.8 threeby) contains choosing mixed(11.14) three derivative harmonic mixed terms(11.11) derivativecoor- in addition terms into additionthe term towith the a term with a = 1 dinates, which bringSection the11.1 equationswe would into like the to≡ form evolve of a system wave equation. that is well-posed,10kl(The reader butkl the may presence of these Retracing the steps in exercise 11.1 weLaplace-like could findLaplace-like out operator how a particularacting operator on choiceγij acting,i.e., of onγH γaffects∂ijl ∂,i.e.,kγij.γ Without∂l ∂kγij these. Without mixed these derivative mixed terms derivative terms wish to review Chapteradditional4.3,and,especially,exercise terms may spoil this property.4.10.) We will return to this strategy the gauge potential ".Similarly,ingeneralrelativitywecanrelaxtheharmonicgaugethe standardthe ADM standard equations ADM could equations be written could as be a written set of wave as a setequations of wave for equations the components for the components These considerations suggest that it would be desirable to eliminate the mixed derivative in SectionGeneralized11.3 below. of the spatialof metric, the spatial which metric,(4) woulda which makea would themi make symmetric them symmetrichyperbolic. hyperbolic. As we discussed As we in discussed in condition with the help of a 4-dimensional gauge2 source function,j ! Hi (t, x ), as we j terms. In electrodynamics, three different approaches= can be taken to eliminate the Di D A j will seeThisCoulomb in Section approach11.3 gauge: has.Thisapproachiscommonlyreferredtoas“generalizedharmonic disadvantages,Section [email protected] A AsSectioni we+ exerciseD wouldjD11.1A like11.1wei to woulddemonstrates,H evolve(t, x like)= a tosystemD evolve thei@t choicethat a system is4⇡ well-posed, (j11.12i that) is well-posed, but the presence but the of presence these of these term: one can make a special gauge choice ; one can bring Maxwell’s equations into a first- coordinates”,forces us and, to satisfy in analogy, a constraint we couldadditional that refer may to the terms beadditional condition inconvenient may spoil terms (11.14 this to may) solve. imposed property. spoil In this on general! property.as the relativity, the analogous harmonicorder gaugesymmetric condition, hyperbolic(4)!a form;0, similarly or, one imposes can introduce a coordinate an auxiliary choice variable – .Inthe “generalized Coulomb gauge”. These considerationsThese= considerations suggest that suggest it would that be it desirable would be to desirable eliminate to the eliminate mixed derivative the mixed derivative the lapse and shiftremainder now have of to this satisfy section equations we will (4.44 discuss) andS briefly1,S (4.452 each). These one ofcoordinates these three may strategies. j j terms. In electrodynamics,terms. In electrodynamics, three different three approaches different approaches can be taken can to eliminate be taken to the eliminateDi D A thej Di D A j not be well-suited to the problemterm: at one hand, canterm: andmake one may a can special lead make togauge acoordinate special choicegauge; singularities. one choice can bring; one One Maxwell’s can bring equations Maxwell’s into equations a first- into a first- way to regain coordinate11.2.2 First-order freedom is hyperbolic not to set formulations! 0asinequation(11.12), but instead to order symmetricorder11.2.1symmetric hyperbolic= Generalized hyperbolicform;J Coulomb or, oneform; gauge can or, introduce one can an introduceauxiliary an variableauxiliary.Inthe variable.Inthe set ! equal to some yet-to-be-chosen gauge source function H(t, xi ), An alternative, gauge-covariant approachremainder to bringing ofremainder this Maxwell’sequations section of this we willsection discuss into we a symmetric will briefly discuss each briefly one of each these one three of these strategies. three strategies. hyperbolic form is to takeThe a most time derivative straightforward of equation approach (11.5)insteadofequation( to eliminating the undesirable11.4), terms is to choose a ! H(t, xi ). (11.14) j which then yields gauge so that the term D=i D A j disappears.M1/M Define2 the quantity Retracing2 the steps inj exercise 11.1 we couldj find out how11.2.1 a particular Generalized11.2.1 choice Generalized Coulomb of H affects gauge Coulomb gauge ∂ Ei Di D ( E j D j ") D j D ( Ei Di ") 4π∂it ji . (11.15) t # D Ai . (11.11) the gauge potential= "−.Similarly,ingeneralrelativitywecanrelaxtheharmonicgauge− − − − −≡ The most straightforwardThe most straightforward approachrad (4) to approach eliminatinga a to eliminating thei undesirable the undesirable terms is to terms choose is a to choose a Usingcondition the constraint with the(11.6 help) we of can a 4-dimensional eliminate the first gauge term source and find function, ah waveab equation! forHE(i t, , x ), as we gauge so that thej term D D j A=disappears. Define the quantity will see in Section 11.32 .Thisapproachiscommonlyreferredtoas“generalizedharmonicgaugej so that the term Di D A j disappears.i j Define the quantity ∂t Ei D j D Ei 4π(∂t ji Di ρe), (11.16) coordinates”, and, in− analogy,+ we could= refer to+ the condition (11.14) imposedi on ! asi the gab = ⌘ab + hab# D Ai .# D Ai . (11.11) (11.11) 10 This“generalized property was first Coulomb realized by De gauge”. Donder (1921)andLanczos (1922). Many of the early hyperbolic formulations≡ ≡ of Einstein’s equations were based on this gauge choice, e.g., Choquet-Bruhat (1952, 1962); Fischer and Marsden (1972). 4 11.2.2 First-order hyperbolic formulationsij = ⌘ij An alternative, gauge-covariant approach to bringing Maxwell’sequations into a symmetric hyperbolic form is to take a time derivative of equationR (11.5= )insteadofequation(11.4), which then yields 1

2 j j ∂ Ei Di D ( E j D j ") D j D ( Ei Di ") 4π∂t ji . (11.15) t = − − − − R−=2M − Using the constraint (11.6) we can eliminate the first term and find a wave equation for Ei , 2 j ∂ Ei D j D Ei 4π(∂t ji RD=0i ρe), (11.16) − t + = +

10 This property was first realized by De Donder (1921)andLanczos (1922). Many of the early hyperbolic formulations of Einstein’s equations were based on this gauge choice, e.g., Choquet-Bruhata (1952, 1962); Fischer and Marsden nae =0 (1972). (i)

