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Black-Hole Binaries, Gravitational Waves, and Numerical Relativity

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Black-Hole Binaries, Gravitational Waves, and Numerical Relativity Black-hole binaries, gravitational waves, and numerical relativity Joan Centrella∗ and John G. Baker† Gravitational Astrophysics Laboratory, NASA/GSFC, 8800 Greenbelt Road, Greenbelt, Maryland 20771, USA Bernard J. Kelly‡ and James R. van Meter§ CRESST and Gravitational Astrophysics Laboratory, NASA/GSFC, 8800 Greenbelt Road, Greenbelt, Maryland 20771, USA and Department of Physics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, Maryland 21250, USA (Dated: November 30, 2010) Understanding the predictions of general relativity for the dynamical interactions of two black holes has been a long-standing unsolved problem in theoretical physics. Black-hole mergers are monu- mental astrophysical events, releasing tremendous amounts of energy in the form of gravitational radiation, and are key sources for both ground- and space-based gravitational-wave detectors. The black-hole merger dynamics and the resulting gravitational waveforms can only be calculated through numerical simulations of Einstein’s equations of general relativity. For many years, nu- merical relativists attempting to model these mergers encountered a host of problems, causing their codes to crash after just a fraction of a binary orbit could be simulated. Recently, however, a series of dramatic advances in numerical relativity has allowed stable, robust black-hole merger simulations. This remarkable progress in the rapidly maturing field of numerical relativity, and the new understanding of black-hole binary dynamics that is emerging is chronicled. Important applications of these fundamental physics results to astrophysics, to gravitational-wave astronomy, and in other areas are also discussed. Contents G. Numerical Approximation Methods 15 H. Extracting the Physics 15 I. Prelude 2 V. Black-Hole Merger Dynamics and Waveforms 16 II. Black-Hole Binaries and Gravitational Waves 2 A. First Glimpses of the Merger: The Lazarus Approach16 A. Basic Properties 3 B. Mergers of Equal-Mass, Nonspinning Black Holes 17 1. Black-Hole Basics 3 1. The First Merger Waveforms 17 2. Gravitational Wave Primer 3 2. Universal Waveform 18 3. Longer Waveforms 20 B. Astrophysical Black Holes 4 C. Mergers of Unequal-Mass, Nonspinning Black Holes 22 C. Gravitational Waves from Black-Hole Binaries 5 1. Mode Analysis and Gravitational Waveforms 22 III. Historical Overview 5 2. A Qualitatively New Feature: Kicks 23 D. Mergers of Spinning Black Holes 24 A. Setting the Stage 5 1. Gravitational Waveforms 25 B. Numerical Relativity Milestones 6 2. Spinning Binary Mergers and Spin Flips 25 C. Breakthroughs and the Gold Rush 7 3. Kicks from Mergers of Spinning Black Holes 26 IV. Numerical Development 8 VI. Interaction of Numerical Relativity with A. Einstein’s Equations 8 Post-Newtonian Theory 27 B. The Cauchy Problem 8 arXiv:1010.5260v2 [gr-qc] 27 Nov 2010 A. Independent Post-Newtonian Dynamics and C. Representing Black Holes in Numerical Spacetimes 9 Waveforms 28 D. Initial Data 10 B. Analytic Full-Waveform Models 30 E. Numerically Friendly Formulations of the Evolution C. Post-Newtonian Models for Numerical Initial Data. 32 Equations 11 D. Post-Newtonian Theory for Interpretation of 1. Hyperbolicity and Well-Posedness 12 Numerical Results. 32 2. Harmonic Formulations 12 3. ADM-based Formulations 12 VII. Applications to Gravitational Wave Data F. Gauge Conditions 13 Analysis 33 1. Choosing the Slicing and Shift 13 A. The Direct Impact of Merger Waveforms in Data 2. Moving Punctures 13 Analysis 35 3. Generalized Harmonic Coordinates 14 B. Developing Analytic Inspiral-Merger-Ringdown 4. Other Coordinate Techniques 14 Gravitational Waveform Templates 37 C. Using Numerical Waveforms in Data Analysis Applications 38 VIII. Impact on Astrophysics 39 ∗Electronic address: [email protected] A. Recoiling Black Holes 39 †Electronic address: [email protected] 1. Predicting the Recoil 39 ‡Electronic address: [email protected] 2. Consequences of Black Hole Recoil 40 §Electronic address: [email protected] B. The Spin of the Final Black Hole 40 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.
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