Numerical relativity: the role of black holes in gravitational wave physics, astrophysics and high-energy physics

Review Article
Part of the following topical collections:
  1. The First Century of General Relativity: GR20/Amaldi10


Black holes play an important role in many areas of physics. Their modeling in the highly-dynamic, strong-field regime of general relativity requires the use of computational methods. We present a review of the main results obtained through numerical relativity simulations of black-hole spacetimes with a particular focus on the most recent developments in the areas of gravitational-wave physics, astrophysics, high-energy collisions, the gauge-gravity duality, and the study of fundamental properties of black holes.


Black holes Numerical relativity Gravitational waves Higher dimensions 


  1. 1.
    Aasi, J., et al.: The NINJA-2 project: Detecting and characterizing gravitational waveforms modelled using numerical binary black hole simulations. Class. Quantum Grav. LIGO-P1300199 (2014). ArXiv:1401.0939 [gr-qc]Google Scholar
  2. 2.
    Aharony, O., Gubser, S.S., Maldacena, J.M., Ooguri, H., Oz, Y.: Large N field theories, string theory and gravity. Phys. Rep. 323, 183–386 (2000). doi:10.1016/S0370-1573(99)00083-6. Hep-th/9905111ADSMathSciNetGoogle Scholar
  3. 3.
    Ajith, P.: Gravitational-wave data analysis using binary black-hole waveforms. Class. Quantum Grav. 25, 114033 (2008). ArXiv:0712.0343 [gr-qc]ADSGoogle Scholar
  4. 4.
    Ajith, P., Hannam, M., Husa, S., Chen, Y., Brügmann, B., et al.: Inspiral-merger-ringdown waveforms for black-hole binaries with non-precessing spins. Phys. Rev. Lett. 106, 241101 (2011). doi:10.1103/PhysRevLett.106.241101. ArXiv:0909.2867 [gr-qc]ADSGoogle Scholar
  5. 5.
    Ajith, P., et al.: Phenomenological template family for black-hole coalescence waveforms. Class. Quantum Grav. 24, S689–S700 (2007). doi:10.1088/0264-9381/24/19/S31. ArXiv:0704.3764 [gr-qc]ADSMATHMathSciNetGoogle Scholar
  6. 6.
    Ajith, P., et al.: A template bank for gravitational waveforms from coalescing binary black holes: I. Non-spinning binaries. Phys. Rev. D 77, 104017 (2008). ArXiv:0710.2335 [gr-qc]ADSMathSciNetGoogle Scholar
  7. 7.
    Ajith, P., et al.: The NINJA-2 catalog of hybrid post-Newtonian/numerical-relativity waveforms for non-precessing black-hole binaries. Class. Quantum Grav. 29, 124001 (2012). doi:10.1088/0264-9381/29/12/124001. ArXiv:1201.5319 [gr-qc]ADSGoogle Scholar
  8. 8.
    Alcubierre, M.: Introduction to 3+1 Numerical Relativity. Oxford University Press, Oxford (2008)MATHGoogle Scholar
  9. 9.
    Alic, D., Bona-Casas, C., Bona, C., Rezzolla, L., Palenzuela, C.: Conformal and covariant formulation of the Z4 system with constraint-violation damping. Phys. Rev. D 85, 064040 (2012). doi:10.1103/PhysRevD.85.064040. ArXiv:1106.2254 [gr-qc]ADSGoogle Scholar
  10. 10.
    Alic, D., Mösta, P., Rezzolla, L., Zanotti, O., Jaramillo, J.L.: Accurate simulations of binary black-hole mergers in force-free electrodynamics. Astrophys. J. 754, 36 (2012). doi:10.1088/0004-637X/754/1/36. ArXiv:1204.2226 [gr-qc]ADSGoogle Scholar
  11. 11.
    Anderson, M., Lehner, L., Megevand, M., Neilsen, D.: Post-merger electromagnetic emissions from disks perturbed by binary black holes. Phys. Rev. D 81, 044004 (2010). doi:10.1103/PhysRevD.81.044004. ArXiv:0910.4969 [astro-ph]ADSGoogle Scholar
  12. 12.
    Ansorg, M., Brügmann, B., Tichy, W.: A single-domain spectral method for black hole puncture data. Phys. Rev. D 70, 064011 (2004). doi:10.1103/PhysRevD.70.064011. Gr-qc/0404056ADSGoogle Scholar
  13. 13.
    Antoniadis, I.: A Possible new dimension at a few TeV. Phys. Lett. B 246, 377–384 (1990). doi:10.1016/0370-2693(90)90617-F ADSMathSciNetGoogle Scholar
  14. 14.
    Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: New dimensions at a millimeter to a Fermi and superstrings at a TeV. Phys. Lett. B 436, 257–263 (1998). doi:10.1016/S0370-2693(98)00860-0. Hep-ph/9804398ADSGoogle Scholar
  15. 15.
    Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: The hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429, 263–272 (1998). doi:10.1016/S0370-2693(98)00466-3. Hep-ph/9803315ADSGoogle Scholar
  16. 16.
    Aylott, B., et al.: Status of NINJA: the Numerical INJection Analysis project. Class. Quantum Grav. 26, 114008 (2009). doi:10.1088/0264-9381/26/11/114008. ArXiv:0905.4227 [gr-qc]ADSGoogle Scholar
  17. 17.
    Aylott, B., et al.: Testing gravitational-wave searches with numerical relativity waveforms: results from the first Numerical INJection Analysis (NINJA) project. Class. Quantum Grav. 26, 165008 (2009). doi:10.1088/0264-9381/26/16/165008. ArXiv:0901.4399 [gr-qc]ADSGoogle Scholar
  18. 18.
    Baker, J.G., Centrella, J., Choi, D.I., Koppitz, M., van Meter, J.: Gravitational-wave extraction from an inspiraling configuration of merging black holes. Phys. Rev. Lett. 96, 111102 (2006). doi:10.1103/PhysRevLett.96.111102. Gr-qc/0511103ADSGoogle Scholar
  19. 19.
    Balasubramanian, V., Kraus, P.: A stress tensor for anti-de Sitter gravity. Commun. Math. Phys. 208, 413–428 (1999). doi:10.1007/s002200050764. Hep-th/9902121ADSMATHMathSciNetGoogle Scholar
  20. 20.
