The numerical relativity breakthrough for binary black holes

The evolution of black-hole (BH) binaries in vacuum spacetimes constitutes the two-body problem in general relativity. The solution of this problem in the framework of the Einstein field equations is a substantially more complex exercise than that of the dynamics of two point masses in Newtonian gravity, but it also presents us with a wealth of new exciting physics. Numerical methods are likely to be the only way to compute the dynamics of BH systems in the fully nonlinear regime and have been pursued since the 1960s, culminating in dramatic breakthroughs in 2005. Here we review the methodology and the developments that finally gave us a solution of this fundamental problem of Einstein's theory and discuss the breakthroughs' implications for the wide range of contemporary BH physics.

[1]  E. Seidel,et al.  Numerical evolution of matter in dynamical axisymmetric black hole spacetimes: I. Methods and tests , 1998, gr-qc/9807017.

[2]  B. Krishnan,et al.  Isolated and Dynamical Horizons and Their Applications , 2004, Living reviews in relativity.

[3]  J. D. Brown,et al.  Trumpet slices of the Schwarzschild-Tangherlini spacetime , 2010, 1010.5723.

[4]  L. Lehner.  Gravitational Radiation from Black Hole Spacetimes , 1998 .

[5]  L. Smarr.  Gauge conditions, radiation formulae and the two black hole collision , 1979 .

[6]  S. Shapiro,et al.  On the numerical integration of Einstein's field equations , 1998, gr-qc/9810065.

[7]  Luciano Rezzolla,et al.  Gauge-invariant non-spherical metric perturbations of Schwarzschild black-hole spacetimes , 2005 .

[8]  Y. Zlochower,et al.  Hangup kicks: still larger recoils by partial spin-orbit alignment of black-hole binaries. , 2011, Physical review letters.

[9]  G. Lovelace,et al.  Momentum flow in black-hole binaries. II. Numerical simulations of equal-mass, head-on mergers with antiparallel spins , 2009, 0907.0869.

[10]  O. Sarbach,et al.  Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations , 2012, Living Reviews in Relativity.

[11]  Helvi Witek,et al.  Initial value formulation of dynamical Chern-Simons gravity , 2014, 1407.6727.

[12]  A multi-domain spectral method for initial data of arbitrary binaries in general relativity , 2006, gr-qc/0612081.

[13]  V. Cardoso,et al.  Exploring New Physics Frontiers Through Numerical Relativity , 2014, Living reviews in relativity.

[14]  Miguel Alcubierre,et al.  Introduction to 3+1 Numerical Relativity , 2008 .

[15]  H. Kreiss,et al.  Testing the well-posedness of characteristic evolution of scalar waves , 2013, 1305.7179.

[16]  Y. Zlochower,et al.  Accurate evolutions of orbiting black-hole binaries without excision. , 2006, Physical review letters.

[17]  山崎 昌男.  M.E.Taylor:Pseudodifferential Operators and Nonlinear PDE,Progress in Mathematics,vol.100 , 1998 .

[18]  John Archibald Wheeler,et al.  Stability of a Schwarzschild singularity , 1957 .

[19]  G. Horowitz,et al.  Further evidence for lattice-induced scaling , 2012, 1209.1098.

[20]  H. Kreiss,et al.  Constraint-preserving Sommerfeld conditions for the harmonic Einstein equations , 2006, gr-qc/0612051.

[21]  B. Bruegmann,et al.  Eccentric black hole mergers and zoom-whirl behavior from elliptic inspirals to hyperbolic encounters , 2012, 1209.4085.

[22]  I. Hinder.  The current status of binary black hole simulations in numerical relativity , 2010, 1001.5161.

[23]  Masaru Shibata,et al.  High-velocity collision of two black holes , 2008, 0810.4735.

[24]  E. Berti,et al.  Universality, maximum radiation, and absorption in high-energy collisions of black holes with spin. , 2012, Physical review letters.

[25]  Frank Herrmann,et al.  Circularization and final spin in eccentric binary-black-hole inspirals , 2007, 0710.5167.

[26]  B. Zupnik.  Reality in noncommutative gravity , 2005, hep-th/0512231.

[27]  B. Szilágyi,et al.  Unambiguous determination of gravitational waveforms from binary black hole mergers. , 2009, Physical review letters.

[28]  Frank Ohme,et al.  Twist and shout: A simple model of complete precessing black-hole-binary gravitational waveforms , 2013, 1308.3271.