⌦ = t a ra

n = ↵⌦ a a

a n ab =0

↵i

i A ni

ni

Ai () ij

Ei () ij

Ai Ei ij Kij 382 Chapter 11 Recasting the evolution equations

382 Chapter 11 Recasting(4) the evolution equations in the form Rab 8π(Tab (1/2)gabT ). Substituting the Ricci tensor as expressed by = − equation(4) (4.46) yields in the form Rab 8π(Tab (1/2)gabT ). Substituting the Ricci tensor as expressed by 1 cd = −(4) c (4) c (4) ed (4) c (4) cd (4) e (4) g ∂d ∂cgab gc(a∂b) # # #(ab)c 2g # #b)cd g # #ecb equation (24.46) yields− − − e(a − ad 382 Chapter 11 Recasting the evolution equations 1 1 cd (4) c (4) c (4)8π Tab 2 edgab(4)T ,c (4) cd (4) e (11.20)(4) 2 g ∂d ∂cgab gc(a∂b) # #= −#(ab)c −2g #e(a #b)cd g #ad #ecb − − ! − " − where(4) we have reintroduced the definition (4.40), in the form Rab 8π(Tab (1/2)8gabπ TT). Substituting1 g T , the Ricci tensor as expressed(11.20) by = − ab 2 ab equation (4.46) yields(4) a bc(4)= a− 1 − ∂ 1/2 ab bc a # g # ! g "g g b cx (11.21) where we have reintroduced≡ the definitionbc = − g (14.40/2 ∂x),b | | = ∇ ∇ 1 gcd∂ ∂ g g ∂ (4)#c (4)#c (4)#| | 2g!ed (4)#c "(4)# gcd (4)#e (4)# 2 d(seec ab exercisesc(a 4.8b) and 4.10 and the discussion(ab)c that followse(a theseb)cd exercises). Thead (4)ecb#a (4)− a bc(4) −a 1 ∂− 1/2 ab bc − a # g # g g g b cx (11.21) are contractions≡ of Christoffelbc = − symbols,g 1/2 and∂xb therefore1 | | do not= transform∇ ∇ like vectors under 8π Tab 2 gabT , (11.20) coordinate transformations.= We382− can| | nowChapter introduce− ! 11 Recasting a gauge" by the setting evolution these quantities equations equal ! a " (4) a where(see exercises weto have some reintroduced4.8 givenandgauge4.10 source theand definition functions the discussion H(4.40, ), that follows these exercises). The # are contractions of Christoffel symbols,in the(4) form anda therefore(4) Ra i do8π not(T transform(1/2)g likeT vectors). Substituting under the Ricci tensor as expressed by Generalized harmonic coordinates 1 # ∂ H (abt, x ). ab ab (11.22) coordinate transformations.(4) a bc(4) Wea can now introduce= a1 gauge/=2 ab by setting−bc thesea quantities equal # g #bc equation (4.46)g yieldsg g b cx (11.21) This approach≡ follows very= closely− g 1/ oura2 ∂ electromagneticxb | | example= ∇ of∇ Section 11.2.1, and to some given gauge source functions| H| , ! " equation (11.22)isthedirectanalogof(1 gcd∂ ∂ g11.14).g Just∂ like(4) the#c choice(4)#c (11.14(4)# ) led to2 theged (4)#c (4)# gcd (4)#e (4)# (see exercises 4.8 and 4.10 and the2 discussiond c ab that followsc(a b) these exercises).(ab) Thec (4)#a e(a b)cd ad ecb elimination of the “mixed derivatives”(4)#a termsH a(t−, inxi Maxwell’s). equations− (11.10), inserting−(11.22) − are contractionsequation of (11.22 Christoffel) into Einstein’s symbols, equations and= therefore (11.20) yields do not a nonlinear transform wave like equation vectors for1 under the 8π Tab 2 gabT , (11.20) This approach follows very12 closely our electromagnetic example= of− Section 11.2.1− , and coordinatespacetime transformations. metric gab. WeAfter can now some introduce manipulations a gauge this byequation setting can these be brought quantities! into equal the " form13 wherea we have reintroduced the definition (4.40), 382 ChapterEinstein 11 Recasting equations:to theequation some evolution given (11.22 equationsgauge)isthedirectanalogof( source functionsDefine H , 11.14 gauge). Just source like the function: choice (11.14) led to the elimination of the “mixedcd derivatives” termscd in Maxwell’s equations(4) d (11.101 ),∂ inserting (4) g ∂d ∂cgab (4)2∂(aag ∂cgba)d (4)2iHa (a,b) bc2(4)Hd a #ab 1/2 ab bc a in the form Rab 8π(Tab (1/2)gabT ). Substituting the Ricci+ # tensorH as( expressedt+, x#). g− by #bc (11.23)(11.22)g g g b cx (11.21) equation (11.22) into Einstein’s equations(4) c (11.20(4) d ) yields a nonlinear waveg equation1/2 ∂xb for the = − 2 #= # 8≡π (2Tab gab=T )−. | | = ∇ ∇ equation (4.46) yields spacetime metric g .12 After some+ manipulationsbd ac = − this equation− can be| brought| ! into the " This approach followsab very closely our electromagnetic example of Section 11.2.1, and (4) a 13 (see exercisesa 4.8 and 4.10 and the discussionb that follows these exercises). The # 1 cd (4) formc (4) Herec (4) we have lowereded (4) thec indices(4) of H withcd (4) the spacetimee (4) metric gab, Ha gab H .Forthe g ∂d ∂cgab gc(a∂b) #equation# (11.22#(ab)isthedirectanalogof()c 2g #e(a #b)cd 11.14g ).#ad Just# likeecb the choice (11.14≡ ) led to the 2 Field− equations− inspecial choice− H a 0werecovertheharmoniccoordinatesofChapterare contractions− of Christoffel symbols,4.3.Moregenerally and therefore do not transform like vectors under elimination of the “mixedcd = derivatives”cd terms in Maxwell’s equations(4) 14d (11.10), inserting we8π referT tog this1 g∂ approachd ∂Tcgab, as2∂coordinate “generalized(a g ∂cgb)d transformations. harmonic2H(a,(11.20)b) coordinates”.2Hd We# canab This now formalism introduce was a gauge by setting these quantities equal generalized harmonic ab 2 ab + + − (11.23) equation= −adopted (11.22) by into−Pretorius Einstein’s(2005a equations,to(4)b) some inc his(4) (11.20 given simulationsd )gauge yields of source a binary nonlinear functionsblack wave hole Hcoalescence equationa, for and the coordinates: ! 12 " 2 #bd #ac 8π (2Tab gabT ) . where we have reintroducedspacetime the definitionmerger, metric (4.40 whichgab),. weAfter will discuss+ some in manipulations more detail= in− Chapter this equation13.− can be brought into the (4) a a i 13 form Identifying equations (11.21) anda (11.22) imposes a new, 4-dimensional# constraintH b(t, x ). (11.22) (4) a bc(4)Herea we have1 lowered∂ the1/2 indicesab of Hbc with thea spacetime metric gab, Ha g=ab H .Forthe # g #bc a g g g b cx (11.21) ≡ ≡New constraintspecial= − choice gequation:1/2 H∂xbcd|0werecovertheharmoniccoordinatesofChapter| = Thiscda∇ approach∇H a gbc follows(4)#a very0. (4) closelyd 4.3 our.Moregenerally electromagnetic(11.24) example of Section 11.2.1, and g ∂d ∂cgab 2∂(a g ∂cgb)d 2H(a,bbc) 2Hd # | | =! +" C ≡ −+ −= ab14 (11.23) we refer to this approach as “generalized(4)equationc (4) harmonicd (11.22)isthedirectanalogof( coordinates”.