    Banks, T., Fischler, W.: A model for high energy scattering in quantum gravity (1999). Report number RU-99–23, UTTP-03-99. Hep-th/9906038Google Scholar
  21. 21.
    Bantilan, H., Pretorius, F., Gubser, S.S.: Simulation of asymptotically AdS5 spacetimes with a generalized harmonic evolution scheme. Phys. Rev. D 85, 084038 (2012). doi:10.1103/PhysRevD.85.084038. ArXiv:1201.2132 [hep-th]ADSGoogle Scholar
  22. 22.
    Baumgarte, T.W., Shapiro, S.L.: On the numerical integration of Einstein’s field equations. Phys. Rev. D 59, 024007 (1998). doi:10.1103/PhysRevD.59.024007. Gr-qc/9810065ADSMathSciNetGoogle Scholar
  23. 23.
    Baumgarte, T.W., Shapiro, S.L.: Numerical Relativity. Cambridge University Press, Cambridge (2010)MATHGoogle Scholar
  24. 24.
    Bayona, C.A., Braga, N.R.: Anti-de Sitter boundary in Poincare coordinates. Gen. Relativ. Gravit. 39, 1367–1379 (2007). doi:10.1007/s10714-007-0446-y. Hep-th/0512182ADSMATHGoogle Scholar
  25. 25.
    Bekenstein, J.D.: Extraction of energy and charge from a black hole. Phys. Rev. D 7, 949–953 (1973). doi:10.1103/PhysRevD.7.949 ADSMathSciNetGoogle Scholar
  26. 26.
    Bekenstein, J.D.: Gravitational-radiation recoil and runaway black holes. Astrophys. J. 183, 657–664 (1973). doi:10.1086/152255 ADSGoogle Scholar
  27. 27.
    Berti, E., Cardoso, V., Gualtieri, L., Horbatsch, M., Sperhake, U.: Numerical simulations of single and binary black holes in scalar-tensor theories: circumventing the no-hair theorem. Phys. Rev. D 87, 124020 (2013). doi:10.1103/PhysRevD.87.124020. ArXiv:1304.2836 [gr-qc]ADSGoogle Scholar
  28. 28.
    Berti, E., Kesden, M., Sperhake, U.: Effects of post-Newtonian spin alignment on the distribution of black-hole recoils. Phys. Rev. D 85, 124049 (2012). doi:10.1103/PhysRevD.85.124049. ArXiv:1203.2920 [astro-ph]ADSGoogle Scholar
  29. 29.
    Bizoń, P., Rostworowski, A.: On weakly turbulent instability of anti-de Sitter space. Phys. Rev. Lett. 107, 031102 (2011). doi:10.1103/PhysRevLett.107.031102. ArXiv:1104.3702 [gr-qc]ADSGoogle Scholar
  30. 30.
    Blanchet, L.: Gravitational radiation from post-Newtonian sources and inspiralling compact binaries. Living Rev. Rel. 9(4) (2006).
  31. 31.
    Blandford, R.D., Znajek, R.L.: Electromagnetic extractions of energy from Kerr black holes. Mon. Not. Roy. Astron. Soc. 179, 433–456 (1977)ADSGoogle Scholar
  32. 32.
    Bode, T., Haas, R., Bogdanović, T., Laguna, P., Shoemaker, D.: Relativistic mergers of supermassive black holes and their electromagnetic signatures. Astrophys. J. 715, 1117–1131 (2010). doi:10.1088/0004-637X/715/2/1117. ArXiv:0912.0087 [gr-qc]ADSGoogle Scholar
  33. 33.
    Bode, T., et al.: Mergers of supermassive black holes in astrophysical environments. Astrophys. J. 744, 45 (2011). doi:10.1088/0004-637X/744/1/45. ArXiv:1101.4684 [gr-qc]ADSGoogle Scholar
  34. 34.
    Bogdanović, T., Reynolds, C.S., Miller, M.C.: Alignment of the spins of supermassive black holes prior to coalescence. Astrophys. J. 661, L147–L150 (2007). Astro-ph/0703054ADSGoogle Scholar
  35. 35.
    Bona, C., Ledvinka, T., Palenzuela, C., Žáček, M.: General-covariant evolution formalism for numerical relativity. Phys. Rev. D 67, 104005 (2003). doi:10.1103/PhysRevD.67.104005. Gr-qc/0302083ADSMathSciNetGoogle Scholar
  36. 36.
    Bona, C., Palenzuela-Luque, C., Bona-Casas, C.: Elements of Numerical Relativity and Relativistic Hydrodynamics. Springer, London, New York (2009)MATHGoogle Scholar
  37. 37.
    Bonnor, W.B., Rotenberg, M.A.: Transport of momentum by gravitational waves: the linear approximation. Proc. R. Soc. Lond. A. 265, 109–116 (1961)ADSMathSciNetGoogle Scholar
  38. 38.
    Brown, D.A., Lundgren, A., O’Shaughnessy, R.: Nonspinning searches for spinning binaries in ground-based detector data: amplitude and mismatch predictions in the constant precession cone approximation. Phys. Rev. D 86, 064020 (2012). doi:10.1103/PhysRevD.86.064020. ArXiv:1203.6060 [gr-qc]ADSGoogle Scholar
  39. 39.
    Brown, J.D., York Jr, J.W.: Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D 47, 1407–1419 (1993). doi:10.1103/PhysRevD.47.1407. Gr-qc/9209012ADSMathSciNetGoogle Scholar
  40. 40.
    Buchel, A., Lehner, L., Liebling, S.L.: Scalar collapse in AdS. Phys. Rev. D 86, 123011 (2012). doi:10.1103/PhysRevD.86.123011. ArXiv:1210.0890 [gr-qc]ADSGoogle Scholar
  41. 41.
    Buchel, A., Liebling, S.L., Lehner, L.: Boson stars in AdS. Phys. Rev. D 87, 123006 (2013). doi:10.1103/PhysRevD.87.123006. ArXiv:1304.4166 [gr-qc]ADSGoogle Scholar
  42. 42.