[29]  P. Chesler,et al.  Horizon formation and far-from-equilibrium isotropization in a supersymmetric Yang-Mills plasma. , 2008, Physical review letters.

[30]  Improved numerical stability of stationary black hole evolution calculations , 2002, gr-qc/0209066.

[31]  L. Randall,et al.  A Large mass hierarchy from a small extra dimension , 1999, hep-ph/9905221.

[32]  Pedro Marronetti,et al.  Simple method to set up low eccentricity initial data for moving puncture simulations , 2010, 1010.2936.

[33]  J. Winicour.  Characteristic Evolution and Matching , 2001, Living reviews in relativity.

[34]  Harald P. Pfeiffer,et al.  Solving Einstein's equations with dual coordinate frames , 2006, gr-qc/0607056.

[35]  Dae-Il Choi,et al.  How to move a black hole without excision: Gauge conditions for the numerical evolution of a moving puncture , 2006 .

[36]  Frank Herrmann,et al.  Comparisons of eccentric binary black hole simulations with post-Newtonian models , 2008, 0806.1037.

[37]  T. Piran,et al.  The Initial Value Problem and Beyond , 1982 .

[38]  L. Lehner,et al.  Dealing with delicate issues in waveform calculations , 2007, 0706.1319.

[39]  D. Garrison.  Numerical Relativity as a Tool for Studying the Early Universe , 2012, 1207.7097.

[40]  J. York,et al.  Time-asymmetric initial data for black holes and black-hole collisions , 1980 .

[41]  Wolfgang Tichy,et al.  Numerical simulation of orbiting black holes. , 2004, Physical review letters.

[42]  M. Ansorg,et al.  Eccentric binary black-hole mergers: The transition from inspiral to plunge in general relativity , 2007, 0710.3823.

[43]  Seidel,et al.  Three-dimensional numerical relativity: The evolution of black holes. , 1995, Physical review. D, Particles and fields.

[44]  C. Palenzuela,et al.  Dual Jets from Binary Black Holes , 2010, Science.

[45]  Edward Seidel,et al.  Numerical evolution of dynamic 3D black holes: Extracting waves. , 1998 .

[46]  Initial Data for Numerical Relativity , 2000, Living reviews in relativity.

[47]  S. Gubser,et al.  Simulation of asymptotically AdS5 spacetimes with a generalized harmonic evolution scheme , 2011, 1201.2132.

[48]  W. Kinnersley.  TYPE D VACUUM METRICS. , 1969 .

[49]  D. Kennefick,et al.  Gravitational radiation reaction for bound motion around a Schwarzschild black hole. , 1994, Physical review. D, Particles and fields.

[50]  Spinning-black-hole binaries: The orbital hang-up , 2006, gr-qc/0604012.

[51]  Thibault Damour,et al.  Transition from inspiral to plunge in binary black hole coalescences , 2000 .

[52]  Mark Hannam,et al.  Status of black-hole-binary simulations for gravitational-wave detection , 2009, 0901.2931.

[53]  R. Janik,et al.  Characteristics of thermalization of boost-invariant plasma from holography. , 2011, Physical review letters.

[54]  Hyperbolic slicings of spacetime: singularity avoidance and gauge shocks , 2002, gr-qc/0210050.

[55]  Simulating merging binary black holes with nearly extremal spins , 2010, 1010.2777.

[56]  Michael Boyle,et al.  Error-analysis and comparison to analytical models of numerical waveforms produced by the NRAR Collaboration , 2013, 1307.5307.

[57]  F. Zerilli.  Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics , 1969 .

[58]  Dean G. Blevins,et al.  Introduction 3-1 , 1969 .

[59]  Oscar A. Reula.  Hyperbolic Methods for Einstein’s Equations , 1998, Living reviews in relativity.

[60]  Cook,et al.  Apparent horizons for boosted or spinning black holes. , 1990, Physical review. D, Particles and fields.

[61]  Frans Pretorius,et al.  High-energy collision of two black holes. , 2008, Physical review letters.

[62]  G. Lovelace,et al.  Binary-black-hole initial data with nearly extremal spins , 2008, 0805.4192.

[63]  Frans Pretorius,et al.  Binary Black Hole Coalescence , 2007, 0710.1338.

[64]  Seidel,et al.  Collision of two black holes. , 1993, Physical review letters.

