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Covariant and gauge-invariant metric-based gravitational-waves extraction in numerical relativity

Joan Fontbuté1,*, Sebastiano Bernuzzi1,†, Simone Albanesi1,2, David Radice3,4,5, Alireza Rashti3,4, William Cook1, Boris Daszuta1, and Alessandro Nagar2,6

  • *Contact author: joan.fontbute@uni-jena.de
  • †Contact author: sebastiano.bernuzzi@uni-jena.de
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Phys. Rev. D 113, 064036 – Published 20 March, 2026

DOI: https://doi.org/10.1103/ndls-mfrn

Abstract

We revisit the problem of gravitational-wave extraction in numerical relativity with gauge-invariant metric perturbation theory of spherical spacetimes. Our extraction algorithm allows the computation of even-parity (Zerilli-Moncrief) and odd-parity (Regge-Wheeler) multipoles of the strain from a (3+1) metric without the assumption that the spherical background is in Schwarzschild coordinates. The algorithm is validated with a comprehensive suite of three-dimensional problems including fluid (f-modes) and spacetime (w-modes) perturbations of neutron stars, gravitational collapse of rotating neutron stars, circular binary black holes mergers and black hole dynamical captures, and binary neutron star mergers. We find that metric extraction is robust in all the considered scenarios and delivers waveforms of overall quality similar to curvature (Weyl) extraction. Metric extraction is particularly valuable in identifying waveform systematics for problems in which the reconstruction of the strain from the Weyl multipoles is ambiguous. Direct comparison of different choices for the gauge-invariant master functions show very good agreement in the even-parity sector. Instead, in the odd-parity sector, assuming the background in Schwarzschild coordinates can minimize gauge effects related to the use of the Γ-driver shift. Moreover, for optimal choices of the extraction radius, a simple extrapolation to null infinity can deliver waveforms compatible to Cauchy-characteristic extrapolated waveforms.

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References (92)

