arXiv:1010.5260v2 [gr-qc] 27 Nov 2010 Contents § ‡ † ∗ r numerical and waves, gravitational binaries, Black-hole lcrncades [email protected] address: Electronic [email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic I.Hsoia Overview Historical III. V ueia Development Numerical IV. I lc-oeBnre n rvttoa Waves Gravitational and Binaries Black-Hole II. .Prelude I. .IiilData Initial D. .StigteStage the Setting A. .BscProperties Basic A. .Enti’ qain 8 Equations Einstein’s A. .Rpeetn lc oe nNmrclSaeie 9 Spacetimes Numerical in Holes Black Representing C. .BekhogsadteGl uh7 Rush Gold the and Breakthroughs C. .GaiainlWvsfo lc-oeBnre 5 Binaries Black-Hole from Waves Gravitational C. .TeCuh rbe 8 Problem Cauchy The B. .NmrclRltvt ietns6 Milestones Relativity Numerical B. .AtohsclBakHls4 Holes Black Astrophysical B. .NmrclyFinl omltoso h Evolution the of Formulations Friendly Numerically E. .GueConditions Gauge F. enr .Kelly J. Bernard G 8800 NASA/GSFC, Laboratory, Astrophysics Gravitational n nohraesaeas discussed. also are areas is other astro to that in results and dynamics physics fundamental binary matur these black-hole of rapidly of applications the understanding co al in new orbit has progress the binary relativity remarkable numerical a This in of advances fraction simulations. dramatic a enco of just mergers series after a these crash to model g codes to their of attempting space-ba equations relativists Einstein’s and merical of ground- gravitati simulations resulting both numerical the for and through dynamics sources merger black-hole key The are ph and amount theoretical tremendous radiation, in releasing problem events, astrophysical unsolved mental long-standing a th been for relativity has general of predictions the Understanding 2010) 30, November (Dated: C Baltimore Maryland, of University USA Physics, of Department NASA/G Laboratory, and Astrophysics USA Gravitational and CRESST onCentrella Joan .AMbsdFruain 12 12 12 Formulations ADM-based 3. Formulations Harmonic Well-Posedness 2. and Hyperbolicity 1. Equations .Mvn ucue 13 14 13 14 Techniques Coordinate Other Coordinates 4. Harmonic Generalized 3. Punctures Shift Moving and Slicing 2. the Choosing 1. .BakHl ais3 3 Primer Wave Gravitational 2. Basics Black-Hole 1. ∗ ‡ n onG Baker G. John and n ae .vnMeter van R. James and † § 11 13 10 5 5 2 3 2 8 yaia neatoso w lc holes black two of interactions dynamical e hsc,t rvttoa-aeastronomy, gravitational-wave to physics, feeg ntefr fgravitational of form the in energy of s oe tbe outbakhl merger black-hole robust stable, lowed nrlrltvt.Frmn er,nu- years, many For relativity. calculated eneral be only can waveforms onal l esmltd eety however, Recently, simulated. be uld nee oto rbes causing problems, of host a untered n edo ueia eaiiy and relativity, numerical of field ing sc.Bakhl egr r monu- are mergers Black-hole ysics. II mato Astrophysics on Impact VIII. mrigi hoild Important chronicled. is emerging I.Apiain oGaiainlWv Data Wave Gravitational to Applications VII. enetRa,Genet ayad271 USA 20771, Maryland Greenbelt, Road, reenbelt I neato fNmrclRltvt with Relativity Numerical of Interaction VI. e rvttoa-aedetectors. gravitational-wave sed ut,10 ilo ice atmr,Mrln 21250, Maryland Baltimore, Circle, Hilltop 1000 ounty, .BakHl egrDnmc n Waveforms and Dynamics Merger Black-Hole V. F,80 rebl od rebl,Mrln 20771, Maryland Greenbelt, Road, Greenbelt 8800 SFC, .NmrclApoiainMtos15 Methods Approximation Numerical G. Analysis otNwoinTheory Post-Newtonian .Ps-etna hoyfrItrrtto of Interpretation for Theory Post-Newtonian D. .Mreso pnigBakHls24 Holes Black Spinning of Mergers D. .TeDrc mato egrWvfrsi Data in Waveforms Merger of Impact Direct The A. .Etatn h hsc 15 Physics the Extracting H. .IdpnetPs-etna yaisand Dynamics Post-Newtonian Independent A. .RciigBakHls39 Holes Black Recoiling A. .FrtGipe fteMre:TeLzrsApproach16 Lazarus The Merger: the of Glimpses First A. .UigNmrclWvfrsi aaAnalysis Data in Waveforms Numerical Using C. .Ps-etna oesfrNmrclIiilDt.32 Data. Initial Numerical for Models Post-Newtonian C. .Mreso nqa-as osinn lc oe 22 Holes Black Nonspinning Unequal-Mass, of Mergers C. .Dvlpn nltcInspiral-Merger-Ringdown Analytic Developing B. .Aayi ulWvfr oes30 Models Full-Waveform Analytic B. .TeSi fteFnlBakHl 40 Hole Black Final the of Spin The B. .Mreso qa-as osinn lc oe 17 Holes Black Nonspinning Equal-Mass, of Mergers B. rvttoa aeomTmlts37 Applications Templates Waveform Gravitational Analysis Waveforms .Peitn h eol39 40 Recoil Hole Black of Consequences Recoil 2. the Predicting 1. ueia eut.32 Results. Numerical .GaiainlWvfrs25 20 25 26 18 23 Holes Black Spinning of Mergers Flips from Spin 17 Kicks and Mergers 3. Binary Spinning 2. Waveforms 22 Gravitational 1. Kicks Feature: New Qualitatively Waveforms A Gravitational and 2. Analysis Mode 1. Waveforms Longer 3. Waveform Universal Waveforms Merger 2. First The 1. elativity 33 27 39 38 35 16 28 2
C. Electromagnetic Counterparts of Black Hole Mergers 41 We begin by setting both the scientific and historical 1. Astrophysical Considerations 41 contexts. In Sec. II we provide a brief overview of astro- 2. Simulations with Magnetic Fields or Gas near the physical black-hole binaries as sources for gravitational- Merging Holes 41 wave detectors. We next turn to a historical overview IX. Frontiers and Future Directions 43 in Sec. III, surveying efforts to evolve black-hole merg- A. Gravitational-Wave Astronomy 43 ers on computers, spanning more than four decades and B. Other Astrophysics 43 culminating with the recent triumphs. Having thus set C. Other Physics 44 D. Strong Gravity as Observational Science 44 the stage, we focus on more in-depth discussions of the key components underlying successful black-hole merger Acknowledgments 45 simulations, discussing computational methodologies in Sec. IV, including numerical-relativity techniques and References 45 black-hole binary initial data. Section V is the heart of this review. Here we discuss the key results from numer- I. PRELUDE ical relativity simulations of black-hole mergers, follow- ing a historical development and concentrating on the The final merger of two black holes in a binary sys- merger dynamics and the resulting gravitational wave- tem releases more power than the combined light from forms. These results have opened up a variety of excit- all the stars in the visible Universe. This vast energy ing applications in general relativity, gravitational waves, comes in the form of gravitational waves, which travel and astrophysics. We discuss synergistic interactions across the Universe at the speed of light, bearing the between numerical relativity and analytic approaches waveform signature of the merger. Today, ground-based to modeling gravitational dynamics and waveforms in gravitational-wave detectors stand poised to detect the Sec. VI, and applications of the results to gravitational- mergers of stellar black-hole binaries, the corpses