    Buonanno, A., Chen, Y., Valisneri, M.: Detection template families for gravitational waves from the final stages of binary-black-hole inspirals: Nonspinning case. Phys. Rev. D 67, 024016 (2003). [Erratum-ibid. 74, 029903 (2006)] gr-qc/0205122.
  43. 43.
    Buonanno, A., Damour, T.: Effective one-body aproach to general relativistic two-body dynamics. Phys. Rev. D 59, 084006 (1999)ADSMathSciNetGoogle Scholar
  44. 44.
    Buonanno, A., Damour, T.: Transition from inspiral to plunge in binary black hole coalescences. Phys. Rev. D 62, 064015 (2000)ADSGoogle Scholar
  45. 45.
    Campanelli, M., Lousto, C.O., Marronetti, P., Zlochower, Y.: Accurate evolutions of orbiting black-hole binaries without excision. Phys. Rev. Lett. 96, 111101 (2006). doi:10.1103/PhysRevLett.96.111101. Gr-qc/0511048ADSGoogle Scholar
  46. 46.
    Campanelli, M., Lousto, C.O., Zlochower, Y.: Spinning-black-hole binaries: the orbital hang up. Phys. Rev. D 74, 041501 (2006). doi:10.1103/PhysRevD.74.041501. Gr-qc/0604012ADSMathSciNetGoogle Scholar
  47. 47.
    Campanelli, M., Lousto, C.O., Zlochower, Y., Merritt, D.: Large merger recoils and spin flips from generic black-hole binaries. Astrophys. J. 659, L5–L8 (2007). doi:10.1086/516712. Gr-qc/0701164ADSGoogle Scholar
  48. 48.
    Campanelli, M., Lousto, C.O., Zlochower, Y., Merritt, D.: Maximum gravitational recoil. Phys. Rev. Lett. 98, 231102 (2007). Gr-qc/0702133ADSGoogle Scholar
  49. 49.
    Cao, Z., Hilditch, D.: Numerical stability of the Z4c formulation of general relativity. Phys. Rev. D 85, 124032 (2012). doi:10.1103/PhysRevD.85.124032. ArXiv:1111.2177 [gr-qc]ADSGoogle Scholar
  50. 50.
    Centrella, J.M., Baker, J.G., Kelly, B.J., van Meter, J.R.: Black-hole binaries, gravitational waves, and numerical relativity. Rev. Mod. Phys. 82, 3069 (2010). doi:10.1103/RevModPhys.82.3069. ArXiv:1010.5260 [gr-qc]Google Scholar
  51. 51.
    Chang, P., Strubbe, L.E., Menou, K., Quataert, E.: Fossil gas and the electromagnetic precursor of supermassive binary black hole mergers. MNRAS 407, 2007–2016 (2010). ArXiv:0906.0825 [astro-ph]ADSGoogle Scholar
  52. 52.
    Chesler, P.M., Yaffe, L.G.: Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma. Phys. Rev. Lett. 102, 211601 (2009). doi:10.1103/PhysRevLett.102.211601. ArXiv:0812.2053 [hep-th]ADSMathSciNetGoogle Scholar
  53. 53.
    Chesler, P.M., Yaffe, L.G.: Boost invariant flow, black hole formation, and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory. Phys. Rev. D 82, 026006 (2010). doi:10.1103/PhysRevD.82.026006. ArXiv:0906.4426 [hep-th]ADSGoogle Scholar
  54. 54.
    Chesler, P.M., Yaffe, L.G.: Holography and colliding gravitational shock waves in asymptotically \(AdS_5\) spacetime. Phys. Rev. Lett. 106, 021601 (2011). doi: 10.1103/PhysRevLett.106.021601. ArXiv:1011.3562 [hep-th]ADSGoogle Scholar
  55. 55.
    Chesler, P.M., Yaffe, L.G.: Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes (2013). ArXiv:1309.1439 [hep-th]Google Scholar
  56. 56.
    Choptuik, M.W.: Universality and scaling in graviational collapse of a massless scalar field. Phys. Rev. Lett. 70, 9–12 (1993). doi:10.1103/PhysRevLett.70.9 ADSGoogle Scholar
  57. 57.
    Choptuik, M.W., Pretorius, F.: Ultra relativistic particle collisions. Phys. Rev. Lett. 104, 111101 (2010). doi:10.1103/PhysRevLett.104.111101. ArXiv:0908.1780 [gr-qc]ADSGoogle Scholar
  58. 58.
    Dai, D.C., Starkman, G., Stojkovic, D., Issever, C., Rizvi, E., et al.: BlackMax: a black-hole event generator with rotation, recoil, split branes, and brane tension. Phys. Rev. D 77, 076007 (2008). doi:10.1103/PhysRevD.77.076007. ArXiv:0711.3012 [hep-ph]ADSGoogle Scholar
  59. 59.
    Damour, T., Nagar, A.: Comparing effective-one-body gravitational waveforms to accurate numerical data. Phys. Rev. D 77, 024043 (2008). ArXiv:0711.2628 [gr-qc]ADSGoogle Scholar
  60. 60.
    Damour, T., Nagar, A., Hannam, M.D., Husa, S., Brügmann, B.: Accurate effective-one-body waveforms of inspiralling and coalescing black-hole binaries. Phys. Rev. D 78, 044039 (2008). ArXiv:[0803.3162]ADSGoogle Scholar
  61. 61.
    Damour, T., Trias, M., Nagar, A.: Accuracy and effectualness of closed-form, frequency-domain waveforms for non-spinning black hole binaries. Phys. Rev. D 83, 024006 (2011). doi:10.1103/PhysRevD.83.024006. ArXiv:1009.5998 [gr-qc]ADSGoogle Scholar
  62. 62.
    Dias, O.J.C., Horowitz, G.T., Marolf, D., Santos, J.E.: On the nonlinear stability of asymptotically anti-de sitter solutions. Class. Quantum Grav. 29, 235019 (2012). doi:10.1088/0264-9381/29/23/235019. ArXiv:1208.5772 [gr-qc]ADSMathSciNetGoogle Scholar
  63. 63.
    Dimopoulos, S., Landsberg, G.: Black Holes at the LHC. Phys. Rev. Lett. 87, 161602 (2001). doi:10.1103/PhysRevLett.87.161602. Hep-th/0106295ADSGoogle Scholar
  64. 64.