(4) a This formalism11.14). Just was like the choice (11.14) led to the (see exercises 4.8 and 4.10 and theEquation discussion (11.23 that) can follows be integrated2 these# exercises). directly# for theThe8π spacetime(2T# g metricT ) .gab. To stabilize the adopted by Pretorius (2005a,b) inelimination hisbd simulationsac of the of “mixed binaryab derivatives” blackab hole coalescence terms in Maxwell’s and equations (11.10), inserting are contractions of Christoffel symbols,system, and it therefore is sometimes do not+ necessary transform to add like linear= vectors− combinations under− of the constraints (11.24) equation (11.22) into Einstein’s equations (11.20) yields a nonlinear wave equation for the coordinate transformations. Wemerger, can nowto which the introduce evolution we will a discuss equationsgauge by in (setting11.23 morea ), detail these i.e., it quantities in is Chapter necessary equal13 to. add terms proportionalb to the Here we have lowered the indices of H with the spacetime12 metric gab, Ha gab H .Forthe to some given gauge source functionsIdentifying H a, equationsa (11.21) andspacetime (11.22) metricimposesgab a. new,After 4-dimensional some manipulations≡ constraint this equation can be brought into the special choice H 0werecovertheharmoniccoordinatesofChapter13 4.3.Moregenerally 12 See Friedrich=(1985); Garfinkle (2002aform). See alsoa the relatedbc (4) “Z4”a formalism suggested14 by Bona et al. (2003). we refer(4) toa this approacha i as “generalized harmonic coordinates”. This formalism was 13 #See PretoriusH ((t2005b, x ).). H g #bc (11.22)0. (11.24) 14 C ≡ − = adopted byThisPretorius= name is somewhat(2005a misleading,,b) in however, his simulations since it does not of singlegcd binary∂ out∂ anyg particular black2∂ hole familygcd of coalescence∂ coordinateg systems.2H and 2H (4)#d d c ab (a a c b)d (a,b) d ab This approach follows veryEquation closely our (Instead,11.23 electromagneticany) canarbitrary be coordinate integrated example system directly can of be Section generated for the by11.2.1 (11.22 spacetime)withasuitablechoiceof, and metric+ gabH.. To stabilize+ the− (11.23) merger, which we will discuss in more detail in Chapter 13. 2 (4)#c (4)#d 8π (2T g T ) . system, it is sometimes necessary to add linear combinations of the constraintsbd ac (11.24) ab ab equation (11.22)isthedirectanalogof(Identifying equations11.14). Just (11.21 like) the and choice (11.22 ()11.14 imposes) led a to new, the 4-dimensional+ constraint= − − elimination of the “mixed derivatives”to the evolution terms equations in Maxwell’s (11.23 equations), i.e., it ( is11.10 necessary), inserting to add terms proportionala to the b Here we have lowered the indices of H with the spacetime metric gab, Ha gab H .Forthe a H a gbc (4)#a 0. (11.24) ≡ equation (11.22) into Einstein’s equations (11.20) yields a nonlinearspecial wave choice equationH abc for0werecovertheharmoniccoordinatesofChapter the 4.3.Moregenerally 12 C ≡ − = spacetime metric g . After12 some manipulations this equation can be brought= into the 14 ab EquationSee Friedrich (11.23(1985)); canGarfinkle be integrated(2002). Seewe directlyalso refer the related to for this “Z4” the approach spacetime formalism suggested as metric “generalized bygBona. Toet al. harmonicstabilize(2003). the coordinates”. This formalism was form13 13 See Pretorius (2005b). ab system,14 This name it is is somewhat sometimes misleading, necessary however,adopted to since add it lineardoes by Pretorius not combinations single out(2005a any particular, ofb) the in family his constraints of simulations coordinate (11.24 systems. of) binary black hole coalescence and cd Instead, anycd arbitrary coordinate system canmerger,(4) be generatedd which by (11.22 we will)withasuitablechoiceof discuss in moreH detaila . in Chapter 13. g ∂d ∂cgabto the2∂( evolutiona g ∂cgb)d equations2H(a,b) (11.232Hd), i.e.,# it is necessary to add terms proportional to the + + − ab (11.23) (4) c (4) d Identifying equations (11.21) and (11.22) imposes a new, 4-dimensional constraint 2 #bd #ac 8π (2Tab gabT ) . + = − − a a bc (4) a 12 See Friedrich (1985); Garfinkle (2002). See also the related “Z4” formalism suggested by BonaHet al. (g2003). # 0. (11.24) a b bc Here we have lowered the indices13 See ofPretoriusH with(2005b the). spacetime metric gab, Ha gab H .Forthe C ≡ − = a 14 ≡ special choice H 0werecovertheharmoniccoordinatesofChapterThis name is somewhat misleading, however,Equation since it4.3 does (.Moregenerally11.23 not single) can out be any integrated particular family directly of coordinate for the systems. spacetime metric gab. To stabilize the = a we refer to this approach as “generalizedInstead, any arbitrary harmonic coordinate coordinates”. system cansystem, be generated14 This it is by formalism sometimes (11.22)withasuitablechoiceof was necessary toH add. linear combinations of the constraints (11.24) adopted by Pretorius (2005a,b) in his simulations of binary blackto the hole evolution coalescence equations and (11.23), i.e., it is necessary to add terms proportional to the merger, which we will discuss in more detail in Chapter 13. Identifying equations (11.21) and (11.22) imposes a new, 4-dimensional constraint 12 See Friedrich (1985); Garfinkle (2002). See also the related “Z4” formalism suggested by Bona et al. (2003). 13 a a bc (4) a See Pretorius (2005b). H g # 0. 14 (11.24) C ≡ − bc = This name is somewhat misleading, however, since it does not single out any particular family of coordinate systems. Instead, any arbitrary coordinate system can be generated by (11.22)withasuitablechoiceofH a . Equation (11.23) can be integrated directly for the spacetime metric gab. To stabilize the system, it is sometimes necessary to add linear combinations of the constraints (11.24) to the evolution equations (11.23), i.e., it is necessary to add terms proportional to the