of mas- wave data analysis in Sec. VII. The impact of merger sim- sive stars. In addition, planning is underway for a space- ulations on astrophysics is presented in Sec. VIII, which based detector that will observe the mergers of massive includes discussions of recoiling black holes and potential black holes, awesome behemoths at the centers of galax- electromagnetic signatures of the final merger. We con- 4 9 clude with a look at the frontiers and future directions of ies, with masses of (10 10 )M⊙, where M⊙ is the mass of the Sun. Since∼ template− matching forms the ba- this field in Sec. IX. sis of most gravitational-wave data analysis, knowledge Before we begin, we mention several other resources of the merger waveforms is crucial. that may interest our readers. The review articles by Calculating these waveforms requires solving the full Lehner (2001) and Baumgarte and Shapiro (2003) pro- Einstein equations of general relativity on a computer in vide interesting surveys of numerical relativity several three spatial dimensions plus time. As you might imag- years before the breakthroughs in black-hole merger sim- ine, this is a formidable task. In fact, numerical rela- ulations. The article by Pretorius (2009) is an early tivists have attempted to solve this problem for many review of the recent successes, covering some of the years, only to encounter a host of puzzling instabili- same topics that we discuss here. Hannam (2009) re- ties causing the computer codes to crash before they viewed the status of black-hole simulations producing could compute any sizable portion of a binary orbit. long waveforms (including at least ten cycles of the dom- Remarkably, in the past few years a series of dramatic inant gravitational-wave mode) and their application to breakthroughs has occurred in numerical relativity (NR), gravitational-wave data analysis. Finally, the text books yielding robust and accurate simulations of black-hole by Bona and Palenzuela (2005) and Alcubierre (2008) mergers for the first time. provide many more details on the mathematical and com- In this article, we review these breakthroughs and putational aspects of numerical relativity than we can the wealth of new knowledge about black-hole mergers include here, and serve as useful supplements to our dis- that is emerging, highlighting key applications to astro- cussions. physics and gravitational-wave data analysis. We focus on comparable-mass black-hole binaries, with component mass ratios 1 q 10, where q = M1/M2 and M1, M2 are the individual≤ ≤ black-hole masses. We will frequently also refer to the symmetric mass ratio II. BLACK-HOLE BINARIES AND GRAVITATIONAL M1M2 q WAVES η 2 = 2 . (1) ≡ (M1 + M2) (1 + q) For simplicity, we choose to set c = 1 and G = 1; with Black holes and gravitational waves are surely among this, we can scale the dynamics and waveforms for black- the most exotic and amazing predictions in all of physics. hole binaries with the total system mass M. In particu- These two offspring of Einstein’s general relativity are lar, we can express both length and time scales in terms of brought together in black-hole binaries, expected to be the mass, giving M 5 10−6M/M s 1.5M/M km. among the strongest emitters of gravitational radiation. ∼ × ⊙ ∼ ⊙ 3
A. Basic Properties Equation (2) requires a M; when a = M the black hole is said to be maximally≤ rotating or “extremal”. No- We begin by presenting some basic properties of black tice that r+ = 2M when a = 0, and that r+ decreases holes and gravitational waves. For fuller discussions and as a increases, thus bringing the location of the horizon more details, see Misner et al. (1973) and Schutz (2009). deeper into the potential well as the black-hole spin in- creases. Photons and test particles in the vicinity of a single 1. Black-Hole Basics black hole can experience either stable or unstable orbits. For a Schwarzschild black hole of mass M, the innermost A black hole forms when matter collapses to infinite stable circular orbit (ISCO) occurs at r =6M for massive density, producing a singularity of infinite curvature in test particles. In the case of a Kerr black hole, the ISCO the fabric of spacetime. Each black hole is surrounded is closer in for co-rotating test particles, and farther out by an event horizon, at which the escape velocity is the for counter-rotating particles. speed of light. The event horizon is a global property of While the concept of an ISCO is strictly defined only the spacetime, since it is defined by the paths of “out- for massive test particles, it has proven useful for stud- going” photons that are the boundary between photon ies of the spacetime around two black holes spiralling trajectories that must fall inward, and those that can es- together on quasicircular orbits. Imagine that you put cape to infinity. The photons defining the event horizon the two black holes on an instantaneously circular orbit hover at finite radius at the surface of the black hole. around each other; at that moment they have neither Since, in principle, mass (energy) can fall into the event nonzero radial velocity nor nonzero radial acceleration. horizon at late times – which will move the location at At any given separation, the black holes have some an- which photon paths can hover – we must know the en- gular momentum. The ISCO is the separation where tire future development of the system to locate the event that angular momentum is a minimum, in analogy to horizon. the test particle definition. Black holes at closer separa- When black holes merge, a single event horizon forms tions would be expected to fall inward, toward the center, whose area is at least as large as the sum of the individ- even without radiating angular momentum via gravita- ual horizons. Since numerical relativists want to know tional radiation. when this occurs during the course of a calculation, they rely on a related concept known as an apparent hori- zon, whose location depends only on the properties of 2. Gravitational Wave Primer the spacetime at any given time (Poisson, 2004). For quiescent black holes, the apparent and event horizons Gravitational waves are ripples in the curvature of coincide; for more general holes, the apparent horizon is spacetime itself. They carry energy and momentum and always inside the event horizon (with restrictions on the travel at the speed of light, bearing the message of dis- behavior of the matter involved). So, in terms of causal- turbances in the gravitational field. ity in a numerically-generated spacetime, any physical As with electromagnetic waves, gravitational waves phenomenon found inside an apparent horizon should not can be decomposed into multipolar contributions that leak out and affect the spacetime outside. reflect the nature of the source that generates them. Re- The simplest black hole is nonrotating and is described call that electromagnetic radiation has no monopole con- by the