[65]  A. Buonanno,et al.  Inspiral, merger and ring-down of equal-mass black-hole binaries , 2006, gr-qc/0610122.

[66]  A. L. Tiec.  The Overlap of Numerical Relativity, Perturbation Theory and Post-Newtonian Theory in the Binary Black Hole Problem , 2012, 1408.5505.

[67]  Peter MacNeice,et al.  Paramesh: A Parallel Adaptive Mesh Refinement Community Toolkit , 2013 .

[68]  Zoom and whirl: Eccentric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction , 2002, gr-qc/0203086.

[69]  Hughes,et al.  Finding black holes in numerical spacetimes. , 1994, Physical review. D, Particles and fields.

[70]  V. Cardoso,et al.  Higher-dimensional puncture initial data , 2011, 1109.2149.

[71]  Ken-ichi Oohara,et al.  General Relativistic Collapse to Black Holes and Gravitational Waves from Black Holes , 1987 .

[72]  J. Maldacena.  The Large N limit of superconformal field theories and supergravity , 1998 .

[73]  N. W. Taylor,et al.  Periastron advance in spinning black hole binaries: Gravitational self-force from numerical relativity , 2013, 1309.0541.

[74]  B. Dewitt,et al.  Collision of two black holes: Theoretical framework , 1976 .

[75]  Dae-Il Choi,et al.  Gravitational-wave extraction from an inspiraling configuration of merging black holes. , 2005, Physical review letters.

[76]  Savas Dimopoulos,et al.  The Hierarchy problem and new dimensions at a millimeter , 1998, hep-ph/9803315.

[77]  C. Lousto,et al.  NR/HEP: roadmap for the future , 2012, 1201.5118.

[78]  K. Nakao,et al.  Black-hole universe: time evolution. , 2013, Physical review letters.

[79]  E. Berti,et al.  Numerical simulations of black-hole binaries and gravitational wave emission , 2011, 1107.2819.

[80]  Frans Pretorius,et al.  Simulation of binary black hole spacetimes with a harmonic evolution scheme , 2006, gr-qc/0602115.

[81]  Dae-Il Choi,et al.  Wave zone extraction of gravitational radiation in three-dimensional numerical relativity , 2005 .

[82]  E. Seidel,et al.  Towards a stable numerical evolution of strongly gravitating systems in general relativity: The conformal treatments , 2000, gr-qc/0003071.

[83]  Seidel,et al.  New formalism for numerical relativity. , 1995, Physical review letters.

[84]  P. C. Peters.  Gravitational Radiation and the Motion of Two Point Masses , 1964 .

[85]  Brandt,et al.  Evolution of distorted rotating black holes. I. Methods and tests. , 1995, Physical review. D, Particles and fields.

[86]  P. Anninos,et al.  Head-On Collision of Two Unequal Mass Black Holes , 1998, gr-qc/9806031.

[87]  The Numerical Evolution of the Collision of Two Black Holes. , 1975 .

[88]  R. Haas,et al.  TIDAL DISRUPTIONS OF WHITE DWARFS FROM ULTRA-CLOSE ENCOUNTERS WITH INTERMEDIATE-MASS SPINNING BLACK HOLES , 2012, 1201.4389.

[89]  Brügmann.  Adaptive mesh and geodesically sliced Schwarzschild spacetime in 3+1 dimensions. , 1996, Physical review. D, Particles and fields.

[90]  Evolution of a periodic eight-black-hole lattice in numerical relativity , 2012, 1204.3568.

[91]  José A. González,et al.  Exploring black hole superkicks , 2007, 0707.0135.

[92]  Duncan A. Brown,et al.  Nonspinning searches for spinning binaries in ground-based detector data: Amplitude and mismatch predictions in the constant precession cone approximation , 2012, 1203.6060.

[93]  Susan G. Hahn,et al.  The two-body problem in geometrodynamics , 1964 .

[94]  L. Rezzolla,et al.  ACCURATE SIMULATIONS OF BINARY BLACK HOLE MERGERS IN FORCE-FREE ELECTRODYNAMICS , 2012, The Astrophysical Journal.

[95]  Y. Zlochower,et al.  Large Merger Recoils and Spin Flips from Generic Black Hole Binaries , 2007, gr-qc/0701164.

[96]  José A. González,et al.  Maximum kick from nonspinning black-hole binary inspiral. , 2007, Physical review letters.

[97]  Lawrence E. Kidder,et al.  Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations , 2001, gr-qc/0105031.