12 See Friedrich (1985); Garfinkle (2002). See also the related “Z4” formalism suggested by Bona et al. (2003). 13 See Pretorius (2005b). 14 This name is somewhat misleading, however, since it does not single out any particular family of coordinate systems. Instead, any arbitrary coordinate system can be generated by (11.22)withasuitablechoiceofH a . 390 Chapter 11 Recasting the evolution equations

390 ChapterBox 11 11.1 Recasting The BSSN the equations evolution equations

In the BSSN formulation of the 3 1 equations the spatial metric γij is decomposed into a + conformally related metricγ ¯ij with determinantγ ¯ 1 (assuming Cartesian coordinates) and a 390 Chapter 11 Recasting the evolution equations = Box 11.1 The BSSN equations 390conformalChapter factor eφ, 11 Recasting the evolution equations In the BSSN formulation of the 3 1 equations the spatial metric γij is decomposed into a 390 Chapter 11 Recasting the evolution equations4φ + Box 11.1 The BSSN equationsconformally relatedγ metricij eγ ¯ijγ¯ijwith. determinantγ ¯ 1 (assuming Cartesian(11.46) coordinates) and a = = conformal factor eφ, In the BSSN formulation of the 3We1 also equations decompose the spatial the extrinsic metric curvatureγij is decomposed intoBox its 11.1trace into and a The traceless BSSN parts equations and conformally + Box 11.1 The BSSN equations conformally related metricγ ¯ with determinantγ ¯ 1 (assuming Cartesian coordinates) and a 4φ ij transform the traceless part as we do the metric, γij e γ¯ij. (11.46) φ = conformal factor e , InIn the the BSSN BSSN formulation formulation of the 3 of1 the equations 3 the1 spatial equations metric= γ theij is decomposed spatial metric into a γij is decomposed into a + 1 390 Chapter 11 Recasting the evolutionconformally relatedequationsWe metric alsoγ decompose ¯ with determinant the4φ extrinsic˜ +γ ¯ curvature1 (assuming into Cartesian its trace coordinates) and traceless and parts a and conformally 4φ ij Kij e Aij γijK. (11.47) conformallyγij e γ¯ij related. φ metricγ ¯=ij with+ determinant3(11.46)= γ ¯ 1 (assuming Cartesian coordinates) and a conformal= factor e transform, the traceless part as we do the metric, = The BSSN formalismIn terms is ofdue these to variables Shibata,φ the Hamiltonian Nakamura, constraint (2.132 )becomes We also decompose the extrinsicconformal curvature into factor its tracee and, traceless parts and conformally4φ 1 γij e γ¯ij. 4φ ˜ (11.46) φ =5φ Kij 5φe Aij γijK. (11.47) transform the traceless part as we doBox the metric,11.1 The BSSNij equationsφ e e ij e= 2 + 35φ Baumgarte and Shapiro 0 γ¯ D¯ i D¯ j e R¯ A˜ij A˜ K 42φπe ρ, (11.48) We also decompose= H = the extrinsic− curvature8 + into8 its trace− 12 and traceless+ parts and conformally 4φ 1 In terms of these variables the Hamiltonianγij constrainte γ¯ (ij2.132. )becomes (11.46) Ktransformij e A˜ theij tracelessγijK. part as we do the metric, (11.47) In the BSSN formulation of thewhile 3 the1 momentum equations3 constraint the (2.133 spatial)becomes metric γij is= decomposed into a = + eφ e5φ e5φ + ij ¯ ¯ 1φ ¯ ˜ ˜ij 2 5φ In terms of these variables the HamiltonianWe also constraint decompose (2.132)becomes the extrinsic0 γ4¯φ ˜D curvature2i D j e R into itsAij A trace andK 2 tracelessπe ρ, parts(11.48) and conformally conformally related metricγ ¯ij with determinantγ ¯ i1 (assuming¯ =K6ijHφ ˜=jie Aij Cartesian6φ ¯ iγ−ijK8. +6 coordinates)φ 8i − 12 and+(11.47) a 0 = D j (e A= ) e+ D3 K 8πe S . (11.49) φ transformeφ e5 theφ tracelesswhile=e5 theφM momentum part= as constraint we− do3 the (2.133 metric,)becomes− conformal factor e , 0 γ¯ ijD¯ D¯ eφ In termsR¯ of theseA˜ A variables˜ij theK 2 Hamiltonian2πe5φρ, constraint(11.48) (2.132)becomes i j The evolution equationij (2.136)forγ splits into two equations, = H = − 8 + 8 − 12 + ij 2 390 Chapter 11 Recasting the evolution equations φ 5φi ¯ 6φ ˜5φji 6φ ¯ i 6φ i whilePrimary the momentum and constraint (2.133)becomes 4φ ij φ e 0 e Dijj (e eA ) 2 e 1D5φK 8πe S . (11.49) 0 γ¯ D¯ i D¯ j e 1 R¯ = Mi A=˜ij A1˜ i 4φ K˜− 3 2πe ρ,− (11.48) γij =eH =γ¯ij.