  1. E. Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity, Lecture Notes in Physics (Springer, Berlin, 2012).
  2. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
  3. F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970).
  4. V. Moncrief, Ann. Phys. (N.Y.) 88, 323 (1974).
  5. V. Moncrief, Ann. Phys. (N.Y.) 88, 343 (1974).
  6. U. H. Gerlach and U. K. Sengupta, Phys. Rev. D 19, 2268 (1979).
  7. C. Gundlach and J. M. Martin-Garcia, Phys. Rev. D 61, 084024 (2000).
  8. J. M. Martin-Garcia and C. Gundlach, Phys. Rev. D 64, 024012 (2001).
  9. O. Sarbach and M. Tiglio, Phys. Rev. D 64, 084016 (2001).
  10. K. Martel and E. Poisson, Phys. Rev. D 71, 104003 (2005).
  11. A. M. Abrahams and R. H. Price, Phys. Rev. D 53, 1963 (1996).
  12. K. Camarda and E. Seidel, Phys. Rev. D 59, 064019 (1999).
  13. E. PazosE. Pazos, E. N. Dorband, A. Nagar, C. Palenzuela, E. Schnetter, and M. Tiglio, Classical Quantum Gravity 24, S341 (2007).
  14. D. Pollney, C. Reisswig, L. Rezzolla, B. Szilagyi, M. Ansorg et al., Phys. Rev. D 76, 124002 (2007).
  15. L. Baiotti, S. Bernuzzi, G. Corvino, R. De Pietri, and A. Nagar, Phys. Rev. D 79, 024002 (2009).
  16. A. Buonanno, Y. Pan, H. P. Pfeiffer, M. A. Scheel, L. T. Buchman, and L. E. Kidder, Phys. Rev. D 79, 124028 (2009).
  17. H. Witek, M. Zilhao, L. Gualtieri, V. Cardoso, C. Herdeiro, A. Nerozzi, and U. Sperhake, Phys. Rev. D 82, 104014 (2010).
  18. C. Reisswig, R. Haas, C. D. Ott, E. Abdikamalov, P. Mösta, D. Pollney, and E. Schnetter, Phys. Rev. D 87, 064023 (2013).
  19. F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005).
  20. J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006).
  21. M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006).
  22. E. Newman and R. Penrose, J. Math. Phys. (N.Y.) 3, 566 (1962).
  23. J. G. Baker, M. Campanelli, and C. O. Lousto, Phys. Rev. D 65, 044001 (2002).
  24. M. Campanelli, B. J. Kelly, and C. O. Lousto, Phys. Rev. D 73, 064005 (2006).
  25. B. Brügmann, J. A. Gonzalez, M. Hannam, S. Husa, U. Sperhake, and W. Tichy, Phys. Rev. D 77, 024027 (2008).
  26. M. A. Scheel, M. Boyle, T. Chu, L. E. Kidder, K. D. Matthews, and H. P. Pfeiffer, Phys. Rev. D 79, 024003 (2009).
  27. W. G. Cook and U. Sperhake, Classical Quantum Gravity 34, 035010 (2017).
  28. C. Reisswig, C. D. Ott, U. Sperhake, and E. Schnetter, Phys. Rev. D 83, 064008 (2011).
  29. W. G. Cook, U. Sperhake, E. Berti, and V. Cardoso, Phys. Rev. D 96, 124006 (2017).
  30. C. Reisswig and D. Pollney, Classical Quantum Gravity 28, 195015 (2011).
  31. S. Albanesi, A. Rashti, F. Zappa, R. Gamba, W. Cook, B. Daszuta, S. Bernuzzi, A. Nagar, and D. Radice, Phys. Rev. D 111, 024069 (2025).
  32. T. Dietrich and S. Bernuzzi, Phys. Rev. D 91, 044039 (2015).
  33. C. O. Lousto, H. Nakano, Y. Zlochower, and M. Campanelli, Phys. Rev. D 82, 104057 (2010).
  34. S. Bernuzzi, M. Thierfelder, and B. Brügmann, Phys. Rev. D 85, 104030 (2012).
  35. H. Nakano, J. Healy, C. O. Lousto, and Y. Zlochower, Phys. Rev. D 91, 104022 (2015).
  36. N. T. Bishop, R. Gomez, L. Lehner, and J. Winicour, Phys. Rev. D 54, 6153 (1996).
  37. N. T. Bishop, R. Gomez, L. Lehner, M. Maharaj, and J. Winicour, Phys. Rev. D 56, 6298 (1997).
  38. M. Babiuc, N. Bishop, B. Szilagyi, and J. Winicour, Phys. Rev. D 79, 084011 (2009).
  39. C. Reisswig, N. T. Bishop, C. W. Lai, J. Thornburg, and B. Szilagyi, Classical Quantum Gravity 24, S327 (2007).
  40. C. Reisswig, N. T. Bishop, D. Pollney, and B. Szilagyi, Classical Quantum Gravity 27, 075014 (2010).
  41. J. Moxon, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. D 102, 044052 (2020).
  42. A. Rashti, R. Gamba, K. Chandra, D. Radice, B. Daszuta, W. Cook, and S. Bernuzzi, Phys. Rev. D 111, 104078 (2025).
  43. R. A. d’Inverno and J. A. Vickers, Phys. Rev. D 54, 4919 (1996).
  44. N. T. Bishop, R. Gomez, L. Lehner, B. Szilagyi, J. Winicour, and R. A. Isaacson, Cauchy characteristic matching, in Black Holes, Gravitational Radiation and the Universe, edited by B. R. Iyer and B. Bhawal (Springer, Dordrecht, 1999), pp. 383–408.
  45. C. Reisswig, N. Bishop, D. Pollney, and B. Szilagyi, Phys. Rev. Lett. 103, 221101 (2009).
  46. S. Ma et al., Phys. Rev. D 109, 124027 (2024).
  47. J. Winicour, Living Rev. Relativity 12, 3 (2009).
  48. H. Friedrich, Commun. Math. Phys. 91, 445 (1983).
