9 Computational Black Hole Dynamics

  • Pablo Laguna
  • Deirdre M. Shoemaker
Part III In Search of the Imprints of Early Universe: Gravitational Waves
Part of the Lecture Notes in Physics book series (LNP, volume 653)


Over the last decade, advances in computer hardware and numerical algorithms have opened the door to the possibility that simulations of sources of gravitational radiation can produce valuable information of direct relevance to gravitational wave astronomy. One source in particular is believed to be of extreme importance: the inspiral and merger of a binary black hole system. Simulations of binary black hole systems involve solving the Einstein equation in full generality. Such a daunting task has been one of the primary goals of the numerical relativity community. This review article focuses on the computational modelling of binary black holes. It provides a basic introduction to the subject and is intended for non-experts in the area of numerical relativity.


Black Hole Einstein Equation Gravitational Wave Apparent Horizon Extrinsic Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1. M. Maggiore, Gravitational Wave Experiments and Early Universe Cosmology, Phys. Reports, 331, 283 (2000).Google Scholar
  2. 2. T.W. Baumgarte and S.L. Shapiro. Numerical relativity and compact binaries. Phys. Reports, 376, 41 2003.Google Scholar
  3. 3. C. Cluter and K.S. Thorne. An overview of gravitational-wave sources. In Proceeding of GR16, 2001.Google Scholar
  4. 4. L. Lehner, Numerical relativity: A review, Class. Quant. Grav. 18, R25 (2001).Google Scholar
  5. 5. J. W. York Jr, Kinematics and dynamics of general relativity, In L. L. Smarr, editor, Sources of gravitational radiation, pages 83–126. Cambridge University Press, Cambridge, (1979).Google Scholar
  6. 6. R. Arnowitt, S. Deser, and C.W. Misner, The dynamics of general relativity, In L. Witten, editor, Gravitation an introduction to current research, pages 227–265. John Wiley, New York, (1962).Google Scholar
  7. 7. G.B. Cook, Living Rev. Rel. 5, 1 (2000).Google Scholar
  8. 8. G. Calabrese and O. Sarbach, J. Math. Phys. 44, 3888 (2003).Google Scholar
  9. 9. M. Alcubierre, Class. Quant. Grav. 20, 607 (2003).Google Scholar
  10. 10. M. Tiglio, L. Lehner, and D. Nilsen, gr-qc/0312001 (2003).Google Scholar
  11. 11. L. Lindblom, M. A. Scheel, L. E. Kidder, H. P. Pfeiffer, D. Shoemaker, and S. A. Teukolsky, gr-qc/0402027, (2004).Google Scholar
  12. 12. M. Anderson amd R.A. Matzner, gr-qc/0307055 (2003).Google Scholar
  13. 13. H. Shinkai and G. Yoneda, gr-qc/0209111, (2002).Google Scholar
  14. 14. O. Reula, Living Rev. Rel. 3, 1 (1998).Google Scholar
  15. 15. T.W. Baumgarte and S.L. Shapiro, Phys. Rev. D 59, 024007 (1999).Google Scholar
  16. 16. M. Shibata and T. Nakamura, Phys. Rev. D 52, 5428 (1995).Google Scholar
  17. 17. L.E. Kidder, M.A. Scheel, and S.A. Teukolsky, Extending the lifetime of 3d black hole computations with a new hyperbolic system of evolution equations, Phys. Rev. D64, 064017 (2001).Google Scholar
  18. 18. J. York, In C.R. Evans and L.S. Finn and D.W. Hobill, editors, Frontiers in Numerical Relativity, Cambridge University Press, Cambridge, (1989).Google Scholar
  19. 19. J. Thornburg, A fast apparent-horizon finder for 3-dimensional cartesian grids in numerical relativity, Class. Quant. Grav. 21, 743 (2004).Google Scholar
  20. 20. M. Alcubierre, B. Brügmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel, and R. Takahashi, Gauge conditions for long-term numerical black hole evolutions without excision, Phys. Rev. D67, 084023 (2003).Google Scholar
  21. 21. G.B. Cook and BBH Alliance, Boosted three-dimensional black hole evolutions with singularity excision, Phys. Rev. Lett. 80, 2512 (1998).Google Scholar
  22. 22. S. Brandt, R. Correll, R. Gómez, M. Huq, P. Laguna L. Lehner, D. Neilsen, R. Matzner, J. Pullin, E. Schnetter, D. Shoemaker, and J. Winicour, Grazing collisions of black holes via the excision of singularities, Phys. Rev. Lett. 85, 5496 (2000).Google Scholar