spherically symmetric Schwarzschild solution to tribution due to the conservation of total charge. By the Einstein equations of general relativity in vacuum analogy, conservation of total mass-energy guarantees (i.e., with no “matter” sources in the spacetime). A that there can be no monopole gravitational radiation. Schwarzschild black hole is fully specified by one quantity, Since dipolar variations of charge and currents are pos- its mass M. The horizon is located at coordinate r =2M sible, electromagnetic waves can have a dipole character. (in Schwarzschild coordinates); its area is 4π(2M)2. However, conservation of linear and angular momenta re- More general black holes can have both charge and moves any possibility of dipolar gravitational waves, so spin. Since a charged black hole in astrophysics will gen- the leading-order contribution to gravitational radiation erally be neutralized rapidly by any surrounding plasma, is quadrupolar. we can consider only rotating, uncharged black holes. Gravitational waves are thus generated by sys- Stationary (i.e., time independent) black holes are de- tems with time-varying mass quadrupole moments scribed by the axisymmetric Kerr solution. A Kerr black (Flanagan and Hughes, 2005; Misner et al., 1973). In the hole is fully specified by two quantities, its mass M wave zone, a gravitational wave is described as a pertur- and its angular momentum per unit mass a. The event bation h to a smooth underlying spacetime. The wave horizon is located at the Boyer-Lindquist (Misner et al., amplitude is 1973) radius r+, where 2 G Q¨ GMquad v 2 2 1/2 h , (3) r = M + (M a ) . (2) 4 2 2 + − ∼ c r ∼ rc c
The area of the event horizon is 8πMr+. where Q is the quadrupole moment of the source, r is 4
Intermediate-mass black holes (IMBHs) have masses 2 4 in the range (10 10 )M⊙. IMBHs may form as the result∼ of multiple− mergers of smaller objects in the centers of dense stellar clusters in the present Universe (Miller and Colbert, 2004; Portegies Zwart and McMillan, 2002), assuming mass loss from stellar winds is not significant (Glebbeek et al., 2009). They may also arise from the evolution of very massive stars early in the history of the Universe, form- FIG. 1 Lines of force for plane gravitational waves propagat- ing black-hole “seeds” in the centers of massive ha- ing along the z axis. The wave on the left is purely in the los (the precursors of the galaxies we see today) early + polarization state, and the one on the right is purely in > the × polarization state. The gravitational waves produce in the history of the Universe, to redshifts z 10 tidal forces in planes transverse to the propagation direction. (Madau and Rees, 2001). Currently the best observa-∼ From (Abramovici et al., 1992). Reprinted with permission tional evidence for IMBHs comes from models of ultra- from AAAS. luminous X-ray sources (Colbert and Miller, 2005). Finally, massive black holes (MBHs) have masses in 4 9 the range (10 10 )M⊙ and are found at the cen- the distance from the source, and Mquad is the mass in ters of galaxies,∼ including− our own Milky Way galaxy. the source that is undergoing quadrupolar changes. This The observational case for the existence of MBHs is quite shows that the strongest gravitational waves will be pro- strong, based on dynamical models of stars and gas be- duced by large masses moving at high velocities, such as lieved to be moving in the potential well of the central binaries of compact stars and black holes. MBH (Desroches et al., 2009; Ferrarese and Ford, 2005; A gravitational wave is purely transverse, acting tidally Kormendy and Richstone, 1995; Richstone et al., 1998). in directions perpendicular to its propagation direction. Black-hole binaries are binary systems in which each When a gravitational wave impinges on a detector of component is a black hole. As mentioned above, we focus length scale L, it produces a length change in that de- here on comparable-mass binaries, which are expected to tector δL/L h/2. By substituting in typical values for produce the strongest gravitational-wave signals. Stel- compact objects∼ in the Universe into Eq. (3), one can lar black-hole binaries may form as the result of binaries see that astrophysical sources typically yield wave ampli- composed of two massive stars; see Bulik and Belczynski tudes of h < 10−21 at the Earth. Consequently, precision (2009) and references therein. Stellar black-hole bina- measurements∼ are needed to make detections. ries may also arise from dynamical processes in which a Gravitational waves have two polarization compo- black hole is captured into an orbit around another black nents, known as h and h for a linearly polarized + × hole in dense stellar environments (Miller and Lauburg, wave. Figure 1 shows the corresponding lines of force 2009; O’Leary et al., 2007). IMBH binaries can also for sinusoidal gravitational waves propagating along the form through dynamical processes in stellar clusters z axis. In the purely + polarization on the left, the wave (G¨urkan et al., 2006) and from mergers of massive halos stretches along one axis and squeezes along the other, at high redshifts. Since both stellar and IMBH binaries alternating sinusoidally as the wave passes. The po- are “dark” – that is, they are generally not surrounded larization wave on the right acts similarly, stretching× and by gas which might produce electromagnetic radiation – squeezing along axes rotated by 45◦. In general, a gravi- we have few observational constraints on these types of tational wave is a superposition of these two states, con- black-hole binaries. This situation will change dramat- veniently written as a complex waveform strain h, where ically, however, with the detection of gravitational ra- diation from these systems (Bulik and Belczynski, 2009; h = h+ + ih×. (4) Miller, 2009), as gravitational waves bring direct informa- tion about the dynamical behavior of the orbiting masses B. Astrophysical Black Holes and do not rely on electromagnetic emissions from nearby matter. Astronomers have found evidence for black holes Since essentially all galaxies are believed to contain an throughout the Universe on a remarkable range of scales. MBH at the center and to undergo a merger with an- The smallest of these, stellar black holes, have masses in other galaxy at least once during the history of the Uni- the range (3 30)M⊙ and form as the end-products verse, MBH binaries can arise when their host galaxies of massive∼ star− evolution. There is good observational merge (Begelman et al., 1980); see also Djorgovski et al. evidence for the existence of stellar black holes, based on (2008) and references therein. However, due to the vast dynamical measurements of the masses of compact ob- cosmic distances involved, and the small angular sepa- jects in transient systems that undergo X-ray outbursts. rations on the sky expected for MBH binaries, only a Since neutron stars cannot