  49. J. Frauendiener, Phys. Rev. D 58, 064002 (1998).
  50. J. Frauendiener, Phys. Rev. D 58, 064003 (1998).
  51. J. Frauendiener, Classical Quantum Gravity 17, 373 (2000).
  52. J. Frauendiener, Living Rev. Relativity 3, 4 (2000).
  53. A. Zenginoğlu, Classical Quantum Gravity 25, 195025 (2008).
  54. D. Hilditch, E. Harms, M. Bugner, H. Rüter, and B. Brügmann, Classical Quantum Gravity 35, 055003 (2018).
  55. E. Gasperin, S. Gautam, D. Hilditch, and A. Vañó Viñuales, Classical Quantum Gravity 37, 035006 (2020).
  56. M. Duarte, J. C. Feng, E. Gasperín, and D. Hilditch, Classical Quantum Gravity 40, 025011 (2023).
  57. C. Peterson, S. Gautam, A. Vañó Viñuales, and D. Hilditch, Phys. Rev. D 110, 124033 (2024).
  58. S. Bernuzzi, J. Fontbuté, S. Albanesi, and A. Zenginoğlu, Phys. Rev. D 112, 084036 (2025).
  59. C. T. Cunningham, R. H. Price, and V. Moncrief, Astrophys. J. 224, 643 (1978).
  60. A. Nagar and L. Rezzolla, Classical Quantum Gravity 22, R167 (2005).
  61. B. Daszuta, F. Zappa, W. Cook, D. Radice, S. Bernuzzi, and V. Morozova, Astrophys. J. Suppl. Ser. 257, 25 (2021).
  62. W. Cook, B. Daszuta, J. Fields, P. Hammond, S. Albanesi, F. Zappa, S. Bernuzzi, and D. Radice, Astrophys. J. Suppl. Ser. 277, 3 (2025).
  63. B. Daszuta, W. Cook, P. Hammond, J. Fields, E. M. Gutiérrez, S. Bernuzzi, and D. Radice, Phys. Rev. D 112, 103006 (2025).
  64. S. Bernuzzi and D. Hilditch, Phys. Rev. D 81, 084003 (2010).
  65. D. Hilditch, S. Bernuzzi, M. Thierfelder, Z. Cao, W. Tichy, and B. Bruegmann, Phys. Rev. D 88, 084057 (2013).
  66. A. Rashti, M. Bhattacharyya, D. Radice, B. Daszuta, W. Cook, and S. Bernuzzi, Classical Quantum Gravity 41, 095001 (2024).
  67. M. C. Babiuc, B. Szilagyi, J. Winicour, and Y. Zlochower, Phys. Rev. D 84, 044057 (2011).
  68. E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M. Hannam, S. Husa, and B. Brügmann, Phys. Rev. D 76, 064034 (2007).
  69. J. S. Read, B. D. Lackey, B. J. Owen, and J. L. Friedman, Phys. Rev. D 79, 124032 (2009).
  70. K. D. Kokkotas and B. F. Schutz, Mon. Not. R. Astron. Soc. 225, 119 (1992).
  71. S. Bernuzzi and A. Nagar, Phys. Rev. D 78, 024024 (2008).
  72. N. Stergioulas, K. D. Kokkotas, and I. Hawke, in Recent Advances in Astronomy Astrophysics, American Institute of Physics Conference Series Vol. 848, edited by N. Solomos (AIP, Lixourion, Kefallinia Island, 2006), pp. 730–737.
  73. L. Baiotti, I. Hawke, and L. Rezzolla, Classical Quantum Gravity 24, S187 (2007).
  74. N. Stergioulas and J. L. Friedman, Astrophys. J. 444, 306 (1995).
  75. M. Davis, R. Ruffini, and J. Tiomno, Phys. Rev. D 5, 2932 (1972).
  76. M. Ansorg, B. Brügmann, and W. Tichy, Phys. Rev. D 70, 064011 (2004).
  77. SXS Gravitational Waveform Database, https://data.black-holes.org/waveforms/index.html.
  78. T. Damour, A. Nagar, E. N. Dorband, D. Pollney, and L. Rezzolla, Phys. Rev. D 77, 084017 (2008).
  79. A. Nagar, D. Chiaramello, R. Gamba, S. Albanesi, S. Bernuzzi, V. Fantini, M. Panzeri, and P. Rettegno, Phys. Rev. D 111, 064050 (2025).
  80. S. Albanesi, R. Gamba, S. Bernuzzi, J. Fontbuté, A. Gonzalez, and A. Nagar, Phys. Rev. D 112, L121503 (2025).
  81. T. Andrade et al., Phys. Rev. D 109, 084025 (2024).
  82. D. Chiaramello and A. Nagar, Phys. Rev. D 101, 101501 (2020).
  83. D. Radice, S. Bernuzzi, and C. D. Ott, Phys. Rev. D 94, 064011 (2016).
  84. T. Dietrich, S. Bernuzzi, and W. Tichy, Phys. Rev. D 96, 121501 (2017).
  85. T. Dietrich, D. Radice, S. Bernuzzi, F. Zappa, A. Perego, B. Brügmann, S. V. Chaurasia, R. Dudi, W. Tichy, and M. Ujevic, Classical Quantum Gravity 35, 24LT01 (2018).
  86. E. Gourgoulhon, P. Grandclement, K. Taniguchi, J.-A. Marck, and S. Bonazzola, Phys. Rev. D 63, 064029 (2001).
  87. S. Bernuzzi, A. Nagar, T. Dietrich, and T. Damour, Phys. Rev. Lett. 114, 161103 (2015).
  88. www.gauss-centre.eu.
  89. J. Mathews, J. Soc. Ind. Appl. Math. 10, 768 (1962).
  90. F. J. Zerilli, J. Math. Phys. (N.Y.) 11, 2203 (1970).
  91. J. N. Goldberg, A. J. MacFarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, J. Math. Phys. (N.Y.) 8, 2155 (1967).
  92. K. S. Thorne, Rev. Mod. Phys. 52, 299 (1980).

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Phys. Rev. D 113, 064036– Published 20 March, 2026
  • Received 6 October 2025
  • Accepted 11 February 2026
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