    Dolan, S.R.: Superradiant instabilities of rotating black holes in the time domain. Phys. Rev. D 87, 124026 (2013). doi:10.1103/PhysRevD.87.124026. ArXiv:1212.1477 [gr-qc]ADSGoogle Scholar
  65. 65.
    Drude, P.: Zur elektronentheorie der metalle. Annalen der Physik 306, 566 (1900). doi:10.1002/andp.19003060312 ADSGoogle Scholar
  66. 66.
    Drude, P.: Zur elektronentheorie der metalle: II. Teil. galvanomagnetische und thermomagnetische effekte. Annalen der Physik 308, 369 (1900). doi:10.1002/andp.19003081102 ADSGoogle Scholar
  67. 67.
    Eardley, D.M., Giddings, S.B.: Classical black hole production in high-energy collisions. Phys. Rev. D 66, 044011 (2002). doi:10.1103/PhysRevD.66.044011. Gr-qc/0201034ADSMathSciNetGoogle Scholar
  68. 68.
    East, W.E., Pretorius, F.: Ultrarelativistic black hole formation. Phys. Rev. Lett. 110, 101101 (2013). doi:10.1103/PhysRevLett.110.101101. ArXiv:1210.0443 [gr-qc]ADSGoogle Scholar
  69. 69.
    Emparan, R., Reall, H.S.: Black holes in higher dimensions. Living Rev. Rel. 11(6) (2008).
  70. 70.
    Farris, B.D., Gold, R., Paschalidis, V., Etienne, Z.B., Shapiro, S.L.: Binary black hole mergers in magnetized disks: simulations in full general relativity. Phys. Rev. Lett. 109, 221102 (2012). doi:10.1103/PhysRevLett.109.221102. ArXiv:1207.3354 [astro-ph]ADSGoogle Scholar
  71. 71.
    Farris, B.D., Liu, Y.T., Shapiro, S.L.: Binary black hole mergers in gaseous environments: ’Binary Bondi’ and ’Binary Bondi-Hoyle-Lyttleton’ accretion. Phys. Rev. D 81, 084008 (2010). doi:10.1103/PhysRevD.81.084008. ArXiv:0912.2096 [asttro-ph]ADSGoogle Scholar
  72. 72.
    Finn, L.S.: Detection, measurement and gravitational radiation. Phys. Rev. D 46, 5236–5249 (1992). Gr-qc/9209010ADSGoogle Scholar
  73. 73.
    Fitchett, M.J.: The influence of gravitational wave momentum losses on the centre of mass motion of a newtonian binary system. MNRAS 203, 1049–1062 (1983)ADSMATHGoogle Scholar
  74. 74.
    Frost, J.A., Gaunt, J.R., Sampaio, M.O.P., Casals, M., Dolan, S.R., et al.: Phenomenology of Production and Decay of Spinning Extra- Dimensional Black Holes at Hadron Colliders. JHEP 10, 014 (2009). doi:10.1088/1126-6708/2009/10/014. ArXiv:0904.0979 [hep-th]ADSGoogle Scholar
  75. 75.
    Garfinkle, D.: Harmonic coordinate method for simulating generic singularities. Phys. Rev. D 65, 044029 (2002). doi:10.1103/PhysRevD.65.044029. Gr-qc/0110013ADSMathSciNetGoogle Scholar
  76. 76.
    Giddings, S.B., Thomas, S.: High energy colliders as black hole factories: the end of short distance physics. Phys. Rev. D 65, 056010 (2002). doi:10.1103/PhysRevD.65.056010. Hep-ph/0106219ADSGoogle Scholar
  77. 77.
    González, J.A., Hannam, M.D., Sperhake, U., Brügmann, B., Husa, S.: Supermassive kicks for spinning black holes. Phys. Rev. Lett. 98, 231101 (2007). doi:10.1103/PhysRevLett.98.231101. Gr-qc/0702052ADSGoogle Scholar
  78. 78.
    González, J.A., Sperhake, U., Brügmann, B., Hannam, M.D., Husa, S.: The maximum kick from nonspinning black-hole binary inspiral. Phys. Rev. Lett. 98, 091101 (2007). doi:10.1103/PhysRevLett.98.091101. Gr-qc/0610154ADSMathSciNetGoogle Scholar
  79. 79.
    Gregory, R., Laflamme, R.: Black strings and p-branes are unstable. Phys. Rev. Lett. 70, 2837–2840 (1993). doi:10.1103/PhysRevLett.70.2837. Hep-th/9301052ADSMATHMathSciNetGoogle Scholar
  80. 80.
    Gregory, R., Laflamme, R.: The Instability of charged black strings and p-branes. Nucl. Phys. B 428, 399–434 (1994). doi:10.1016/0550-3213(94)90206-2. Hep-th/9404071ADSMATHMathSciNetGoogle Scholar
  81. 81.
    Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105–114 (1998). doi:10.1016/S0370-2693(98)00377-3. Hep-th/9802109ADSMathSciNetGoogle Scholar
  82. 82.
    Guedes, J., Madau, P., Mayer, L., Callegari, S.: Recoiling massive black holes in gas-rich galaxy mergers. Astrophys. J. 729, 125 (2011). ArXiv:1008.2032 [astro-ph]ADSGoogle Scholar
  83. 83.
    Gundlach, C., Calabrese, G., Hinder, I., Martín-García, J.M.: Constraint damping in the Z4 formulation and harmonic gauge. Class. Quantum Grav. 22, 3767–3773 (2005). doi:10.1088/0264-9381/22/17/025 MATHGoogle Scholar
  84. 84.
    Haiman, Z., Loeb, A.: What is the highest plausible redshift of luminous quasars? Astrophys. J. 552, 459–463 (2001)ADSGoogle Scholar
  85. 85.
    Hannam, M., et al.: Twist and shout: a simple model of complete precessing black-hole-binary gravitational waveforms (2013). ArXiv:1308.3271 [gr-qc]Google Scholar
  86. 86.
    de Haro, S., Solodukhin, S.N., Skenderis, K.: Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence. Commun. Math. Phys. 217, 595–622 (2001). doi:10.1007/s002200100381. Hep-th/0002230ADSMATHGoogle Scholar
  87. 87.