[98]  José A González,et al.  Supermassive recoil velocities for binary black-hole mergers with antialigned spins. , 2007, Physical review letters.

[99]  Well-posedness of formulations of the Einstein equations with dynamical lapse and shift conditions , 2006, gr-qc/0604035.

[100]  D. Shoemaker,et al.  Unequal mass binary black hole plunges and gravitational recoil , 2006, gr-qc/0601026.

[101]  Larry Smarr,et al.  Kinematical conditions in the construction of spacetime , 1978 .

[102]  Measuring gravitational waves from binary black hole coalescences. I. Signal to noise for inspiral, merger, and ringdown , 1997, gr-qc/9701039.

[103]  Helmut Friedrich,et al.  On the hyperbolicity of Einstein's and other gauge field equations , 1985 .

[104]  Black holes and sub-millimeter dimensions , 1998, hep-th/9808138.

[105]  G. Horowitz,et al.  Optical conductivity with holographic lattices , 2012, 1204.0519.

[106]  S. Dimopoulos,et al.  THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NEW DIMENSIONS AT A MILLIMETER TO A FERMI AND SUPERSTRINGS AT A TeV , 2005 .

[107]  The structure of general relativity with a numerical illustration: The collision of two black holes , 1975 .

[108]  Vincent Moncrief,et al.  Gravitational perturbations of spherically symmetric systems. I. The exterior problem , 1974 .

[109]  D. Christodoulou.  Reversible and Irreversible Transformations in Black-Hole Physics , 1970 .

[110]  A new general purpose event horizon finder for 3D numerical spacetimes , 2003, gr-qc/0305039.

[111]  Mark A. Scheel,et al.  Simulations of Binary Black Hole Mergers Using Spectral Methods , 2009, 0909.3557.

[112]  R. W. Lindquist,et al.  INTERACTION ENERGY IN GEOMETROSTATICS , 1963 .

[113]  J. Thornburg.  Event and Apparent Horizon Finders for 3 + 1 Numerical Relativity , 2005, Living reviews in relativity.

[114]  I. Hinder,et al.  Constraint damping in the Z4 formulation and harmonic gauge , 2005, gr-qc/0504114.

[115]  C. Bona-Casas,et al.  Elements of Numerical Relativity and Relativistic Hydrodynamics , 2009 .

[116]  Frans Pretorius,et al.  Evolution of binary black-hole spacetimes. , 2005, Physical review letters.

[117]  M. Ruiz,et al.  Outer boundary conditions for Einstein's field equations in harmonic coordinates , 2007, 0707.2797.

[118]  Scott H. Hawley,et al.  Evolutions in 3D numerical relativity using fixed mesh refinement , 2003, gr-qc/0310042.

[119]  G. Landsberg,et al.  Black holes at the Large Hadron Collider. , 2001 .

[120]  M. Campanelli.  Understanding the fate of merging supermassive black holes , 2004, astro-ph/0411744.

[121]  V. Cardoso,et al.  Dynamics of black holes in de Sitter spacetimes , 2012, 1204.2019.

[122]  G. Lovelace,et al.  Geometrically motivated coordinate system for exploring spacetime dynamics in numerical-relativity simulations using a quasi-Kinnersley tetrad , 2012, 1208.0630.

[123]  José A. González,et al.  Inspiral, merger, and ringdown of unequal mass black hole binaries: A multipolar analysis , 2007, gr-qc/0703053.

[124]  M. Parashar,et al.  Boosted Three-Dimensional Black-Hole Evolutions with Singularity Excision , 1997, gr-qc/9711078.

[125]  F. Ohme,et al.  Towards models of gravitational waveforms from generic binaries: II. Modelling precession effects with a single effective precession parameter , 2014, 1408.1810.

[126]  Thibault Damour,et al.  Improved effective-one-body description of coalescing nonspinning black-hole binaries and its numerical-relativity completion , 2012, 1212.4357.

[127]  Binary Black Hole Mergers in 3d Numerical Relativity , 1997, gr-qc/9708035.

[128]  P. Bizoń,et al.  Weakly turbulent instability of anti-de Sitter spacetime. , 2011, Physical review letters.

[129]  C. Misner.  Wormhole Initial Conditions , 1960 .

[130]  D. Shoemaker,et al.  Late inspiral and merger of binary black holes in scalar–tensor theories of gravity , 2011, 1112.3928.