have masses > 3M⊙, com- few candidates are currently known through electromag- pact objects more massive than this must∼ be black holes netic observations (Komossa, 2003; Komossa et al., 2003; (Remillard and McClintock, 2006). Rodriguez et al., 2006). When they form, MBH bina- 5 ries typically have relatively wide separations, and the 1957; Teukolsky, 1973; Zerilli, 1970) form the basis gravitational radiation they emit is very weak and insuf- of ringdown calculations, producing gravitational wave- ficient to cause the binary to coalesce within the age of forms in the shape of exponentially damped sinusoids the Universe. However, various processes such as gaseous (Berti et al., 2009; Leaver, 1986). dissipation and N-body interactions with stars can re- The characteristic gravitational-wave frequency of a move orbital energy from the binary and cause the black quasicircular black-hole binary, produced by the domi- holes to spiral together (Armitage and Natarajan, 2002; nant (highest-order) quadrupole component, is Gould and Rix, 2000); see also Berentzen et al. (2009) 3 1/2 and Colpi et al. (2009) and references therein. Eventu- fGW 2forb (M/R ) , (5) ally, the black holes reach separations at which gravita- ∼ ∼ tional radiation reaction becomes the dominant energy- where forb is the orbital frequency. Astrophysical black- loss mechanism, leading to the final coalescence of the hole binaries produce gravitational waves that span three black holes and the emission of strong gravitational waves frequency regimes, depending on the black-hole masses (Sesana et al., 2009b). (Flanagan and Hughes, 2005). Stellar black-hole binaries and the lower mass end of the IMBH binaries radiate in 4 the high frequency band, fGW (10 10 )Hz, which is already being observed by ground-based∼ − laser interferom- C. Gravitational Waves from Black-Hole Binaries eter detectors such as LIGO (Abbott et al., 2009b), and will be observed by the advanced detectors by 2016 Mergers of comparable-mass black-hole binaries are ex- (Smith, 2009). Low frequency gravitational waves∼ cover pected to be among the strongest sources of gravitational the band f (10−5 1)Hz and will be observed by waves. This final death spiral of a black-hole binary en- GW the space-based∼ laser interferometer− LISA, currently un- compasses three stages: inspiral, merger, and ringdown der development (Jennrich, 2009). MBH binaries with (Flanagan and Hughes, 1998; Hughes, 2009). 4.5 7 masses M (10 10 )M⊙ will be very strong sources In the early stages of the inspiral, the orbits of most for LISA, with∼ the lower− mass systems visible out to red- astrophysical black-hole binaries will circularize due to shifts z > 10 (Arun et al., 2009a); the inspirals of IMBH the emission of gravitational radiation (Peters, 1964; binaries∼ will also be detectable (Miller, 2009). Finally, Peters and Mathews, 1963). During the inspiral, the or- −9 −7 the very low frequency band fGW (10 10 )Hz bital time scale is much shorter than the time scale on will be observed by pulsar timing arrays∼ (Verbiest− et al., which the orbital parameters change; consequently, the 2009). This band is expected to dominated by gravita- black holes spiral together on quasicircular orbits. Since tional waves from a very large population of unresolved the black holes have wide separations, they can be treated MBH binaries (Sesana et al., 2008) with possibly a few as point particles. The inspiral dynamics and waveforms discrete sources (Sesana et al., 2009a). can be calculated using post-Newtonian (PN) equations, which result from a systematic expansion of the full Ein- 2 2 2 stein equations in powers of ǫ v /c GM/Rc , where III. HISTORICAL OVERVIEW R is the binary separation (Blanchet,∼ 2006).∼ The inspiral phase produces gravitational waves in the characteristic The quest to calculate the gravitational-wave signals form of a chirp, which is a sinusoid with both frequency from the merger of two black holes spans more than four and amplitude increasing with time. decades and encompasses key developments in theoreti- As the black holes spiral inward, they eventually reach cal and experimental general relativity, astrophysics, and the strong-field, dynamical regime of general relativity. computational science. In this section, we begin by de- In this merger stage, the orbital evolution is no longer lineating these threads in general terms, and then turn quasi-adiabatic; rather, the black holes plunge together to a more detailed account of select milestones along the and coalesce into a single, highly distorted remnant black path toward successful simulations of black-hole mergers. hole, surrounded by a common horizon. Since the point- particle and PN approximations break down, numerical relativity simulations of the Einstein equations in three A. Setting the Stage dimensions are needed to calculate the merger. Due to the difficulty of these simulations, the resulting gravi- At the end of the 18th century Michell (1784) and tational waveforms were completely unknown until re- Laplace (1796) first speculated, using Newtonian grav- cently. ity, that a star could become so compact that the es- Finally, the highly distorted remnant black hole settles cape velocity from its surface would exceed the speed of down into a quiescent rotating Kerr black hole by shed- light. In the 20th century, scientists realized that such ding its nonaxisymmetric modes through gravitational- black holes could form as the final state of total gravita- wave emission. We call this process the “ringdown,” tional collapse in general relativity (Harrison et al., 1965; in analogy to how a bell that has been struck sheds its Oppenheimer and Snyder, 1939); John Wheeler would distortions as sound waves. Various analytic techniques later popularize the term “black hole” to describe such an of black-hole perturbation theory (Regge and Wheeler, object (Misner et al., 2009; Ruffini and Wheeler, 1971). 6