    Harris, C.M., Richardson, P., Webber, B.R.: CHARYBDIS: a black hole event generator. JHEP 0308, 033 (2003). Hep-ph/0307305ADSGoogle Scholar
  88. 88.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)MATHGoogle Scholar
  89. 89.
    Healy, J., Bode, T., Haas, R., Pazos, E., Laguna, P., Shoemaker, D.M., Yunes, N.: Late inspiral and merger of binary black holes in scalar-tensor theories of gravity (2011). ArXiv:1112.3928 [gr-qc]Google Scholar
  90. 90.
    Heller, M.P., Janik, R.A., Witaszczyk, P.: A numerical relativity approach to the initial value problem in asymptotically anti-de Sitter spacetime for plasma thermalization—an ADM formulation. Phys. Rev. D 85, 126002 (2012). doi:10.1103/PhysRevD.85.126002. ArXiv:1203.0755 [hep-th]ADSGoogle Scholar
  91. 91.
    Heller, M.P., Janik, R.A., Witaszczyk, P.: The characteristics of thermalization of boost-invariant plasma from holography. Phys. Rev. Lett. 108, 201602 (2012). doi:10.1103/PhysRevLett.108.201602. ArXiv:1103.3452 [hep-th]ADSGoogle Scholar
  92. 92.
    Heller, M.P., Mateos, D., van der Schee, W., Trancanelli, D.: Strong coupling isotropization of non-abelian plasmas simplified. Phys. Rev. Lett. 108, 191601 (2012). doi:10.1103/PhysRevLett.108.191601. ArXiv:1202.0981 [hep-th]ADSGoogle Scholar
  93. 93.
    Hemberger, D.A., Lovelace, G., Loredo, T.J., Kidder, L.E., Scheel, M.A., Szilágyi, B., Taylor, N.W., Teukolsky, S.A.: Final spin and radiated energy in numerical simulations of binary black holes with equal masses and equal, aligned or anti-aligned spins. Phys. Rev. D 88, 064014 (2013). doi:10.1103/PhysRevD.88.064014. ArXiv:1305.5991 [gr-qc]ADSGoogle Scholar
  94. 94.
    Hilditch, D., Bernuzzi, S., Thierfelder, M., Cao, Z., Tichy, W., Brügmann, B.: Compact binary evolutions with the Z4c formulation. Phys. Rev. D 88, 084057 (2013). doi:10.1103/PhysRevD.88.084057. ArXiv:1212.2901 [gr-qc]ADSGoogle Scholar
  95. 95.
    Hinder, I., et al.: Error-analysis and comparison to analytical models of numerical waveforms produced by the NRAR collaboration . Class. Quant. Grav. 31, 025012 (2014). ArXiv:1307.5307 [gr-qc]Google Scholar
  96. 96.
    Horbatsch, M.W., Burgess, C.P.: Cosmic black-hole hair growth and quasar OJ287. J. Cosmol. Astropart. Phys. 1205, 010 (2012). doi:10.1088/1475-7516/2012/05/010. ArXiv:1111.4009 [gr-qc]ADSGoogle Scholar
  97. 97.
    Horowitz, G.T., Hubeny, V.E.: Quasinormal modes of AdS black holes and the approach to thermal equilibrium. Phys. Rev. D 62, 024027 (2000). doi:10.1103/PhysRevD.62.024027. Hep-th/9909056ADSMathSciNetGoogle Scholar
  98. 98.
    Horowitz, G.T., Santos, J.E., Tong, D.: Further evidence for lattice-induced scaling. JHEP 1211, 102 (2012). doi:10.1007/JHEP11(2012)102. ArXiv:1209.1098 [hep-th]
  99. 99.
    Horowitz, G.T., Santos, J.E., Tong, D.: Optical conductivity with holographic lattices. JHEP 1207, 168 (2012). doi:10.1007/JHEP07(2012)168. ArXiv:1204.0519 [hep-th]
  100. 100.
    Jałmużna, J., Rostworowski, A., Bizoń, P.: A Comment on AdS collapse of a scalar field in higher dimensions. Phys. Rev. D 84, 085021 (2011). doi:10.1103/PhysRevD.84.085021. ArXiv:1108.4539 [gr-qc]ADSGoogle Scholar
  101. 101.
    Kanti, P.: Black holes at the LHC. Lect. Notes Phys. 769, 387–423 (2009). doi:10.1007/978-3-540-88460-6_10. ArXiv:0802.2218 [hep-th]ADSMathSciNetGoogle Scholar
  102. 102.
    Kesden, M., Sperhake, U., Berti, E.: Final spins from the merger of precessing binary black holes. Phys. Rev. D 81, 084054 (2010). ArXiv:1002.2643 [astro-ph]ADSGoogle Scholar
  103. 103.
    Kesden, M., Sperhake, U., Berti, E.: Relativistic suppression of black hole recoils. Astrophys. J. 715, 1006–1011 (2010). ArXiv:1003.4993 [astro-ph]ADSGoogle Scholar
  104. 104.
    Klebanov, I.R.: TASI lectures: introduction to the AdS/CFT correspondence. pp. 615–650 (2000). Hep-th/0009139Google Scholar
  105. 105.
    Komossa, S., Zhou, H., Lu, H.: A recoiling supermassive black hole in the quasar SDSSJ092712.65+294344.0? Astrophys. J. 678, L81 (2008). ArXiv:0804.4585 [astro-ph]ADSGoogle Scholar
  106. 106.
    Lehner, L.: Numerical relatvity: a review. Class. Quantum Grav. 18, R25–R86 (2001). doi:10.1088/0264-9381/18/17/202. Gr-qc/0106072ADSMATHMathSciNetGoogle Scholar
  107. 107.
    Lehner, L., Pretorius, F.: Black strings, low viscosity fluids, and violation of cosmic censorship. Phys. Rev. Lett. 105, 101102 (2010). doi:10.1103/PhysRevLett.105.101102. ArXiv:1006.5960 [hep-th]ADSMathSciNetGoogle Scholar
  108. 108.
    Li, Y., et al.: Formation of \(z\tilde{\;}6\) quasars from hierarchical galaxy mergers. Astrophys. J. 665, 187–208 (2007). Astro-ph/0608190ADSGoogle Scholar
  109. 109.