[131]  Lev Davidovich Landau,et al.  Classical theory of fields , 1952 .

[132]  Harald P. Pfeiffer,et al.  Measuring orbital eccentricity and periastron advance in quasicircular black hole simulations , 2010, 1004.4697.

[133]  M. Ruiz,et al.  Multipole expansions for energy and momenta carried by gravitational waves , 2008 .

[134]  Roger Penrose,et al.  An Approach to Gravitational Radiation by a Method of Spin Coefficients , 1962 .

[135]  Edward Seidel,et al.  Black Hole Excision for Dynamic Black Holes , 2001 .

[136]  Michael Boyle,et al.  High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions , 2007, 0710.0158.

[137]  L. Lehner,et al.  Black strings, low viscosity fluids, and violation of cosmic censorship. , 2010, Physical review letters.

[138]  M. Campanelli,et al.  Accurate black hole evolutions by fourth-order numerical relativity , 2005 .

[139]  T. Damour,et al.  Effective one-body approach to general relativistic two-body dynamics , 1999 .

[140]  C. Lousto,et al.  New conformally flat initial data for spinning black holes , 2002 .

[141]  David Merritt,et al.  Maximum gravitational recoil. , 2007, Physical review letters.

[142]  B. Bruegmann,et al.  Numerical black hole initial data with low eccentricity based on post-Newtonian orbital parameters , 2009, 0901.0993.

[143]  A. Buonanno,et al.  The complete non-spinning effective-one-body metric at linear order in the mass ratio , 2011, 1111.5610.

[144]  Circular orbits and spin in black-hole initial data , 2006, gr-qc/0605053.

[145]  M. Shibata.  NUMERICAL RELATIVITY , 2015 .

[146]  J. Novak,et al.  Spectral Methods for Numerical Relativity , 2007, Living reviews in relativity.

[147]  R. Kerr,et al.  Gravitational field of a spinning mass as an example of algebraically special metrics , 1963 .

[148]  Harmonic coordinate method for simulating generic singularities , 2001, gr-qc/0110013.

[149]  Roger Penrose,et al.  Asymptotic properties of fields and space-times , 1963 .

[150]  Thibault Damour,et al.  New effective-one-body description of coalescing nonprecessing spinning black-hole binaries , 2014, 1406.6913.

[151]  D. Shoemaker,et al.  Decoding the final state in binary black hole mergers , 2014, 1407.5989.

[152]  E. Witten.  Anti-de Sitter space and holography , 1998, hep-th/9802150.

[153]  Edward Seidel,et al.  Gravitational collapse of gravitational waves in 3D numerical relativity , 2000 .

[154]  Hermann Bondi,et al.  Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system , 1962, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[155]  Paul O'Brien,et al.  The transient gravitational-wave sky , 2013, 1305.0816.

[156]  Edward Seidel,et al.  Three-dimensional simulations of distorted black holes. I: Comparison with axisymmetric results. , 1999 .

[157]  R. Sachs.  Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time , 1962, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[158]  Hiroyuki Nakano,et al.  Intermediate-mass-ratio black-hole binaries: numerical relativity meets perturbation theory. , 2010, Physical review letters.

[159]  Seidel,et al.  Head-on collision of two equal mass black holes. , 1995, Physical review. D, Particles and fields.

[160]  M. Choptuik.  The Binary Black Hole Grand Challenge Project , 1997 .

[161]  E. Poisson,et al.  The Motion of Point Particles in Curved Spacetime , 2003, Living reviews in relativity.

[162]  F. Ohme.  Analytical meets numerical relativity: status of complete gravitational waveform models for binary black holes , 2011, 1111.3737.

[163]  John Shalf,et al.  The Cactus Framework and Toolkit: Design and Applications , 2002, VECPAR.

[164]  Z. Etienne,et al.  Binary black-hole mergers in magnetized disks: simulations in full general relativity. , 2012, Physical review letters.

[165]  P. Chesler,et al.  Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes , 2013, 1309.1439.

[166]  J. York,et al.  Kinematics and dynamics of general relativity , 1979 .

[167]  Thorne,et al.  Spin-induced orbital precession and its modulation of the gravitational waveforms from merging binaries. , 1994, Physical review. D, Particles and fields.

[168]  C. Ott,et al.  Gravitational wave extraction in simulations of rotating stellar core collapse , 2010, 1012.0595.

[169]  Y. Zlochower,et al.  Orbital evolution of extreme-mass-ratio black-hole binaries with numerical relativity. , 2010, Physical review letters.