Beginning in the 1960s, many highly energetic astrophys- tion variables.” Quantities based on the extrinsic curva- ical phenomena were discovered with physical properties ture Kij , which is roughly the time derivative of γij , play pointing to extremely strong gravitational fields as un- the role of “conjugate momenta.” Variation of an action derlying mechanisms; among these are quasars and X-ray with respect to γij produces a set of six first-order evo- binaries such as Cygnus X-1 (Overbeck and Tananbaum, lution equations for the conjugate momenta; varying the 1968; Overbeck et al., 1967), the first credible black-hole momenta gives six first-order evolution equations for γij . candidate. As discussed in Sec. II.B, today astrophysical ADM also introduced four Lagrange multipliers as freely black holes are believed to exist on a vast range of scales specifiable gauge or coordinate conditions, representing throughout the Universe, and black-hole binaries are con- the four coordinate degrees of freedom in general relativ- sidered to be strong sources of gravitational waves. ity. Variation of these Lagrange multipliers yields four Einstein’s equations of general relativity form a cou- equations that must hold on each slice: a Hamiltonian pled set of nonlinear partial differential equations, in constraint, and three momentum constraints. which dynamic curved spacetime takes the role of New- Originally intended as a tool for quantizing gravity, the ton’s gravitational field and interacts nonlinearly with ADM formalism later became the basis for most work in massive bodies. Gravitational waves were first recognized numerical relativity. As we discuss in Sec. IV below, as solutions to the linearized, weak-field Einstein equa- key elements in this approach are solving the Cauchy tions early in the past century. By mid-century, gravi- problem, beginning with the initial data on a spacelike tational waves were recognized as real physical phenom- slice, and then evolving that data forward in time. The ena, carrying energy and being capable of producing a constraint equations form the basis for this initial value response when impinging on a detector. This develop- problem. Appropriate choices for the gauge conditions ment spawned a major branch of experimental general are crucial ingredients for today’s successful black-hole relativity, with concepts for the first gravitational-wave merger simulations. detectors appearing in the 1960s; see Camp and Cornish (2004) for a review. The discovery of two neutron stars in a binary system by Hulse and Taylor (1975) provided B. Numerical Relativity Milestones an astrophysical laboratory for the first indirect detec- tion of gravitational radiation. Decades of observa- Hahn and Lindquist (1964) made the first known at- tion revealed the binary orbit to be shrinking by pre- tempt to simulate the head-on collision of two equal- cisely the amount expected if the system were emit- mass black holes on a computer in 1964, using a two- ting gravitational waves according to general relativ- dimensional axisymmetric approach. Their simulation ity (Weisberg and Taylor, 2005); Hulse and Taylor were ran for 50 timesteps to a duration of 1.8M; at awarded the Nobel Prize in 1993. Today, the progno- this point, they decided that the simulation∼ was no sis for direct detection of gravitational waves is excellent, longer accurate enough to warrant continued evolution, with the first events expected from the advanced ground- and stopped the code. Smarr and Eppley (Eppley, based interferometers around the middle of this coming 1975; Smarr, 1975, 1977; Smarr et al., 1976) returned decade. to this problem in the mid-1970s, again employing two- As you might expect, Einstein’s equations pose dimensional axisymmetry but now using the ADM for- formidable obstacles to anyone who would dare to probe malism, specialized coordinates, and improved coordi- the physics within. Throughout most of the 20th cen- nate conditions. Although they encountered problems tury, relativists uncovered a fairly small number of exact with instabilities and large numerical errors, they man- solutions by exploiting symmetries, and made progress aged to evolve the collision and extract information about toward more general problems using various perturba- the emitted gravitational waves. Smarr and Eppley had tive expansions. By the 1960s, computers had become used the most powerful computers of their day. Going to powerful enough to encourage attempts at solving Ein- the next step, orbiting black holes in three dimensions, stein’s equations numerically, to uncover physics beyond was deemed to be not feasible at the time, due to un- the realm of perturbation theory. The subsequent devel- resolved questions about the instabilities and insufficient opment of numerical relativity has been made possible computer power. Consequently, the black-hole merger in part by continued increases in computer power and problem lay dormant for over a decade. advances in algorithms and computational methods. In the 1990s, attention returned to black-hole mergers Most numerical-relativity simulations start with the as the LIGO project began to move forward, and black- idea of decomposing four-dimensional spacetime into hole mergers were recognized as the strongest sources a stack of curved three-dimensional spacelike slices for this detector. Since the signal-to-noise ratio for threaded by a congruence of timelike curves (York Jr., such ground-based detectors is fairly modest, having a 1979). Arnowit, Deser, and Misner (ADM) pioneered this template for the merger waveform is a key part of the “3+1” approach as the basis for a canonical formulation data analysis strategy. Numerical relativists revisited the of the dynamics of general relativity (Arnowitt et al., problem of colliding two black holes head-on with modern 1962). In this Hamiltonian formulation, the three-metric techniques and more powerful computers (Anninos et al., γij on the spatial slices takes the role of the “configura- 1993; Bernstein et al., 1994). In the mid-1990s, the 7
National Science Foundation funded a Computational coordinate conditions that keep the slices from Grand Challenge grant for a large multi-institution col- • crashing into singularities and the spatial coordi- laboration aimed at evolving black-hole mergers in three nates from falling into the black holes as the evolu- dimensions and calculating the resulting gravitational tion proceeds (Alcubierre et al., 2003; Bona et al., waveforms. Around the same time, a large and very ac- 1995); tive numerical relativity group arose at the newly-formed the Cactus Computational Toolkit1, which pro- Albert-Einstein Institut (AEI) in Potsdam, Germany. • During the late 1990s and into the early 2000s, two de- vided a framework for developing numerical- velopments on the experimental side further increased relativity codes and analysis tools used by many the desire for black-hole merger simulations: the ground- groups; based gravitational-wave detectors started taking data, modern adaptive mesh refinement finite-difference and interest grew in LISA and its potential for observ- • (including Carpet2 with Cactus, BAM (Br¨ugmann, ing gravitational waves from massive black-hole binary 1999), and Paramesh (MacNeice et al., 2000)), and mergers. multi-domain spectral (Pfeiffer et al., 2003) infras- While no one expected the task at hand – develop- tructures for numerical relativity. ing computer codes to solve the full Einstein equations in three dimensions for the final few orbits and merger Throughout this period, the length of black-hole evo- of two black holes – to be simple, numerical relativists lutions gradually increased. Improvements in the for- found that the problem was far more difficult than antic- malisms allowed simulations of single black holes and, ipated. Producing a waveform useful for gravitational- later, two black holes to increase in