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Maximum Entropy Conjecture for Black Hole Mergers

Monica Rincon-Ramirez1,2,*, Nathan K. Johnson-McDaniel3,†, Eugenio Bianchi1,2, Ish Gupta4,5,6,1,2, Vaishak Prasad1,2, and B. S. Sathyaprakash1,2

  • *Contact author: mramirez@alumni.psu.edu
  • †Contact author: nkjm.physics@gmail.com
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Phys. Rev. Lett. 137, 021406 – Published 7 July, 2026

DOI: https://doi.org/10.1103/hvp6-ydbq

Abstract

The final state of a binary black hole merger is predicted with high precision by numerical relativity, but could there be a simple thermodynamic principle within general relativity that governs the selection of the remnant? Using post-Newtonian relations between the mass M (including the binding energy) and angular momentum J of quasicircular, nonspinning binaries, we uncover a puzzling result: When the binary’s instantaneous M and J are mapped to those of a hypothetical Kerr black hole, the corresponding entropy exhibits a maximum during the evolution. This maximum occurs at values of M and J, strikingly close to those of the final remnant predicted by numerical relativity. Consistent behavior is observed when using the relation between M and J obtained from numerical relativity evolution. Although this procedure is somewhat ad hoc, the agreement between the masses and spins of the final state obtained from numerical relativity and the results of this maximum entropy procedure is remarkable, with agreement to within a few percent when using either post-Newtonian or numerical relativity results for M and J. These findings allow us to propose an entropy maximization conjecture for binary black hole mergers, hinting that thermodynamic principles may govern the selection of the final black hole state.

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References (47)

  1. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), GWTC-4.0: Updating the gravitational-wave transient catalog with observations from the first part of the fourth LIGO-Virgo-KAGRA observing run, arXiv:2508.18082.
  2. T. W. Baumgarte and S. L. Shapiro, Numerical Relativity: Solving Einstein’s Equations on the Computer (Cambridge University Press, Cambridge, England, 2010), 10.1017/CBO9781139193344.
  3. M. A. Scheel et al., The SXS collaboration’s third catalog of binary black hole simulations, Classical Quantum Gravity 42, 195017 (2025).
  4. M. A. Scheel, M. Boyle, T. Chu, L. E. Kidder, K. D. Matthews, and H. P. Pfeiffer, High-accuracy waveforms for binary black hole inspiral, merger, and ringdown, Phys. Rev. D 79, 024003 (2009).

  5. We work in units with c=1, G=1, and kB=1, keeping track of Planck’s constant ℏ to distinguish quantum from classical phenomena.

  6. A. Buonanno, L. E. Kidder, and L. Lehner, Estimating the final spin of a binary black hole coalescence, Phys. Rev. D 77, 026004 (2008).
  7. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Properties of the binary black hole merger GW150914, Phys. Rev. Lett. 116, 241102 (2016).
  8. P. C. W. Davies, Thermodynamics of black holes, Proc. R. Soc. A 353, 499 (1977).
  9. F. Hofmann, E. Barausse, and L. Rezzolla, The final spin from binary black holes in quasi-circular orbits, Astrophys. J. Lett. 825, L19 (2016).
  10. J. Healy and C. O. Lousto, Remnant of binary black-hole mergers: New simulations and peak luminosity studies, Phys. Rev. D 95, 024037 (2017).
  11. X. Jiménez-Forteza, D. Keitel, S. Husa, M. Hannam, S. Khan, and M. Pürrer, Hierarchical data-driven approach to fitting numerical relativity data for nonprecessing binary black holes with an application to final spin and radiated energy, Phys. Rev. D 95, 064024 (2017).
  12. V. Varma, S. E. Field, M. A. Scheel, J. Blackman, D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeiffer, Surrogate models for precessing binary black hole simulations with unequal masses, Phys. Rev. Res. 1, 033015 (2019).
  13. J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31, 161 (1973).
  14. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  15. S. W. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett. 26, 1344 (1971).
  16. A. Ashtekar and B. Krishnan, Quasi-local black hole horizons: Recent advances, Living Rev. Relativity 28, 8 (2025).
  17. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), GW250114: Testing Hawking’s area law and the Kerr nature of black holes, Phys. Rev. Lett. 135, 111403 (2025).
  18. E. Fermi, Thermodynamics (Dover Publications, New York, 1956) reprint of lectures delivered at Columbia University.
  19. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1991).
  20. L. Blanchet, Post-Newtonian theory for gravitational waves, Living Rev. Relativity 27, 4 (2024).
  21. L. Blanchet, D. Langlois, and E. Ligout, Innermost stable circular orbit of arbitrary-mass compact binaries at fourth post-Newtonian order, Phys. Rev. D 112, 064025 (2025).
  22. M. Cabero, A. B. Nielsen, A. P. Lundgren, and C. D. Capano, Minimum energy and the end of the inspiral in the post-Newtonian approximation, Phys. Rev. D 95, 064016 (2017).