[170]  Scott H. Hawley,et al.  Dynamical evolution of quasicircular binary black hole data , 2004, Physical Review D.

[171]  J. Marsden,et al.  The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I , 1972 .

[172]  José A. González,et al.  Reducing eccentricity in black-hole binary evolutions with initial parameters from post-Newtonian inspiral , 2007, 0706.0904.

[173]  J. Maldacena,et al.  Large N Field Theories, String Theory and Gravity , 1999, hep-th/9905111.

[174]  S. Teukolsky,et al.  Geometry of a Black Hole Collision , 1995, Science.

[175]  Compact binary evolutions with the Z4c formulation , 2012, 1212.2901.

[176]  H. Shinkai.  Formulations of the Einstein Equations for Numerical Simulations , 2008, 0805.0068.

[177]  Michael Boyle,et al.  Effective-one-body model for black-hole binaries with generic mass ratios and spins , 2013, Physical Review D.

[178]  Dae-Il Choi,et al.  Getting a Kick Out of Numerical Relativity , 2006, astro-ph/0603204.

[179]  J. D. Brown.  Probing the puncture for black hole simulations , 2009, 0908.3814.

[180]  Bernard F. Schutz,et al.  Physics, Astrophysics and Cosmology with Gravitational Waves , 2009, Living reviews in relativity.

[181]  Frans Pretorius,et al.  Numerical relativity using a generalized harmonic decomposition , 2005 .

[182]  D. Shoemaker,et al.  Grazing collisions of black holes via the excision of singularities. , 2000, Physical review letters.

[183]  Strongly hyperbolic second order Einstein's evolution equations , 2004, gr-qc/0402123.

[184]  H. Pfeiffer,et al.  Revisiting event horizon finders , 2008, 0809.2628.

[185]  R. Gold,et al.  Radiation from low-momentum zoom-whirl orbits , 2009, 0911.3862.

[186]  Schrödinger representation for the polarized Gowdy model , 2006, gr-qc/0607084.

[187]  Christopher Beetle,et al.  Towards wave extraction in numerical relativity: foundations and initial value formulation , 2004 .

[188]  Oliver Elbracht,et al.  Using curvature invariants for wave extraction in numerical relativity , 2008, 0811.1600.

[189]  M. Scheel,et al.  Testing outer boundary treatments for the Einstein equations , 2007, 0704.0782.

[190]  S. Hartnoll.  Lectures on holographic methods for condensed matter physics , 2009, 0903.3246.

[191]  Adalbert Kerber,et al.  The Cauchy Problem , 1984 .

[192]  Nonexistence of conformally flat slices in Kerr and other stationary spacetimes. , 2003, Physical review letters.

[193]  Merger of binary neutron stars of unequal mass in full general relativity , 2003, gr-qc/0310030.

[194]  M. Alcubierre,et al.  Simple excision of a black hole in 3 + 1 numerical relativity , 2000, gr-qc/0008067.

[195]  E. Berti,et al.  Numerical simulations of single and binary black holes in scalar-tensor theories: Circumventing the no-hair theorem , 2013, 1304.2836.

[196]  Marcus Ansorg,et al.  Single-domain spectral method for black hole puncture data , 2004 .

[197]  José A. González,et al.  Beyond the Bowen–York extrinsic curvature for spinning black holes , 2006, gr-qc/0612001.

[198]  Michael Taylor,et al.  Pseudodifferential Operators and Nonlinear PDE , 1991 .

[199]  Coalescence remnant of spinning binary black holes , 2003, astro-ph/0305287.

[200]  Luis Lehner,et al.  Numerical relativity: a review , 2001 .

[201]  C. Gundlach,et al.  Pseudospectral apparent horizon finders: An efficient new algorithm , 1997, gr-qc/9707050.

[202]  STABLE CHARACTERISTIC EVOLUTION OF GENERIC THREE-DIMENSIONAL SINGLE-BLACK-HOLE SPACETIMES , 1998, gr-qc/9801069.

[203]  B. Dewitt,et al.  Maximally slicing a black hole. , 1973 .

[204]  Michael Boyle,et al.  The NINJA-2 catalog of hybrid post-Newtonian/numerical-relativity waveforms for non-precessing black-hole binaries , 2012, 1201.5319.

[205]  J. York.  Energy and Momentum of the Gravitational Field , 1980 .