duration to > 10M. wave detection purposes typically would require running The addition of new slicing and shift conditions∼ again a simulation for a duration of several hundred M. How- increased the evolution times to > 30M. The Lazarus ever, a variety of instabilities plagued the codes, causing project took a novel approach to∼ combine these rela- them to crash well before any significant portion of an tively short-duration binary simulations with perturba- orbit could be achieved. tion techniques for the late-time behavior to produce Nevertheless, during a period encompassing a little a hybrid model for a black-hole merger (Baker et al., over a decade, much important work was accomplished 2000, 2001, 2002a,b). By late 2003, Br¨ugmann et al. that laid the foundations for later success. Key mile- (2004) carried out the first complete orbit of two equal- stones include these developments: mass, nonspinning black holes. While this simulation lasted 100M, the code crashed shortly after the or- ∼ initial data for binary black holes near the bit was completed and the gravitational waves were not • ISCO (Baumgarte, 2000; Cook, 1994, 2002; extracted. Overall, progress was slow, difficult, and in- Cook and Pfeiffer, 2004); cremental. However, the situation was about to change dramatically. new methods for representing black holes • on computational grids such as punctures C. Breakthroughs and the Gold Rush (Brandt and Br¨ugmann, 1997) and excision (Alcubierre and Br¨ugmann, 2001; Anninos et al., In early 2005, Frans Pretorius electrified the relativ- 1995b; Seidel and Suen, 1992; Shoemaker et al., ity community when he achieved the first evolution of 2003); an equal-mass black-hole binary through its final or- bit, merger, and ringdown using techniques very differ- recognition of the importance of hyperbol- • ent from those employed by the rest of the community icity in formulating the Einstein equations (Pretorius, 2005a). Later in 2005 the groups at the Uni- for numerical solution (Abrahams et al., 1997; versity of Texas at Brownsville (UTB) and NASA’s God- Anderson et al., 1997; Bona and Mass´o, 1992; dard Space Flight Center independently and simultane- Friedrich and Rendall, 2000); ously discovered a new method, called “moving punc- tures,” that also produced successful black-hole merg- improved formulations of the Einstein equations • ers (Baker et al., 2006b; Campanelli et al., 2006a). Their (Baumgarte and Shapiro, 1998; Nakamura et al., presentations were given back-to-back at a workshop on 1987; Shibata and Nakamura, 1995); numerical relativity, to the amazement of each other and the assembled participants. fully three-dimensional evolution codes and their • Since the moving-puncture approach was based on un- use in evolving distorted black holes (Brandt et al., derlying techniques used by several other groups, it was 2003; Camarda and Seidel, 1999), boosted black holes (Cook et al., 1998), head-on collisions (Sperhake et al., 2005), and grazing collisions (Alcubierre et al., 2001a; Brandt et al., 2000; 1 http://www.cactuscode.org/ Br¨ugmann, 1999); 2 http://www.carpetcode.org/ 8 rapidly and readily adopted by most in the community, The metric characterizes the geometry of spacetime by producing a growing avalanche of results. 2006 was a year giving the infinitesimal spacetime interval ds through the of many firsts: the first simulations of unequal-mass black following definition: holes and the accompanying recoil of the remnant hole 2 ν (Baker et al., 2006c), the first mergers of spinning black ds = g ν dx dx , (7) holes (Campanelli et al., 2006d), the first long waveforms where we use the Einstein summation convention which ( 7 orbits) (Baker et al., 2007c), the first comparisons with∼ PN results (Baker et al., 2007d; Buonanno et al., implies summation over all values of a given index that appears twice in an expression. Many physical impli- 2007a), and the first systematic parameter study in nu- merical relativity (Gonz´alez et al., 2007b). The year cations of the metric are immediately apparent from Eq. (7). For example, when ds2 = 0, the resulting metric- 2007 opened with the discovery of “superkicks” – re- coil velocities exceeding 1000 km s−1 (Campanelli et al., determined relationship between the time and space co- ordinates yields the paths that light rays must follow in 2007b; Gonz´alez et al., 2007a), and in 2008 a black-hole binary merger with mass ratio q = 10 was accomplished this spacetime. The dependence of Einstein’s tensor on the metric (Gonz´alez et al., 2009), while Campanelli et al. (2009) carried out the first long-term evolution of generic spin- can be simply illustrated in coordinates known as “har- monic”; in vacuum Eq. (6) takes the form: ning and precessing black-hole binaries. As this article was being written in 2009 the state of the art contin- 2g ν t ν =0, (8) ues to advance, with the first simulations of black-hole − mergers using spectral numerical techniques (Chu et al., where 2 is the flat-space wave-operator and t ν repre- 2009; Scheel et al., 2009; Szil´agyi et al., 2009), and the sents all terms nonlinear in the metric. If t ν is inter- first steps towards modeling the flows of gas around the preted as an effective source term, this is a simple wave merging black holes (van Meter et al., 2010b). Applica- equation. The familiar form of this equation suggests tions of the merger results in areas such as comparisons that its Cauchy problem can be solved by specifying the with PN expressions for the waveforms, astrophysical metric and its first derivative on an initial, spatial sur- computations of black-hole merger rates, and the devel- face and then integrating in time, as for an ordinary wave opment of templates for gravitational-wave data analysis equation. have accompanied these technical developments in the Einstein’s equations admit various formulations and simulations. The study of black-hole mergers using nu- coordinate conditions, which should be tailored to the merical relativity is thriving indeed. problem at hand – in this case, numerical simulation of black-hole spacetimes. Regardless of these choices, current numerical practices universally involve an initial IV. NUMERICAL DEVELOPMENT three-dimensional slice of spacetime that is evolved for- ward in time. Here, we review the history and current In this section, we discuss the mathematical and nu- most common choices of initial data, black-hole repre- merical foundations underlying current black-hole merger sentations, formulation, coordinate conditions, and some simulations, highlighting the key issues involved in details of the numerics. The impetus, and most impor- achieving successful evolutions. For more detailed treat- tant outcome, of all these developments is the ability to ments, we direct the interested reader to Alcubierre generate gravitational waveforms from black-hole binary (2008) or (Gourgoulhon, 2007). sources over many cycles.