    Lindblom, L.: Use and abuse of the model waveform accuracy standards. Phys. Rev. D 80, 064019 (2009). ArXiv:0907.0457 [gr-qc]ADSGoogle Scholar
  110. 110.
    Lindblom, L., Baker, J.G., Owen, B.J.: Improved time-domain accuracy standards for model gravitational waveforms. Phys. Rev. D 82, 084020 (2010). ArXiv:1008.1803 [gr-qc]ADSGoogle Scholar
  111. 111.
    MacDonald, I., Nissanke, S., Pfeiffer, H.P.: Suitability of post-Newtonian/numerical-relativity hybrid waveforms for gravitational wave detectors. Class. Quantum Grav. 28, 134002 (2011). doi:10.1088/0264-9381/28/13/134002. ArXiv:1102.5128 [gr-qc]ADSGoogle Scholar
  112. 112.
    Magorrian, J., Tremaine, S., Richstone, D., Bender, R., Bower, G., Dressler, A., Faber, S.M., Gebhardt, K., Green, R., Grillmair, C., Kormendy, J., Lauer, T.: The demography of massive dark objects in galaxy centers. Astron. J 115, 2285–2305 (1998). Astro-ph/9708072ADSGoogle Scholar
  113. 113.
    Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1997). Hep-th/9711200ADSMathSciNetGoogle Scholar
  114. 114.
    Maliborski, M., Rostworowski, A.: Time-periodic solutions in Einstein AdS—massless scalar field system. Phys. Rev. Lett. 111, 051102 (2013). doi:10.1103/PhysRevLett.111.051102. ArXiv:1303.3186 [gr-qc]ADSGoogle Scholar
  115. 115.
    Megevand, M.: Perturbed disks get shocked. Binary black hole merger effects on accretion disks. Phys. Rev. D80, 024012 (2009). doi:10.1103/PhysRevD.80.024012. ArXiv:0905.3390 [astro-ph]ADSGoogle Scholar
  116. 116.
    Merritt, D., Milosavljević, M., Favata, M., Hughes, S., Holz, D.: Consequences of gravitational radiation recoil. Astrophys. J. 607, L9–L12 (2004). Astro-ph/0402057ADSGoogle Scholar
  117. 117.
    Mösta, P., et al.: Vacuum electromagnetic counterparts of binary black-hole mergers. Phys. Rev. D 81, 064017 (2010). doi:10.1103/PhysRevD.81.064017. ArXiv:0912.2330 [gr-qc]ADSGoogle Scholar
  118. 118.
    Mroué, A.H., et al.: A catalog of 171 high-quality binary black-hole simulations for gravitational-wave astronomy. Phys. Rev. Lett. 111, 241104 (2013). ArXiv:1304.6077 [gr-qc]Google Scholar
  119. 119.
    Nastase, H.: Introduction to AdS-CFT (2007). Report number TIT-HEP-578. ArXiv:0712.0689 [hep-th].Google Scholar
  120. 120.
    Okawa, H., Nakao, K.I., Shibata, M.: Is super-Planckian physics visible? Scattering of black holes in 5 dimensions. Phys. Rev. D 83, 121501 (2011). ArXiv:1105.3331 [gr-qc]ADSGoogle Scholar
  121. 121.
    Palenzuela, C., Anderson, M., Lehner, L., Liebling, S.L., Neilsen, D.: Stirring, not shaking: binary black holes’ effects on electromagnetic fields. Phys. Rev. Lett. 103, 081101 (2009). doi:10.1103/PhysRevLett.103.081101. ArXiv:0905.1121 [astro-ph]ADSGoogle Scholar
  122. 122.
    Palenzuela, C., Garrett, T., Lehner, L., Liebling, S.L.: Magnetospheres of black hole systems in force-free plasma. Phys. Rev. D 82, 044045 (2010). doi:10.1103/PhysRevD.82.044045. ArXiv:1007.1198 [gr-qc]ADSGoogle Scholar
  123. 123.
    Palenzuela, C., Lehner, L., Liebling, S.L.: Dual jets from binary black holes. Science 329, 927 (2010). doi:10.1126/science.1191766. ArXiv:1005.1067 [astro-ph]ADSGoogle Scholar
  124. 124.
    Palenzuela, C., Lehner, L., Yoshida, S.: Understanding possible electromagnetic counterparts to loud gravitational wave events: binary black hole effects on electromagnetic fields. Phys. Rev. D 81, 084007 (2010). doi:10.1103/PhysRevD.81.084007. ArXiv:0911.3889 [gr-qc]ADSGoogle Scholar
  125. 125.
    Pan Yi. and Buonanno, A., Boyle, M., Buchman, L.T., Kidder L, E., Pfeiffer, H.P., Scheel, M.A.: Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism. Phys. Rev. D 84, 124052. doi:10.1103/PhysRevD.84.124052. ArXiv:1106.1021 [gr-qc]
  126. 126.
    Pan, Y., Buonanno, A., Buchman, L.T., Chu, T., Kidder, L.E., Pfeiffer, H.P., Scheel, M.A.: Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of nonprecessing, spinning, equal-mass black holes. Phys. Rev. D 81, 084041 (2010). doi:10.1103/PhysRevD.81.084041. ArXiv:0912.3466 [gr-qc]ADSGoogle Scholar
  127. 127.
    Pani, P., Cardoso, V., Gualtieri, L., Berti, E., Ishibashi, A.: Black hole bombs and photon mass bounds. Phys. Rev. Lett. 109, 131102 (2012). doi:10.1103/PhysRevLett.109.131102. ArXiv:1209.0465 [gr-qc]ADSGoogle Scholar
  128. 128.
    Pekowsky, L., O’Shaughnessy, R., Healy, J., Shoemaker, D.: Comparing gravitational waves from nonprecessing and precessing black hole binaries in the corotating frame. Phys. Rev. D 88, 024040 (2013). doi:10.1103/PhysRevD.88.024040. ArXiv:1304.3176 [gr-qc]ADSGoogle Scholar
  129. 129.
    Peres, A.: Classical radiation recoil. Phys. Rev. 128, 2471–2475 (1962)ADSMATHMathSciNetGoogle Scholar
  130. 130.