∂t φ − α8K +β ∂8iKφ ij ∂i−βe,12 Aij+ γijK(11.46). (11.50) (11.47) = −6 + + 6 secondary fields i 6φ ji 2 =6φ Thei evolution6φ equationi (2.136)forγij=splits into two+ equations,3 0 D¯Boxwhilej (e 11.1A˜ the) momentum Thee BSSND¯ equationsK constraint8πe S (2.133. )becomes (11.49) = M = − 3 − k k k 2 k ∂t γ¯ij 2α A˜ij β ∂k γ¯ij γ¯ik∂ j β γ¯kj∂1i β γ¯ij∂k β , 1 (11.51) We also decomposeIn the BSSN the formulation extrinsic ofIn the curvature terms 3 1 equations of these into the variables spatial its390 trace metricChapter theγ andij Hamiltonianis 11 decomposed traceless Recasting2 the into parts constraint evolution a andi equations (conformally2.132i )becomes The evolution equation (2.136)forγij splits into two equations,= − i+ ¯ 6φ+˜ ji ∂t φ+6φ ¯ i αK− 36βφ ∂i φ ∂i β , (11.50) + 0 D j (e A ) e =D−K6 8π+e S . + 6 (11.49) conformally related metricγ ¯ij with determinantγ ¯ 1 (assuming= M = Cartesian coordinates)− 3 and− a transform the traceless part as we dowhile the the evolution metric, equation= (2.135) for Kij splits into the twoφ equations5φ 5φ φ 1 i 1 i 2 conformal factor e , ∂ φ Theα evolutionK β ∂ equationφ ∂ (β2.136, )forγij splits intoφ two˜ (11.50) equations,e k Box 11.1e Thek BSSNij equationsk e 2k 5φ t i i ij∂t γ¯ij 1¯ 2α Aij β ∂k¯γ¯ij γ¯ik∂ j β˜ γ¯˜kj∂i β γ¯ij∂k β , (11.51) = −6 + +ij06 γ˜¯ ˜ijDi =D−j 2e + R + i Aij+ A − 3 K 2πe ρ, (11.48) ∂t K 4φγ D j Di αIn1 theα(A BSSNij A formulationK ) 4πα of(ρ the 3S) 1β equations∂i K, the spatial(11.52) metric γ is decomposed into a γij =4eφ− γ¯ij. = H+=2 +1 3 +i− 8(11.46)1+ +i + 8 − 12 +ij ∂ γ¯ 2α A˜ βkK∂ γ¯ γ¯ e=∂ βkA˜whileγ¯ ∂ theβk evolutionγ ∂γt¯φK∂.β equationk , αK (2.135β ∂(11.51)i φ) for K∂iijβsplits+, into the two equations(11.47)(11.50) t ij ij k ijij ik j ijkj i conformallyij ij =k − related6 + metricγ ¯+ij 6with determinantγ ¯ 1 (assuming Cartesian coordinates) and a Evolution = − + ∂ A˜+ e 4φ + (D D α−)TF3 α(RTF 8π STF) α(K A˜ 2A˜ A˜l ) = We also decompose the extrinsicwhile curvaturet the=ij momentum into− its trace+i andj3 constraint tracelessij parts (2.133φ andij conformally)becomes1ij il j = − conformal+ factorij −e , ˜+ ˜ij −2 2 i while theequations evolution equation (2.135) for Kij splits into the two equations∂t K˜ γk D j Di α 2 α(kAij A k K ) 4παk(ρ S) (11.53)β ∂i K, (11.52) transform the traceless part as we do the metric, k ∂t γ¯˜ij ˜2α Aijk= −β˜ ∂k γ¯ijk γ¯ik+∂˜j β kγ¯kj∂+i β3 +γ¯ij∂k β , + + (11.51) In terms of these variables the Hamiltonian constraintβ ∂!k Aij= −A (ik2.132∂ j β + A)becomeskj∂i β + Aij"∂k+β . − 3 4φ 1 + + + − 3 γij2 e γ¯ij. (11.46) ij ˜ ˜ij 2 1 ˜ i 4φ i ¯ TF 6φ ˜TFji TF=6φ ¯ i ˜ ˜ 6˜lφ i ∂t K γ D j Di α α(AijwhileA theK evolution4φ) ˜ 4πα equation(ρ ∂St )A (ij2.135β0∂ei)−K for, Kij(Dsplitsi D jDα into)(11.52)j (e theα two(ARij equations) 8π Sije) Dα(KAij 82πAileA j )S . (11.49) = − + In theK lastij+ 3 equatione A+ij theγij superscript+K. + = TF denotes− the trace-free+(11.47) part− of a tensor,+ e.g., R−TF =φ + 35φ We also= decomposeM5φ = the extrinsic curvature− 32 into its trace− andij traceless parts and(11.53) conformally e e k e ˜ 1 ˜ k ˜ φ k ˜ φ k = ∂ A˜ e 4φ (ijD D¯ α)¯TFR φα(RγTFR/3.8π¯ WeSTF also) ij splitα(K˜ theA˜ Ricci˜ij2A˜˜ tensorβA˜∂!˜lkij)Aij into A2R2ik∂ j β R¯ Akj5Rφ∂i β,wherei ARij"∂kcanβ . be found by In0 termst ij of these− variablesγ¯ iDji theD j Hamiltonianeij ijij constraint∂Rt K ij γ (2.132DAj D)becomestransformiji αijA α+(Ail theij A tracelessj +KK partij) 4 as2πα+πij we(eρ doij theSρ) , metric,−β3∂i Kij, (11.48)(11.52) = − The+ − evolution− = equation− + + (−2.136)for+ 3 γ +¯=splits+ into+ + two equations, = H = inserting− φ8 ln2ψ+into8 equation (3.10−). We12 expressij+(11.53)Rij in terms of the conformal connection βk ∂! A˜ A˜ ∂ βk A˜ ∂ φβ=k 5A˜φ "∂ βk . 