  23. It is important to note that the post-Newtonian expansion is an asymptotic series that is not guaranteed to monotonically approach its limiting value [20, 24, 25].

  24. N. Yunes and E. Berti, Accuracy of the post-Newtonian approximation: Optimal asymptotic expansion for quasicircular, extreme-mass ratio inspirals, Phys. Rev. D 77, 124006 (2008); 83, 109901(E) (2011).
  25. T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Improved filters for gravitational waves from inspiralling compact binaries, Phys. Rev. D 57, 885 (1998).
  26. T. Damour, P. Jaranowski, and G. Schäfer, Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems, Phys. Rev. D 89, 064058 (2014).
  27. R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).
  28. A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Living Rev. Relativity 7, 10 (2004).
  29. M. A. Scheel, M. Giesler, D. A. Hemberger, G. Lovelace, K. Kuper, M. Boyle, B. Szilágyi, and L. E. Kidder, Improved methods for simulating nearly extremal binary black holes, Classical Quantum Gravity 32, 105009 (2015).
  30. A. Gupta, B. Krishnan, A. B. Nielsen, and E. Schnetter, Dynamics of marginally trapped surfaces in a binary black hole merger: Growth and approach to equilibrium, Phys. Rev. D 97, 084028 (2018).
  31. T. Mädler and J. Winicour, Bondi-Sachs formalism, Scholarpedia 11, 33528 (2016).
  32. J. L. Friedman, K. Uryu, and M. Shibata, Thermodynamics of binary black holes and neutron stars, Phys. Rev. D 65, 064035 (2002); 70, 129904(E) (2004).
  33. A. Le Tiec and P. Grandclément, Horizon surface gravity in corotating black hole binaries, Classical Quantum Gravity 35, 144002 (2018).
  34. A. Le Tiec, L. Blanchet, and B. F. Whiting, The first law of binary black hole mechanics in general relativity and post-Newtonian theory, Phys. Rev. D 85, 064039 (2012).
  35. LIGO-Virgo-KAGRA Algorithm Library—lalsuite, free software (GPL), 10.7935/GT1W-FZ16 (2025).
  36. J. D. Hunter, matplotlib: A 2D graphics environment, Comput. Sci. Eng. 9, 90 (2007).
  37. C. R. Harris et al., Array programming with numpy, Nature (London) 585, 357 (2020).
  38. M. Boyle, K. Mitman, M. A. Scheel, and L. C. Stein, The sxs package, 10.5281/zenodo.4034006 (2025).
  39. N. K. Johnson-McDaniel and M. Rincon-Ramirez, BBH Entropy Conjecture Scripts (2026), https://gitlab.com/johnsonmcdaniel/bbh_entropy_conjecture_scripts.
  40. T. Damour, A. Nagar, D. Pollney, and C. Reisswig, Energy versus angular momentum in black hole binaries, Phys. Rev. Lett. 108, 131101 (2012).
  41. A. Nagar, T. Damour, C. Reisswig, and D. Pollney, Energetics and phasing of nonprecessing spinning coalescing black hole binaries, Phys. Rev. D 93, 044046 (2016).
  42. S. Ossokine, T. Dietrich, E. Foley, R. Katebi, and G. Lovelace, Assessing the energetics of spinning binary black hole systems, Phys. Rev. D 98, 104057 (2018).
  43. M. Boyle et al., The SXS Collaboration catalog of binary black hole simulations, Classical Quantum Gravity 36, 195006 (2019).
  44. M. Rincon-Ramirez, N. K. Johnson-McDaniel, I. Gupta, V. Prasad, E. Bianchi, and B. S. Sathyaprakash, Examining the maximum entropy principle for spinning black hole mergers (in preparation).
  45. J. L. Jaramillo, R. P. Macedo, P. Moesta, and L. Rezzolla, Towards a cross-correlation approach to strong-field dynamics in Black Hole spacetimes, AIP Conf. Proc. 1458, 158 (2012).
  46. Y. Chen et al., Multipole moments on the common horizon in a binary-black-hole simulation, Phys. Rev. D 106, 124045 (2022).
  47. V. Prasad, Tidal deformation of dynamical horizons in binary black hole mergers and its imprint on gravitational radiation, Phys. Rev. D 109, 044033 (2024).

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Phys. Rev. Lett. 137, 021406– Published 7 July, 2026
  • Received 3 February 2026
  • Accepted 13 May 2026
Article part of CHORUS
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