    Peters, P.C.: Gravitational radiation and the motion of two point masses. Phys. Rev. 136, B1224–B1232 (1964)ADSGoogle Scholar
  131. 131.
    Pfeiffer, H.P.: Numerical simulations of compact object binaries. Class. Quantum Grav. 29, 124004 (2012). doi:10.1088/0264-9381/29/12/124004. ArXiv:1203.5166 [gr-qc]ADSGoogle Scholar
  132. 132.
    Pretorius, F.: Evolution of binary black-hole spacetimes. Phys. Rev. Lett. 95, 121101 (2005). doi:10.1103/PhysRevLett.95.121101. Gr-qc/0507014ADSMathSciNetGoogle Scholar
  133. 133.
    Pretorius, F.: Numerical relativity using a generalized harmonic decomposition. Class. Quantum Grav. 22, 425–452 (2005). doi:10.1088/0264-9381/22/2/014. Gr-qc/0407110ADSMATHMathSciNetGoogle Scholar
  134. 134.
    Pretorius, F.: Binary black hole coalescence. In: Colpi, M., et al. (ed.) Physics of relativistic objects in compact binaries: from birth to coalescence. Springer, New York (2009). ArXiv:0710.1338 [gr-qc]Google Scholar
  135. 135.
    Pretorius, F., Khurana, D.: Black hole mergers and unstable circular orbits. Class. Quantum Grav. 24, S83–S108 (2007). doi:10.1088/0264-9381/24/12/S07. Gr-qc/0702084ADSMATHMathSciNetGoogle Scholar
  136. 136.
    Pürrer, M., Hannam, M., Ajith, P., Husa, S.: Testing the validity of the single-spin approximation in inspiral-merger-ringdown waveforms. Phys. Rev. D 88, 064007 (2013). doi:10.1103/PhysRevD.88.064007. ArXiv:1306.2320 [gr-qc]ADSGoogle Scholar
  137. 137.
    Randall, L., Sundrum, R.: A large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 3370–3373 (1999). doi:10.1103/PhysRevLett.83.3370. Hep-ph/9905221ADSMATHMathSciNetGoogle Scholar
  138. 138.
    Randall, L., Sundrum, R.: An alternative to compactification. Phys. Rev. Lett. 83, 4690–4693 (1999). doi:10.1103/PhysRevLett.83.4690. Hep-th/9906064ADSMATHMathSciNetGoogle Scholar
  139. 139.
    Rees, M.J.: Accretion and the quasar phenomenon. Phys. Scr. 17, 193–200 (1978)ADSGoogle Scholar
  140. 140.
    Rinne, O., Lindblom, L., Scheel, M.A.: Testing outer boundary treatments for the Einstein equations. Class. Quantum Grav. 24, 4053–4078 (2007). doi:10.1088/0264-9381/24/16/006. ArXiv:0704.0782 [gr-qc]ADSMATHMathSciNetGoogle Scholar
  141. 141.
    Ruiz, M., Rinne, O., Sarbach, O.: Outer boundary conditions for Einstein’s field equations in harmonic coordinates. Class. Quantum Grav. 24, 6349–6378 (2007). doi:10.1088/0264-9381/24/24/012. ArXiv:0707.2797 [gr-qc]ADSMATHMathSciNetGoogle Scholar
  142. 142.
    Santamaria, L.: L.: Matching post-newtonian and numerical relativity waveforms: systematic errors and a new phenomenological model for non-precessing black hole binaries. Phys. Rev. D 82, 064016 (2010). ArXiv:1005.3306ADSGoogle Scholar
  143. 143.
    Sasaki, M., Tagoshi, H.: Analytic black hole perturbation approach to gravitational radiation. Living Rev. Rel. 6(6) (2003).
  144. 144.
    Schnittman, J.D.: Spin-orbit resonance and the evolution of compact binary systems. Phys. Rev. D 70, 124020 (2004). Astro-ph/0409174ADSGoogle Scholar
  145. 145.
    Shibata, M., Nakamura, T.: Evolution of three-dimensional gravitational waves: harmonic slicing case. Phys. Rev. D 52, 5428–5444 (1995). doi:10.1103/PhysRevD.52.5428 ADSMATHMathSciNetGoogle Scholar
  146. 146.
    Shibata, M., Okawa, H., Yamamoto, T.: High-velocity collisions of two black holes. Phys. Rev. D 78, 101501(R) (2008). doi:10.1103/PhysRevD.78.101501. ArXiv:0810.4735 [gr-qc]ADSGoogle Scholar
  147. 147.
    Shibata, M., Yoshino, H.: Bar-mode instability of rapidly spinning black hole in higher dimensions: numerical simulation in general relativity. Phys. Rev. D 81, 104035 (2010). doi:10.1103/PhysRevD.81.104035. ArXiv:1004.4970 [gr-qc]
  148. 148.
    Shibata, M., Yoshino, H.: Nonaxisymmetric instability of rapidly rotating black hole in five dimensions. Phys. Rev. D 81, 021501 (2010). doi:10.1103/PhysRevD.81.021501. ArXiv:0912.3606 [gr-qc]ADSGoogle Scholar
  149. 149.
    Sorkin, E., Oren, Y.: On Choptuik’s scaling in higher dimensions. Phys. Rev. D 71, 124005 (2005). doi:10.1103/PhysRevD.71.124005. Hep-th/0502034ADSGoogle Scholar
  150. 150.
    Sperhake, U.: Numerical relativity in higher dimensions. Int. J. Mod. Phys. D 22, 1330005 (2013). doi:10.1142/S021827181330005X. ArXiv:1301.3772 [gr-qc]ADSMathSciNetGoogle Scholar
  151. 151.
    Sperhake, U., Berti, E., Cardoso, V.: Numerical simulations of black-hole binaries and gravitational wave emission. C. R. Phys. 14, 306–317 (2013). doi:10.1016/j.crhy.2013.01.004. ArXiv:1107.2819 [gr-qc]ADSGoogle Scholar
  152. 152.
    Sperhake, U., Berti, E., Cardoso, V., Pretorius, F.: Universality, maximum radiation and absorption in high-energy collisions of black holes with spin. Phys. Rev. Lett. 111, 041101 (2013). doi:10.1103/PhysRevLett.111.041101. ArXiv:1211.6114 [gr-qc]ADSGoogle Scholar
  153. 153.