5φ 1 TF k ij ikij j φ kj ei¯ i˜ ejk ¯ij4iInφk theij lasteij equationTF2 the5φTF superscriptTF TF denotes˜ the4φ ˜˜ trace-free˜l part of a tensor, e.g., Rij + 0 + γ¯ D¯ functionsD¯ +e ∂)t AR¯ij−γ¯3 e)−Ajk˜ A˜ (D∂ijDγ¯ j α,K)which2α yieldsπ(eRijρ, 8π Sij )(11.48)α(KKijAij e 2AAilijA j ) γijK. = (11.47) i j = ij − + − + − ¯ 3 φ φ while the momentum constraint= H = (2.133−)becomes8 ≡+ 8 Rij= −−γijR12/3. We+ also split the Ricci1 tensor into= Rij R+ij 1R ,where(11.53)R can be found by k − k k 2 TF k i = + ij i ij In the last equation the superscript TF denotes the1 trace-freeβ ∂!k A˜ij partA˜ik∂ ofj β a tensor,A˜kj∂∂tiφβ e.g., RijA˜ij"∂kαβ K. β ∂i φ ∂i β , (11.50) while the momentum constraint (2.133)becomes¯ insertinglm Inφ termsln ofψ ¯ theseintok equation¯ variablesk ¯ ( the3.10lm= Hamiltonian). We¯ k express¯ constraintR¯¯ijk in¯ terms (2.132 of)becomes the conformal connection Rij +γ¯ ¯∂m ∂l γ¯+ijφ γ¯k(i ∂ j))+φ ) )(ij=−)k 3−γ¯6 2)l(i+) j)km )im)+klj 6. (11.54) Rij γijR/3. We also split the Ricci tensor into= R−ij2 Rij Rij+,wherei= jkRiji +can be foundij+ by + functions2 )¯ γ¯ )¯ ∂ j γ¯ , which yields φ 5φ 5φ − Constraint i In the last6φ equationji 2¯= the+ superscript6φ i TFjk denotes6φ thei trace-free parte of ae tensor, e.g.,e RTF inserting φ ln ψ into0 equation (3.10i D¯).¯¯i We(e6 expressφ ˜ jiA˜ R) ij6inφ ¯ termsi e ofD¯≡ the6φK conformali 0=8π− e connectionγS¯ ijD.¯ D!¯ eφ R¯ A˜ "A˜(11.49)ij ijK 2 2πe5φρ, (11.48) = 0 The )Djjare(e nowA ) treatede asD independentK 8πe S functions. that satisfy(11.49)i theirj ownφ evolution equations,φ ij = 2 ¯ i jk ¯ i =ijM =R γ R/3.− We3 also split− the Ricci1 tensor˜ = intoH =Rk R¯ −R ,where8 + k8R can be− found12k by+ k functionsequations) γ¯ ) jk = M∂ j γ¯ ,=whichij yieldsij − ∂3t γ¯ijR¯ 2αγ−¯ lmA∂ij∂ γ¯ β γij¯∂k∂γ¯)ijij¯ k )¯ijγk¯)ik¯ ∂ j β γ¯ lmij 2γ)¯¯kjk )∂¯ i β )¯ k )¯ γ¯ij.∂k β(11.54), (11.51) ≡ = − − ij 2 m l ij k¯(=i j) + (ij)k l(i j)km im klj The evolution equation (2.136)forinsertingγij splits¯ iφ intoln twoψ˜ij equations,into equationwhile=¯= thei (−3.10−˜ momentumkj2 ). Weij express+ constraint+ Rij inij (2.133 terms++ )becomes of˜ij the+ conformal+ connection+− 3 1 ∂t ) = 2A ∂ j α 2α ) jk A γ¯ ∂ j K 8πγ¯ S j 6A ∂ j φ The evolution equation¯ (2.136lm )forγ ¯ splitsk ¯ )k¯=¯i into−γ jk)¯ twolmi +i ¯ equations,k∂ ¯γ ij, ¯−k ¯3 − + ! " Rij γ¯ ∂m ∂l γ¯ij γ¯k(i ∂functionsijj)) ) )(ij)k ¯ Theγ¯ jk )¯2)arel(i )j now¯j)km treatedwhich2)im) yields asklj independent. 1 (11.54) functions that satisfy2 their own(11.55) evolution equations, = −2 + while the+1 evolution≡ +i j ¯ i =1 equation−¯ j i# i + (¯2.135i j ) forli K i j splits¯ lj 6φ into˜ jii $ the6φ two¯ i equations6φ i ∂t φ αK ββ∂i∂φj ) ∂)i β∂,j β ) ∂ j β 0γ¯ ∂l(11.50)∂ jijβ Dγ¯j (e∂ j ∂Al β ). e D K 8πe S . (11.49) i = −6 ++ 1 +!−6 + 3 "+ 3 = M =+2 − 3 − The )¯ are now treated as independent functionsR¯ that satisfyγ¯ lm their∂ ∂ ownγ¯¯ i evolutionγ¯ ∂ ˜ij)¯ k equations,)¯ k )¯ ¯ i γ˜¯kjlm 2)¯ k ij)¯ )¯ k )¯ij . ˜(11.54)ij 1 ij i m ∂l t )ij 1k(i2Aj) ∂i j α 2α(ij))k jk A γl¯(i ∂jj)kmK 8πimγ¯ kljS j 6A ∂ j φ = −2 The evolution+= −2 equation+ + (2.136+)for−γ1ij3 splits into−+ two equations,+ ∂ γ¯ ∂2αt φA˜ βk ∂2γ¯ αKγ¯ ∂ ββk ∂γ¯i φij∂ βk ∂γ¯ i ∂ββ,k , ij(11.51)2 2 1 (11.50) i (11.55) ¯ i ˜ijt ij ¯ iij ˜kj i k ijij∂ Kik j ijγkj i D˜ijD αijj k iα(A˜j # A˜i !i Kj ) li4παj(ρ" lj S) i $ β ∂ K, (11.52) ∂t ) 2A ∂ j α = 2−α ) jkTheA=+ −)¯ are6γ¯ now+∂tj K treated+8πγ+¯ asS independentj +6A−j 6∂3jiφβ functions∂ j )¯ )¯ thatij∂ j β satisfy their)¯ ∂ j β own evolutionγ¯ ∂l ∂ j β equations,γ¯ ∂ j ∂l β . i = − + − 3 −= − + + + − + 3+ 3 1+ 3+ i +1+ i + 2 1 (11.55)∂t φ αK β ∂i φ ∂i β , (11.50) while the evolutionj equation¯ i ¯ j # (2.135i ) for¯ i Kijjsplits intoli the twoj equationslj i $ 2 = −6 + + 6 β ∂ j ) ) ∂ j β ) ∂ j β ¯ i γ¯ ∂l˜∂ijj β γ¯ ∂ j ∂¯l βi .