B. The Cauchy Problem A. Einstein’s Equations The Cauchy problem concerns solution of the field The central task of numerical relativity is solving Ein- equations given initial data specified on an initial (typi- stein’s field equations cally spatial) hypersurface. In practice, the Cauchy prob- lem is more conveniently investigated in a “3+1” formu- G ν =8πT ν , (6) lation, explicitly based on a foliation of the spacetime into three-dimensional spatial slices parametrized by a where Einstein’s tensor G represents the curvature of ν time coordinate. A common 3+1 formulation inspired spacetime, the energy-momentum tensor T ν contains by ADM (Arnowitt et al., 1962) divides up the compo- the matter sources, and ,ν = 0, 1, 2, 3. By conven- nents of the metric according to their relationships with tion, an index = 0 selects a “ time” component, and space and time, such that the line element takes the form: =1, 2, 3 selects a “space” component. G ν depends on 2 2 i 2 i i j the first and second derivatives of the metric tensor g ν . ds = ( α + β βi)dt +2βidtdx + γij dx dx , (9) For vacuum black-hole spacetimes, T ν = 0. Note that − all of the tensor fields discussed here are symmetric in where α is called the lapse function, βi the shift vector, the indices, e.g. g ν = gν . and γij = gij is the spatial three-metric. We write the 9 time coordinate x0 = t and the spatial coordinate indices i, j =1, 2, 3. Note that contraction with g ν or its inverse na ta g ν is used to lower or raise indices of four-dimensional ij a a a tensors, respectively, while γij or its inverse γ is used to x − β dt x lower or raise indices of three-dimensional tensors. The t = dt lapse and shift represent coordinate freedom in the met- ric; we can choose these quantities arbitrarily. However, αdt βa since the three-metric γ (and its first and second spa- ij xa tial derivatives) determines the intrinsic curvature of the t =0 slice, it carries the information about the gravitational field and thus is constrained by the physics. The meaning of the lapse and shift can be understood by considering two successive spatial slices separated by FIG. 2 The 3+1 split into space and time. Two spatial slices an infinitesimal time interval dt (Fig. 2). An observer at t = 0 and t = dt are depicted. α is the lapse, and αdt along a vector normal to the first slice will measure an represents the proper time lapse between slices. βa is the elapsed proper time of dτ = αdt in evolving to the second shift, and βadt represents the amount by which the spatial slice, and a change in spatial coordinate of dxi = βidt. coordinates shift between slices. na is normal to the slice at − a Since Einstein’s equations are second order in time, we t = 0. If a ray parallel to n intersects the t = 0 slice at a a a − a must also specify the initial time derivative of the three- point x , then it will intersect the t = dt slice at x β dt. a ≡ a a metric. Rather than specifying this derivative directly, t αn + β is a coordinate time vector. If a ray parallel to ta intersects the t = 0 slice at point xa, then it will also we define a new quantity: intersect the t = dt slice at xa. 1 Kij = (∂tγij Diβj Dj βi) , (10) −2α − − where (3)R is the three-dimensional Ricci scalar associ- ated with the three-metric γ , and K traceK = where ∂ = ∂/∂t is an ordinary partial derivative and D ij ij t i γij K , sometimes called the mean curvature≡ . These con- is the spatial covariant derivative. Note that the space- ij straint equations must be solved in order to obtain an time covariant derivative is a partial derivative with a initial spatial slice consistent with Einstein’s equations. correction such that it transforms∇ as a vector and satisfies λg ν = 0. Di is the projection of onto the spatial slice∇ and is equivalent to a three-dimensional∇ covariant C. Representing Black Holes in Numerical Spacetimes derivative formed from γij . For the case of Euclidean normal coordinates, α = 1 and βi = 0, Eq. (10) reduces How does one represent an exotic object such as a to the simple expression K = (1/2)∂ γ . ij − t ij black hole in a numerical simulation? In particular, how If we define a unit vector n normal to the spatial slice, can one use finite computational methods repeatedly to we can show that Eq. (10) is equivalent to K = D n . ij − i j model an object which, analytically, contains physical As suggested by this expression, Kij is a measure of the and/or coordinate singularities? Fortunately, two suc- change of the normal vector as it is transported along the cessful strategies have emerged to meet this challenge. slice. In this way K gives an extrinsic measure of the ij The unusual topology of black holes offers one way out. curvature of a three-dimensional spatial slice with respect As Einstein and Rosen (1935) originally showed, a black to its embedding in four-dimensional spacetime. It is hole can be considered a “bridge” or “wormhole” con- therefore called the extrinsic curvature. Depending on necting one Universe or “worldsheet”, to a second world- the formulation, the extrinsic curvature might or might sheet (see Fig. 3). Exploiting this topology, a continuous not come into the evolution equations; however, K is ij spatial slice which avoids the physical singularity con- almost universally utilized when calculating initial data. tained within the event horizon of each black hole can Of the ten component equations of Eq. (6), six deter- be constructed as follows. Starting with a spatial slice mine the time-evolution of the metric, while four must of Schwarzschild spacetime, remove the interior of the be satisfied on a spatial slice at any given time, and are event horizon. Identify the resulting spherical bound- thus constraint equations. With an appropriate choice of ary with the spherical boundary of an identical copy of time coordinate, and assuming vacuum spacetime, these this space. The two-dimensional analog would be to take four constraint equations are equivalent to the condition two sheets of paper, cut out a disk from each, and then G0ν = 0. The time-time component G00 = 0 is called the glue the resulting circular edges together. Each sheet, Hamiltonian constraint, and the time-space components or copy of Schwarzschild spacetime in this example, is G0i = 0 are called the momentum constraint. These take called a worldsheet. The identified spherical boundaries the form of conditions on the extrinsic curvature: connecting the worldsheets form what is referred to as the “throat” of the wormhole. (3)R + K2 K Kij = 0 (11) − ij To complete this construction, we require an appropri- D (Kij γij K)=0 (12) ate coordinate system to continuously cover both world- j − 10 sheets. Brill and Lindquist discovered coordinates that will prove convenient, in which the three-metric of a par- ticular spatial slice of Schwarzschild spacetime is given by (Brill and Lindquist, 1963) r′ → ∞ m 4 γ = 1+ δ , (13) ij 2r ij where r is a radial coordinate. In these coordinates, the ′ event horizon is at r = m/2. We can consider each of throat r = rh the worldsheets described above as being separately la- beled by such coordinates. Designate one worldsheet as A and the other as B, and call their radial coordinates rA and rB respectively. Since the interior of the event hori- zon has been removed from each worldsheet, assume that ′ rA m/2 and rB m/2. The metric on each worldsheet r → 0 has≥ the form of Eq.