    Sperhake, U., Brügmann, B., Müller, D., Sopuerta, C.F.: 11-orbit inspiral of a mass ratio 4:1 black-hole binary. Class. Quantum Grav. 28, 134004 (2011). doi:10.1088/0264-9381/28/13/134004. ArXiv:1012.3173 [gr-qc]
  154. 154.
    Sperhake, U., Cardoso, V., Pretorius, F., Berti, E., González, J.A.: The high-energy collision of two black holes. Phys. Rev. Lett. 101, 161101 (2008). doi:10.1103/PhysRevLett.101.161101. ArXiv:0806.1738 [gr-qc]ADSGoogle Scholar
  155. 155.
    Sperhake, U., Cardoso, V., Pretorius, F., Berti, E., Hinderer, T., Yunes, N.: Cross section, final spin and zoom-whirl behavior in high-energy black hole collisions. Phys. Rev. Lett. 103, 131102 (2009). doi:10.1103/PhysRevLett.103.131102. ArXiv:0907.1252 [gr-qc]ADSGoogle Scholar
  156. 156.
    Sturani, R., et al.: Complete phenomenological gravitational waveforms from spinning coalescing binaries. J. Phys. Conf. Ser. 243, 012007 (2010). ArXiv:1005.0551 [gr-qc]ADSGoogle Scholar
  157. 157.
    Taracchini, A.: A.: A prototype effective-one-body model for non-precessing spinning inspiral-merger-ringdown waveforms. Phys. Rev. D 86, 024011 (2012). doi:10.1103/PhysRevD.86.024011. ArXiv:1202.0790 [gr-qc]ADSGoogle Scholar
  158. 158.
    Taracchini, A, et al.: Effective-one-body model for black-hole binaries with generic mass ratios and spins (2013). ArXiv:1311.2544 [gr-qc]Google Scholar
  159. 159.
    van Paradijs, J.: Gamma-ray Bursts. In: American Astronomical Society Meeting Abstracts. Bulletin of the American Astronomical Society, vol. 30, pp. 1291-+ (1998). ArXiv:astro-ph/9802177.Google Scholar
  160. 160.
    Volonteri, M., Gültekin, K., Dotti, M.: Gravitational recoil: effects on massive black hole occupation fraction over cosmic time. MNRAS 404, 2143–2150 (2010). ArXiv:1001.1743 [astro-ph]ADSGoogle Scholar
  161. 161.
    Witek, H., Cardoso, V., Gualtieri, L., Herdeiro, C., Sperhake, U., Zilhão, M.: Head-on collisions of unequal mass black holes in \(d=5\) dimensions. Phys. Rev. D 83, 044017 (2011). ArXiv:1011.0742 [gr-qc]ADSGoogle Scholar
  162. 162.
    Witek, H., Cardoso, V., Ishibashi, A., Sperhake, U.: Superradiant instabilities in astrophysical systems. Phys. Rev. D 87, 043513 (2013). doi:10.1103/PhysRevD.87.043513. ArXiv:1212.0551 [gr-qc]ADSGoogle Scholar
  163. 163.
    Witek, H., Zilhão, M., Gualtieri, L., Cardoso, V., Herdeiro, C., Nerozzi, A., Sperhake, U.: Numerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction. Phys. Rev. D 82, 104014 (2010). doi:10.1103/PhysRevD.82.104014. ArXiv:1006.3081 [gr-qc]ADSGoogle Scholar
  164. 164.
    Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998). Hep-th/9802150ADSMATHMathSciNetGoogle Scholar
  165. 165.
    Yoshino, H., Shibata, M.: Higher-dimensional numerical relativity: current status. Prog. Theor. Phys. Suppl. 189, 269–310 (2011). doi:10.1143/PTPS.189.269 ADSMATHGoogle Scholar
  166. 166.
    Zel’dovich, Y.B.: Pis’ma. Zh. Eksp. Teor. Fiz. 14, 270 (1971)Google Scholar
  167. 167.
    Zel’dovich, Y.B.: Zh. Eksp. Teor. Fiz 62, 2076 (1972)Google Scholar
  168. 168.
    Zilhão, M.: New frontiers in Numerical Relativity. Ph.D. thesis, University of Porto (2012). ArXiv:1301.1509 [gr-qc]Google Scholar
  169. 169.
    Zilhão, M., Ansorg, M., Cardoso, V., Gualtieri, L., Herdeiro, C., Sperhake, U., Witek, H.: Higher-dimensional puncture initial data. Phys. Rev. D 84, 084039 (2011). ArXiv:1109.2149 [gr-qc]ADSGoogle Scholar
  170. 170.
    Zilhao, M., Cardoso, V., Gualtieri, L., Herdeiro, C., Sperhake, U., Witek, H.: Dynamics of black holes in de Sitter spacetimes. Phys. Rev. D 85, 104039 (2012). doi:10.1103/PhysRevD.85.104039. ArXiv:1204.2019 [gr-qc]ADSGoogle Scholar
  171. 171.
    Zilhao, M., Cardoso, V., Herdeiro, C., Lehner, L., Sperhake, U.: Collisions of charged black holes. Phys. Rev. D 85, 124062 (2012). doi:10.1103/PhysRevD.85.124062. ArXiv:1205.1063 [gr-qc].
  172. 172.
    Zilhão, M., Cardoso, V., Herdeiro, C., Lehner, L., Sperhake, U.: Collisions of oppositely charged black holes. Phys. Rev. D 89, 044008 (2014). ArXiv:1311.6483 [gr-qc]Google Scholar
  173. 173.
    Zilhão, M., Witek, H., Sperhake, U., Cardoso, V., Gualtieri, L., Herdeiro, C., Nerozzi, A.: Numerical relativity for D dimensional axially symmetric space-times: formalism and code tests. Phys. Rev. D 81, 084052 (2010). doi:10.1103/PhysRevD.81.084052. ArXiv:1001.2302 [gr-qc]ADSGoogle Scholar
  174. 174.
  175. 175.

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical SciencesUniversity of CambridgeCambridge UK

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