[206]  D. Shoemaker,et al.  MERGERS OF SUPERMASSIVE BLACK HOLES IN ASTROPHYSICAL ENVIRONMENTS , 2011, 1101.4684.

[207]  Jonathan Thornburg,et al.  Coordinates and boundary conditions for the general relativistic initial data problem , 1987 .

[208]  P. Chesler,et al.  Holography and Colliding gravitational shock waves in asymptotically AdS5 spacetime. , 2010, Physical review letters.

[209]  G. Lovelace,et al.  High-accuracy gravitational waveforms for binary black hole mergers with nearly extremal spins , 2011, 1110.2229.

[210]  Joan M. Centrella,et al.  Black-hole binaries, gravitational waves, and numerical relativity , 2010, 1010.5260.

[211]  J. York.  Gravitational degrees of freedom and the initial-value problem , 1971 .

[212]  R. Arnowitt,et al.  Republication of: The dynamics of general relativity , 2004 .

[213]  A. Čadež.  Colliding Black Holes. , 1971 .

[214]  Moving black holes via singularity excision , 2003, gr-qc/0301111.

[215]  Towards a realistic neutron star binary inspiral: Initial data and multiple orbit evolution in full general relativity , 2003, gr-qc/0312030.

[216]  E. Schnetter,et al.  Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision , 2005 .

[217]  Harald P. Pfeiffer,et al.  Simulations of unequal-mass black hole binaries with spectral methods , 2012, 1206.3015.

[218]  E. Seidel,et al.  Gauge conditions for long-term numerical black hole evolutions without excision , 2002, gr-qc/0206072.

[219]  Michael Boyle,et al.  Catalog of 174 binary black hole simulations for gravitational wave astronomy. , 2013, Physical review letters.

[220]  J. W. York.  ROLE OF CONFORMAL THREE-GEOMETRY IN THE DYNAMICS OF GRAVITATION. , 1972 .

[221]  Lawrence E. Kidder,et al.  High-accuracy waveforms for binary black hole inspiral, merger, and ringdown , 2008, 0810.1767.

[222]  Bernd Bruegmann,et al.  A Simple Construction of Initial Data for Multiple Black Holes , 1997 .

[223]  S. Komossa.  Recoiling Black Holes: Electromagnetic Signatures, Candidates, and Astrophysical Implications , 2012, 1202.1977.

[224]  Scott C. Noble,et al.  Dynamic fisheye grids for binary black hole simulations , 2013, 1309.2960.

[225]  Harald P. Pfeiffer,et al.  Numerical simulations of compact object binaries , 2012, 1203.5166.

[226]  Conformal ``thin sandwich'' data for the initial-value problem of general relativity , 1998, gr-qc/9810051.

[227]  F. Ohme,et al.  Wormholes and trumpets: Schwarzschild spacetime for the moving-puncture generation , 2008, 0804.0628.

[228]  C. Lousto,et al.  The Lazarus project : A pragmatic approach to binary black hole , 2001, gr-qc/0104063.

[229]  F. Pretorius,et al.  Black hole mergers and unstable circular orbits , 2007, gr-qc/0702084.

[230]  Harald P. Pfeiffer,et al.  A Multidomain spectral method for solving elliptic equations , 2002, gr-qc/0202096.

[231]  Y. Zlochower,et al.  Characteristic extraction tool for gravitational waveforms , 2010, 1011.4223.

[232]  F. Pirani,et al.  Gravitational waves in general relativity III. Exact plane waves , 1959, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[233]  Colliding black holes: The close limit. , 1994, Physical review letters.

[234]  P. Romatschke,et al.  Shock wave collisions in AdS5: approximate numerical solutions , 2011, 1108.3715.

[235]  Nakamura,et al.  Evolution of three-dimensional gravitational waves: Harmonic slicing case. , 1995, Physical review. D, Particles and fields.

[236]  C. Palenzuela,et al.  Understanding possible electromagnetic counterparts to loud gravitational wave events: Binary black hole effects on electromagnetic fields , 2009, 0911.3889.

[237]  M. Choptuik,et al.  Universality and scaling in gravitational collapse of a massless scalar field. , 1993, Physical review letters.

[238]  Frans Pretorius,et al.  Cross section, final spin, and zoom-whirl behavior in high-energy black-hole collisions. , 2009, Physical review letters.

[239]  O. Rinne.  Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations , 2006, gr-qc/0606053.