  23. 23. R. Gómez, L. Lehner, R. Marsa, and J. Winicour, Moving black holes in 3d, Phys. Rev. D57, 4778 (1998).Google Scholar
  24. 24. E. Seidel and W. Suen, Phys. Rev. Lett. 69, 1845 (1992).Google Scholar
  25. 25. P. Anninos, G. Gaues, J. Masso, E. Seidel, and L. Smarr, Horizon boundary condition for black hole spacetimes, Phys. Rev. D51, 5562 (1995).Google Scholar
  26. 26. R. Marsa and M. Choptuik, Black hole–scalar field interactions in spherical symmetry, Phys. Rev. D54, 4929 (1996).Google Scholar
  27. 27. M. Scheel, T. Baumgarte, G. Cook, S. Shapiro, and S. Teukolsky, Numerical evolution of black holes with a hyperbolic formulation of general relativity, Phys. Rev. D56, 6320 (1997).Google Scholar
  28. 28. M. Alcubierre and B. Brügmann, Simple excision of a black hole in 3+1 numerical relativity, Phys. Rev. D63, 104006 (2001).Google Scholar
  29. 29. H. Yo, T. Baumgarte, and S. Shapiro, A numerical testbed for singularity excision in moving black hole spacetimes, Phys. Rev. D64, 124011 (2001).Google Scholar
  30. 30. D. Shoemaker, K. Smith, U. Sperhake, P. Laguna, E. Schnetter, and D. Fiske, Class. Quant. Grav. 20, 3729 (2003).Google Scholar
  31. 31. U. Sperhake, K. Smith, B. Kelly, P. Laguna, and D. Shoemaker, Phys. Rev. D69, 024012 (2004).Google Scholar
  32. 32. G. Calabrese, L. Lehner, D. Neilsen, J. Pullin, O. Reula, O. Sarbach, and M. Tiglio, Novel finite-differencing techniques for numerical relativity: application to black hole excision, Class. Quant. Grav. 20, L245 (2003).Google Scholar
  33. 33. A. Ashtekar, C. Beetle, and S. Fairhurst, Mechanics of isolated horizons, Class. Quant. Grav. 17, 253 (2000).Google Scholar
  34. 34. A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski, and J. Wisniewski, Generic isolated horizons and their applications, Phys. Rev. Lett. 85, 3564 (2000).Google Scholar
  35. 35. A. Ashtekar and B. Krishnan, Dynamical horizons: Energy, angular momentum, fluxes and balance laws, Phys. Rev. Lett. 89, 261101 (2002).Google Scholar
  36. 36. A. Ashtekar and B. Krishnan, Dynamical horizons and their properties, Phys. Rev. D68, 104030 (2003).Google Scholar
  37. 37. O. Dreyer, B. Krishnan, E. Schnetter, and D. Shoemaker, Introduction to isolated horizons in numerical relativity, Phys. Rev. D67, 024018 (2003).Google Scholar
  38. 38. A. Ashtekar, S. Fairhurst, and B. Krishnan, Isolated horizons: Hamiltonian evolution and the first law, Phys. Rev. D62, 104025 (2000).Google Scholar
  39. 39. A. Ashtekar, C. Beetle, and J. Lewandowski, Mechanics of rotating isolated horizons, Phys. Rev. D64, 044016 (2001).Google Scholar
  40. 40. S. Brandt and B. Brügmann, A simple construction of initial data for multiple black holes, Phys. Rev. Lett. 78, 3606 (1997).Google Scholar
  41. 41. C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation. Freeman: San Francisco, (1973).Google Scholar
  42. 42. R.A. Matzner, M.F. Huq, and D. Shoemaker, Initial data and coordinates for multiple black hole systems, Phys. Rev. D59, 024015 (1999).Google Scholar
  43. 43. Harald P. Pfeiffer and Gregory B. Cook and Saul A. Teukolsky, Comparing initial-data sets for binary black holes, Phys. Rev. D66, 024047 (2002).Google Scholar
  44. 44. P. Anninos, K. Camarda J. Massó, E. Seidel, W. Suen, and J. Towns, Three dimensional numerical relativity: the evolution of black holes, Phys. Rev. D52, 2059 (1995).Google Scholar
  45. 45. R. Gómez et al, Stable characteristic evolution of generic three-dimensional single black hole space-times, Phys. Rev. Lett. 80, 3915 (1998).Google Scholar
  46. 46. M. Alcubierre, B. Brügmann, D. Pollney, E. Seidel, and R. Takahashi, Black hole excision for dynamic black holes, Phys. Rev. D64, 061501 2001.Google Scholar
  47. 47. H.-J. Yo, T. W. Baumgarte, and S. L. Shapiro, Phys. Rev. D66, 084026 (2002).Google Scholar