≥ (13). As noted by Brill and Lindquist, this form of the metric is unchanged by the transforma- tion r′ = m2/4r, and r′ = r when r = m/2. So if we define a new coordinate r′ by FIG. 3 In the wormhole representation of a black hole, the initial slice typically just touches the horizon. The upper m m2 m “sheet” represents the exterior space, while the lower sheet is r = r′ for r′ , r = for r′ , (14) a duplicate, joined to the upper sheet by a “throat”. A ≥ 2 B 2r′ ≤ 2 mapping spatial infinity on worldsheet B to r′ = 0, we obtain a single continuous coordinate system that applies be used by the Caltech-Cornell dual-coordinate spectral to worldsheet A for r′ m/2 and worldsheet B for r′ code (Scheel et al., 2009, 2006). ≥ ≤ m/2. Either of these methods can be useful in representing This avoidance of the physical singularity comes at black holes on the initial data slice. Surprisingly, we will the expense of a coordinate singularity in the metric also see that either of these representations can be made at r′ = 0, called a “puncture”. However, it turns out robust enough to persist as the black holes evolve. this coordinate singularity can be confined to a single scalar variable, as suggested in Eq. (13). With a suitable change of variables and other means, numerical simu- D. Initial Data lations haven proven capable of handling this irregular scalar field. Thus the “puncture” method is one way to The starting point for successful simulations of black- represent a black hole that is amenable to computation. hole mergers is finding initial data for astrophysically re- Another strategy is excision, first proposed by Unruh alistic inspiralling black holes. If we were simulating the (Thornburg, 1993). Given that no physical information orbits of stars in Newtonian gravity, this would be a sim- can escape an event horizon to influence the exterior, ple procedure. For example, we could simply specify the the interior of a black hole can in principle be excised masses and spins of the stars, along with their positions from the computational domain3. This relies on the fact and velocities on orbits derived from the dynamics of that all physical information propagates inwards from the point particles, and then evolve the system numerically event horizon towards the physical singularity, i.e. light- to allow it to “relax” into orbits appropriate for bodies of cones tilt inwards, and nothing physical propagates out- finite size. In general relativity, however, the initial data ward from the horizon. Extrapolation can be used for the must satisfy the constraint equations (11) - (12), which boundary condition of the excised region, and any non- are cast in terms of the 3-metric γ and the extrinsic physical numerical error that escapes the horizon should ij curvature K . Since there is no obvious, or natural, be negligible. This approach is not constrained to the ij connection between these field variables and the astro- particular coordinates required by the puncture method, physical properties of inspiralling black holes, obtaining but it can be more difficult due to the need for precise suitable initial data is a major challenge. positioning of the excision boundary and accurate ex- trapolation. Excision was used successfully for orbiting Building on the earlier work of Lichnerowicz (1944), binary simulations by Pretorius (2005a) and continues to York developed a general procedure for solving the con- straint equations to produce initial data for the Cauchy problem in the 1970s (see York Jr. (1979) for a review). This approach generally requires solving a coupled el- 3 As unphysical coordinate “gauge modes” may couple to physical liptic system of four nonlinear field equations. We can modes, we also assume no superluminal coordinate effects are break this problem down into more manageable pieces present. with some simplifying assumptions. While these choices 11 do come at the cost of some loss of generality and astro- enforce the constraints and optionally the choice of slic- physical realism in the initial data for two black holes, ing condition. Boundary conditions may additionally be it has been seen that for sufficiently long evolutions the supplied to enforce rotational motion. final orbits and waveform signatures from the black-hole Either approach provides an ansatz for constructing evolution are largely insensitive to this level of detail in three-dimensional binary black-hole initial field data for the initial data, at least in the case of equal-mass non- a specified choice of particle-like parameters, masses, po- spinning black holes. sitions, momenta, and spins. Inevitably there are differ- The first simplification is to choose traceless extrinsic ences from the nearly quiescent evolving systems that we curvature, K = 0. With this, the Hamiltonian constraint seek to represent. Generally there is some level of spu- is decoupled from the momentum constraints, and can rious radiation generated from a period of initial tran- be solved separately. To find solutions corresponding to sient dynamics through which the system relaxes to be- multiple black holes, we generally further assume that come quiescent on sub-orbital time scales. In particu- the initial slice is conformally flat. That is, the three- lar, extraneous radiation content is an unavoidable conse- metric is the product of a scalar conformal factor with a quence of conformal flatness, which post-Newtonian anal- flat metric, ysis has shown must deviate from the physically relevant inspiralling binary solution (Damour et al., 2000). This 4 γij = ψ δij . (15) spurious radiation is seen in plots of the gravitational waveforms produced by mergers; see Fig. 9 and other With this, the problem reduces to solving a (typically waveform plots in Sec. V.B.2. Often the simulations will nonlinear) equation for the scalar field ψ. also undergo a period of initial gauge-evolution which, Brill and Lindquist (1963) found a simple solution, though physically inconsequential, may affect the quality representing N black holes momentarily at rest, which of the simulation numerically. For puncture initial data gives Hannam et al. (2009b) analyzed some of these gauge dy- namics, and developed a promising approach (“trumpet” N m data) to mitigate it (see Sec. IV.F.2 for related gauge is- ψ =1+ i (16) 2r sues). However, for simulations lasting for several orbits i i these modest transient effects are generally negligible. Most black-hole binary simulation studies are designed where mi is the mass associated with the ith black hole to represent the astrophysical population of systems, and ri represents its coordinate center. Each ri corre- sponds to the location of a puncture, as described in the which have circularized before the gravitational radi- last section. This is a valuable solution, but not gen- ation becomes observationally significant. These sim- erally useful, because these black holes lack momentum ulations begin with circularly inspiralling initial data and spin. configurations. Even before reliable numerical relativ- An explicit solution by Bowen for the momentum con- ity simulations were possible, considerable attention had straint was the last crucial step in defining a procedure for been given to prescribing initial data for these near- calculating initial data for multiple black holes with spec- circular configurations (see Cook (2000) for a review). ified linear and angular momenta (Bowen and York Jr., Within the CTS approach, it is particularly natural to 1980). The Bowen-York prescription employed a two- impose initially circular motion upon the system, result- sheeted topology found by Misner for the black-hole inte- ing in an initial data prescription for quasicircular or- riors (Misner, 1963). Later Brandt and Br¨ugmann gen- bits (Caudill et al., 2006; Cook, 2002; Cook and Pfeiffer, eralized the procedure for the Brill-Lindquist topology 2004). For either the CTS or Brandt-Br¨ugmann data, (Brandt and Br¨ugmann, 1997). The Brandt-Br¨ugmann quasicircular parameters may be chosen by constraints puncture data for arbitrary momenta and spin is now on either an effective gravitational potential or total widely used because of its ease of implementation. energy of the system (Caudill et al., 2006; Cook, 1994; More recently York developed another modeling Gourgoulhon et al., 2002). For more realistic inspi- ansatz, known as the Conformal-Thin-Sandwich (CTS) ralling trajectories, the PN approximation may be used approach (Pfeiffer and York Jr., 2005; York Jr., 1999), (Husa et al., 2008b), or, for higher accuracy, an it- which has certain additional advantages. For example, erative procedure involving short numerical evolutions it turns out the conventional Brandt-Br¨ugmann punc- (Pfeiffer et al., 2007). ture cannot yield a spin parameter (a/M) greater than 0.93 (Dain et al., 2002), while CTS data can go higher (Lovelace∼ et al., 2008). As discussed above, the spatial E. Numerically Friendly Formulations of the Evolution metric is chosen to be conformally flat, then instead of Equations providing an ansatz for the extrinsic curvature Kij , the initial time-derivative of the conformal metric is speci- The Einstein evolution equations, which determine the fied (generally to vanish). In addition, a condition can time-development of the initial data, form a set of at least be imposed on the slicing (see below). The result is a six coupled, nonlinear propagation equations. The exact coupled system of elliptic equations which is solved to formulation of these equations depends on the choice of 12