˜kj ij 2 ij ˜ij + − + 3 k ∂t )+ 3 2A 4∂φj α+ k2α ) jk A TFk γ¯ ∂ j K 8πTFγ¯ Skj 6A ∂TFj φ l ∂t γ¯ij 2α A˜ij β ∂∂kt γA¯˜ijij1 = −γ¯eik−∂ j β+ (Dγi¯Dkj∂j αi−β) 3 αγ−¯(ijR∂k β ,+8π S ) (11.51)α(K A˜ij2 2A˜il A˜ ) ij ˜ ˜ij 2 i 2 ∂ γ¯ 1 2α A˜ij βk ∂ γ¯ ijγ¯ ∂ βk γ¯ ∂ (11.55)βk γ¯ ∂ βk , j (11.51) ∂t K = −γ D j Di α +α(Aij A K+=) 4j πα¯(iρ −¯S+j)# βi ∂i K,¯ i t −ijj +3 li(11.52)ij j − kljij iik$j + kj i −ij k = − + + 3 +β ∂ j ) +) ∂+j β ) ∂ j β = −γ¯ ∂l ∂ j β+ γ¯ ∂ j+∂l β . + − 3 (11.53) + k − + 3 k+ 3 +k 2 k ˜ 4φ TF TF TF β ∂!whileA˜ ˜ the evolutionA˜ ˜∂l equationβ (A2.135˜ ∂)β for Kij splitsA˜ into"∂ theβ two. equations while the evolution∂t A equationij e− ((2.135Di D j α) ) forα(KR splits8π S ) intoαk( theK Aijij two2Ailik equationsA )j kj i ij k = − + ijij− +ij + +− j + − 3 2 (11.53) k ˜ ˜ k ˜ k ˜ k ij ij 1 2 i β ∂!k Aij Aik∂ j β Akj∂i β Aij"∂k β . ∂t K γ D j Di α α(A˜ij A˜ K ) 4πα(ρ S) β ∂i K, (11.52) + + In the+ last equation1− 3 the superscript= − TF+denotes+ the3 trace-free+ + part+ of a tensor, e.g., RTF ∂ K γ ijD D α α(A˜ A˜ij K 2) 4πα(ρ S) βi ∂ K, (11.52) ij t In the last equationj i the superscriptijTF denotes the trace-free part˜ of a tensor,4φ e.g., RiTFTF TF TF φ˜ ˜ ˜l φ = ∂t Aij e− (Di D j αij) α(Rij 8π Sij ¯) α(K Aij 2Ail A j ) = − + Rij γijR+/3.3 We also+ splitφ the=+ φ Ricci+− tensor=+ into R−ij Rij+ R ,where− R can be found by R γ R/3. We also split the Ricci tensor into R R¯ R ,whereR can be found by 2 ij ij (11.53) ij ij − ij ij ij ij k ˜ ˜ k ˜ k = ˜ +k − 4φ TF TF ¯= TF+ β ∂!k Aij Aik∂ j β l Akj∂i β ¯ Aij"∂k β . ∂t A˜ij insertinge− φ ln(Dψ iintoD j equationα) inserting (3.10α).( WeRφ expressln8Rψπij Sininto terms) equation of theα( conformal+K A (˜3.10ij connection+).2 WeA˜il A˜ express+ ) −Rij3 in terms of the conformal connection ¯ i= jk ¯ i ij ij = ij j =functions ) − γ¯ ) jk ∂ j γ¯ ,+which yieldsi − jk i + ij − TF functions )¯ γ¯ )¯ In the last∂ equationγ¯ , which the superscript yieldsTF denotes the trace-free part of a tensor, e.g., Rij ≡ = − 2 jk j (11.53) = k ˜ 1 ˜ k ˜ ≡k R˜ =γ−R/k3. We also split the Ricci tensor into R R¯ Rφ ,whereRφ can be found by β ∂!¯ k Aij lmAik∂ j β ¯Ak kj∂¯ ki β¯ lm Aijij¯ k"∂¯kijβ . ¯ k ¯ ij ij ij ij Rij γ¯ ∂m ∂l γ¯ij γ¯k(i ∂ j)) ) )(ij)k γ¯ 2)−l(i ) j)km )im)klj . (11.54) ¯= + + = −2+ + + + −+ 13insertinglm φ +ln ψ into equationk (3.10).k We express Rlmij in termsk of the conformalk connection ¯ ¯ i= jk ¯ i ¯ ij ¯ ¯ ¯ ¯ ¯ ¯ ¯ i Rij γ¯functions! ∂m ∂l)γ¯ij γ¯ )γ¯k("i ∂ j)∂)j γ¯ , which) ) yields(ij)k γ¯ 2)l(i ) j)km )im)klj . (11.54) In the last equationThe ) are now the treated superscript as independentTF functionsdenotes that= satisfy−2 the their trace-free own evolution≡ + equations, partjk = − of a+ tensor, e.g.,+ RTF + 1 ij i ij i kj 2 ij ij ¯ij lm ¯ k ¯ k ¯ lm ¯=k ¯ ¯ k ¯ ∂t )¯ 2A˜ ∂ j α 2α )¯ ¯A˜i γ¯ ∂ j K 8πγ¯ S j 6AR˜ ij∂ j φ φγ¯ ∂m ∂l γ¯ij γ¯k(i ∂φj)) ) )(ij)k γ¯ ! 2)l(i ) j)km )im)klj . (11.54)" Rij γijR/3. We also= split− the+ RicciThe jk) tensorare− 3 now into treated− Rij as+R independent¯ij = −R2 ,where functions+ R thatcan+ satisfy be found+ their by own evolution+ equations, 2 1 ij (11.55) ij − j ¯ i ¯ j # i ¯ i j li j =¯ i lj +i $ ! " inserting φ ln ψ into equationβ ∂ j ) ) (∂3.10j β ).) We∂ j β expressγ¯ ∂l ∂ jTheβR¯ )γ¯arein∂ j now∂l termsβ . treated of as independent the conformal functions that connection satisfy their own evolution equations, + − + 3 i + 3 ij+ij i kj 2 ij ij ij = ∂ )¯ 2A˜ ∂ α 2α )¯ A˜ γ¯ ∂ 2K 8πγ¯ S 6A˜ ∂ φ ¯ i jk ¯ i ij t j ¯ i ˜ijjk ¯ i ˜kj j ij ij j ˜ij j functions ) γ¯ ) ∂ j γ¯ , which yields= − ∂+t ) 2A ∂ j α 2−α )3jk A γ¯ −∂ j K 8πγ¯ S j +6A ∂ j φ ≡ jk = − = − +2 − 31 − + (11.55) (11.55) j i j # j i i j #i i j2 i j li1 li j j ljlj i $ i $ ¯ ¯ β ∂ j )¯ )¯ ¯∂ j β )¯ ∂ j β γ¯ ∂l ∂ j β γ¯ ∂ j ∂l β . β ∂ j ) ) ∂+j β − ) ∂ j +β 3 γ¯+ 3∂l ∂ j β + γ¯ ∂ j ∂l β . 1 lm k k + lm− k + 3 k + 3 + R¯ij γ¯ ∂m ∂l γ¯ij γ¯k(i ∂ j))¯ )¯ )¯ (ij)k γ¯ 2)¯ )¯ j)km )¯ )¯ klj . (11.54) = −2 + + + l(i + im The )¯ i are now treated as independent functions that satisfy their! own evolution equations," 2 ¯ i ˜ij ¯ i ˜kj ij ij ˜ij ∂t ) 2A ∂ j α 2α ) jk A γ¯ ∂ j K 8πγ¯ S j 6A ∂ j φ = − + − 3 − + (11.55) j i j # i 2 i j 1 li j lj i $ β ∂ j )¯ )¯ ∂ j β )¯ ∂ j β γ¯ ∂l ∂ j β γ¯ ∂ j ∂l β . + − + 3 + 3 + The computational details get complicated fast