  48. 48. S. Brandt, K. Camarda, E. Seidel, and R. Takahashi, Three dimensional distorted black holes, Class. Quant. Grav. 20, 1 (2003).Google Scholar
  49. 49. R. Gómez, Gravitational waveforms with controlled accuracy, Phys. Rev. D64, 024007 (2001).Google Scholar
  50. 50. J. Baker, S. Brandt, M. Campanelli, C. Lousto, E. Seidel, and R. Takahashi, Nonlinear and perturbative evolution of distorted black holes. ii. odd-parity modes, Phys. Rev. D62, 127701 (2000).Google Scholar
  51. 51. G. Allen, K. Camarda, and E. Seidel, Black hole spectroscopy: Determining waveforms from 3d excited black holes, gr-qc/9806036 (1998).Google Scholar
  52. 52. D.R. Briland and R.W. Lindquist, Phys. Rev. 131, 471 (1963).Google Scholar
  53. 53. P. Papadopoulos, Nonlinear harmonic generation in finite amplitude black hole oscillations, Phys. Rev. D65, 084016 2002.Google Scholar
  54. 54. Y. Zlochower, R Gómez, S. Husa, L. Lehner, and J. Winicour, Mode coupling in the nonlinear response of black holes, Phys. Rev. D68, 084014 (2003).Google Scholar
  55. 55. S. Hahn and R. Lindquist, The two body problem in geometrodynamics, Annals of Physics 29, 304 (1964).Google Scholar
  56. 56. L. L. Smarr, Gauge conditions, radiation formulae and the two black hole collisions, In L. L. Smarr, editor, Sources of gravitational radiation, page 275. Cambridge University Press, Cambridge, (1979).Google Scholar
  57. 57. B. Brügmann, Binary black hole mergers in 3d numerical relativity, Int. J. Mod. Phys. D8, 85 (1999).Google Scholar
  58. 58. P. Marronetti, M. Huq, P. Laguna, L. Lehner, R. Matzner, and D. Shoemaker, Approximate analytical solutions to the initial data problem of black hole binary systems, Phys. Rev. D62, 024017 (2000).Google Scholar
  59. 59. M. Alcubierre, W. Benger, B. Brügmann, G. Lanfermann, L. Nerger, E. Seidel, and R. Takahashi, The 3d grazing collision of two black holes, Phys. Rev. Lett. 87, 271103 (2001).Google Scholar
  60. 60. J. Baker, M. Campanelli, C.O. Lousto, and R. Takahashi, The lazarus project: A pragmatic approach to binary black hole evolutions, Phys. Rev. D 65, 124012 (2002).Google Scholar
  61. 61. J. Baker, B. Brügmann, M. Campanelli, and C.O. Lousto, Gravitational waves from black hole collisions via an eclectic approach, Class. Quant. Grav. 17, L149 (2000).Google Scholar
  62. 62. J. Baker, B. Brügmann, M. Campanelli, C.O. Lousto, and R. Takahashi, Plunge waveforms from inspiralling binary black holes, Phys. Rev. Lett. 87, 121103 (2001).Google Scholar
  63. 63. B. Brügmann, W. Tichy, and N. Jansen, Numerical simulation of orbiting black holes, gr-qc/0312112 (2003).Google Scholar
  64. 64. E. Schnetter, S. Hawley, and I. Hawke, Evolutions in 3d numerical relativity using fixed mesh refinement, Class. Quant. Grav 21, 1465 (2004).Google Scholar
  65. 65. B. Imbiriba, J. Baker, D. Choi, J. Centrella, D. Fiske, J. Brown, J. van Meter, and K. Olson, Evolving a puncture black hole with fixed mesh refinement, gr-qc/0403048 (2004).Google Scholar
  66. 66. L. Kidder, M. Scheel, S. Teukolsky, E. Carlson, and G. Cook, Black hole evolution by spectral methods, Phys. Rev. D62, 084032 (2000).Google Scholar
  67. 67. S. Bonazzola, E. Gourgoulhon, and J. Marck, Spectral methods in general relativistic astrophysics, J. Comput. Appl. Math. 109, 892 (1999).Google Scholar
  68. 68. P. Grandclément, S. Bonazzola, E. Gourgoulhon, and J. Marck, A multi-domain spectral method for scalar and vectorial poisson equations with non-compact sources, J. Comput. Phys. 170, 231 (2001).Google Scholar

Authors and Affiliations

  • Pablo Laguna
    • 1
  • Deirdre M. Shoemaker
    • 2
  1. 1.Department of Astronomy and Astrophysics, Institute for Gravitational Physics and Geometry, Center for Gravitational Wave Physics, Penn State University, University Park, PA 16802USA
  2. 2.Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853USA

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