Adaptive mesh 1 refinement

R B

RS 2 4

Multipatch spherical regions 3 Fig: Reisswig et al. Event horizons cannot be used in simulations

Region of spacetime that null Black hole: geodesics cannot escape

2+1 dim hypersurface defined by Event horizon: future directed null geodesics

2 dim surface defined on slice by Apparent horizon: future directed null geodesics

Must be inside the event horizon Outline

History and formalism Initial data

Evolution of field equations Wave extraction

The tools we use 1

@2A + D DjA H(t, xi)=D @ 4⇡j t i j i i t i

S1,S2

J

M1/M2 Wave extraction requires a known, perturbed

rad background hab

rad Wave zone: gab = ⌘ab + hab Strong gravity, no approx.

4 ij = ⌘ij

R = 1

R =2M

R =0

a nae(i) =0

⌦ = t a ra

n = ↵⌦ a a

a n ab =0

↵i Fig: jpl.nasa.gov/

i A ni

ni

Ai () ij

Ei () ij

Ai Ei ij Kij After wave extraction, we build a correspondence

1

S1,S2 1 1 J 1 S1,S2

M1/M2 S1,S2

S1J,S2 rad hab J M1J/M2

gab = ⌘Doab +it 249,999hab more times!M1/M2 rad Mh1ab/M2 4 rad ij = ⌘ij hab gab = ⌘radab + hab hab

R = gab = ⌘ab + hab 1 4 ij = ⌘ij gab = ⌘ab + hab 4 R =2M ij = ⌘ij R = 4 ij = 1⌘ij R =0 R = R =21M R = 1 a nae(i) =0 R =2M R =0 R =2M ⌦ = t R =0 a a a r nae =0 R(i=0) a na = ↵⌦a nae(i) =0 ⌦ = t a a ra nae(i) =0 a n ab =0 ⌦a = at na = r↵⌦a ⌦ = t a ra i ↵ naa= ↵⌦a n ab=0 n = ↵⌦ a a i a A n n ab =0 i ↵i a n ab =0 i i n ↵i A ni ↵i i i A A ni i () ij n i A ni i i E ij Ai n () () ij ni i Ai Ei ij Kij AEi ij ()() ij Ai () ij i A EE ij K i i () ij ij Ei () ij Ai Ei ij Kij

Ai Ei ij Kij Outline

History and formalism Initial data

Evolution of field equations Wave extraction

The tools we use The Einstein Toolkit is free, open source, and actively developed

• 113 Members • 69 Groups

einsteintoolkit.org There are tutorials for new users

• You need a cluster • Binary black holes • Relativistic magnetohydrodynamics • ~5000 cpu hrs / run

einsteintoolkit.org SimulationTools is a free and open source package for analyzing simulation data in Mathematica

• The simulation is only half the effort ➡ Post processing! • Ian Hinder & Barry Wardell

simulationtools.org You can simulate the GW150914 event yourself!

• Barry Wardell & Ian Hinder • Parameter files for simulation • 16,108 cpu hrs • Post processing scripts

einsteintoolkit.org/about/gallery/gw150914/ You can simulate the GW150914 event yourself!

Trajectories Waveform

Common Horizons horizon You can simulate the GW150914 event yourself! These were some of the main points

ANNALS OF PHYSICS: f8, 304-331 (1964) S1 S2 P2 The Two-Body Problem in Geometrodynamics

SUSAN G. HAHN

International Business Machines Corporation, New York, New York

AND

RICHARD W. LINDQUIST

Adelphi University, Garden City, New York P1

The problem of two interacting masses is investigated within the framework of geometrodynamics. It is assumed that the space-time continuum is free of all real sources of mass or charge; particles are identified with multiply con- nected regions of empty space. Particular attention is focused on an asymp- totically flat space containing a “handle” or “wormhole.” When the two “mouths” of the wormhole are well separated, they seem to appear as two cen- ters of gravitational attraction of equal mass. To simplify the problem, it isThank you! assumed that the metric is invariant under rotations about the axis of sym- metry, and symmetric with respect to the time t = 0 of maximum separation of the two mouths. Analytic initial value data for this case have been ob- tained by Misner; these contain two arbitrary parameters, which are uniquely determined when the mass of the two mouths and their initial separation have been specified. We treat a particular case1 in which the ratio of mass to initial separation is approximately one-half. To determine a unique solution of the remaining (dynamic) field equations, the coordinate conditions go- = -& are imposed; then the set of second order equations is transformed into a quasi- linear first order system and the difference scheme of Friedrichs used to ob- tain Ra Bnumerical solution. Its behavior agrees qualitatively with that of the one-body problem, and can be interpreted as a mutual attraction and pinching- off of the two mouths of the wormhole. RS I. INTRODUCTION Wheeler (1, 2) has used the term “geometrodynamics” to characterize2 those solutions of the field4 equations for gravitation and electromagnetism’ 41 = R,v - ?4 g& = 2(F,,FP - Pi gj,.F,sF=B) (l.la) FPu;v = 0 (l.lb)

1 Throughout this paper Greek subscripts and superscripts range from 0 to 3 and Latin ones from 1 to 3. Also, units are chosen so that G (universal gravitation constant) = c = 1. 3043