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Slowly rotating black holes in degenerate higher order scalar-tensor theories

Hugo Candan1,2, Karim Noui2, and David Langlois3

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Phys. Rev. D 114, 024086 – Published 29 July, 2026

DOI: https://doi.org/10.1103/8d8w-hnjg

Abstract

We study slowly rotating black hole solutions within degenerate higher order scalar tensor (DHOST) theories. Starting from a static, spherically symmetric metric solution of a DHOST theory, we employ the Hartle-Thorne Ansatz to model a slowly rotating spacetime. We show that the differential equation governing the frame-dragging function ω (which is supposed to depend on the radial coordinate only) is integrable for any DHOST theory allowing us to obtain its explicit form. We also consider angular dependence in ω and show that, for specific DHOST theories, regularity at the horizon and at infinity forbids it, as in general relativity. As an illustration of the formalism introduced here, we study the slowly rotating version of black hole solutions with primary hair obtained recently, examining the influence of the rotation on the innermost stable circular orbit (ISCO) and on the circular light trajectories in the equatorial plane.

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References (59)

  1. D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. (N.Y.) 12, 498 (1971).
  2. D. Langlois and K. Noui, Degenerate higher derivative theories beyond Horndeski: Evading the Ostrogradski instability, J. Cosmol. Astropart. Phys. 02 (2016) 034.
  3. D. Langlois and K. Noui, Hamiltonian analysis of higher derivative scalar-tensor theories, J. Cosmol. Astropart. Phys. 07 (2016) 016.
  4. J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and G. Tasinato, Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order, J. High Energy Phys. 12 (2016) 100.
  5. A. De Felice, D. Langlois, S. Mukohyama, K. Noui, and A. Wang, Generalized instantaneous modes in higher-order scalar-tensor theories, Phys. Rev. D 98, 084024 (2018).
  6. D. Langlois, Dark energy and modified gravity in degenerate higher-order scalar–tensor (DHOST) theories: A review, Int. J. Mod. Phys. D 28, 1942006 (2019).
  7. B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26, 331 (1971).
  8. R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).
  9. L. Hui and A. Nicolis, No-hair theorem for the Galileon, Phys. Rev. Lett. 110, 241104 (2013).
  10. L. Capuano, L. Santoni, and E. Barausse, Black hole hairs in scalar-tensor gravity and the lack thereof, Phys. Rev. D 108, 064058 (2023).
  11. E. Babichev, C. Charmousis, and A. Lehébel, Asymptotically flat black holes in Horndeski theory and beyond, J. Cosmol. Astropart. Phys. 04 (2017) 027.
  12. E. Babichev, C. Charmousis, and N. Lecoeur, Exact Black Hole Solutions in Higher-Order Scalar-Tensor Theories (Springer, Cham, 2024).
  13. A. Bakopoulos, C. Charmousis, P. Kanti, N. Lecoeur, and T. Nakas, Black holes with primary scalar hair, Phys. Rev. D 109, 024032 (2024).
  14. C. Charmousis, M. Crisostomi, R. Gregory, and N. Stergioulas, Rotating black holes in higher order gravity, Phys. Rev. D 100, 084020 (2019).
  15. T. Anson, E. Babichev, C. Charmousis, and M. Hassaine, Disforming the Kerr metric, J. High Energy Phys. 01 (2021) 018.
  16. J. Ben Achour, H. Liu, H. Motohashi, S. Mukohyama, and K. Noui, On rotating black holes in DHOST theories, J. Cosmol. Astropart. Phys. 11 (2020) 001.
  17. J. B. Achour, A. De Felice, M. A. Gorji, S. Mukohyama, and M. C. Pookkillath, Disformal map and Petrov classification in modified gravity, J. Cosmol. Astropart. Phys. 10 (2021) 067.
  18. E. Babichev, C. Charmousis, D. D. Doneva, G. N. Gyulchev, and S. S. Yazadjiev, Testing disformal noncircular deformation of Kerr black holes with LISA, J. Cosmol. Astropart. Phys. 06 (2024) 065.
  19. J. Ben Achour, A. Cisterna, and H. Roussille, A circular Disformal Kerr black hole, J. Cosmol. Astropart. Phys. 04 (2026) 041.
  20. E. Babichev and J. Mazza, Properties of general stationary axisymmetric spacetimes: Circularity and beyond, J. Cosmol. Astropart. Phys. 10 (2025) 011.
  21. K. Van Aelst, E. Gourgoulhon, P. Grandclément, and C. Charmousis, Hairy rotating black holes in cubic Galileon theory, Classical Quantum Gravity 37, 035007 (2020).
  22. P. Grandclément, Fully consistent rotating black holes in the cubic Galileon theory, Classical Quantum Gravity 41, 025012 (2024).
  23. B. Kleihaus, J. Kunz, and E. Radu, Rotating black holes in Dilatonic Einstein-Gauss-Bonnet theory, Phys. Rev. Lett. 106, 151104 (2011).
  24. B. Kleihaus, J. Kunz, S. Mojica, and E. Radu, Spinning black holes in Einstein-Gauss-Bonnet-Dilaton theory: Nonperturbative solutions, Phys. Rev. D 93, 044047 (2016).
  25. J. F. M. Delgado, C. A. R. Herdeiro, and E. Radu, Spinning black holes in shift-symmetric Horndeski theory, J. High Energy Phys. 04 (2020) 180.
  26. C. Charmousis, A. Lehébel, E. Smyrniotis, and N. Stergioulas, Astrophysical constraints on compact objects in 4D Einstein-Gauss-Bonnet gravity, J. Cosmol. Astropart. Phys. 02 (2022) 033.
  27. M. Gammon and R. B. Mann, Slowly rotating black holes in 4D Einstein-Gauss-Bonnet gravity, Phys. Rev. D 110, 044004 (2024).
  28. A. Maselli, H. O. Silva, M. Minamitsuji, and E. Berti, Slowly rotating black hole solutions in Horndeski gravity, Phys. Rev. D 92, 104049 (2015).
  29. J. Lense and H. Thirring, On the influence of the proper rotation of a central body on the motion of the planets and the moon, according to Einstein’s theory of gravitation, Z. Phys. 19, 156 (1918).
  30. J. B. Hartle, Slowly rotating relativistic stars. 1. Equations of structure, Astrophys. J. 150, 1005 (1967).
  31. J. B. Hartle and K. S. Thorne, Slowly rotating relativistic stars. II. Models for neutron stars and supermassive stars, Astrophys. J. 153, 807 (1968).
  32. C. Charmousis, S. Iteanu, D. Langlois, and K. Noui, Axial perturbations of black holes with primary scalar hair, J. Cosmol. Astropart. Phys. 05 (2025) 102.
  33. LIGO Scientific Collaboration and Virgo Collaboration, Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  34. LIGO Scientific Collaboration, Virgo Collaboration, Fermi-GBM Collaboration, and INTEGRAL Collaboration, Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017).
  35. Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, Astrophys. J. Lett. 875, L1 (2019).
  36. LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration, Black hole spectroscopy and tests of general relativity with GW250114, Phys. Rev. Lett. 136, 041403 (2026).
  37. LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, GW250114: Testing Hawking’s area law and the Kerr nature of black holes, Phys. Rev. Lett. 135, 111403 (2025).
  38. E. Barausse et al., Prospects for fundamental pwith LISA, Gen. Relativ. Gravit. 52, 81 (2020).
  39. ET collaboration, The science of the Einstein telescope, J. Cosmol. Astropart. Phys. 03 (2026) 081.
  40. T. Kobayashi, Horndeski theory and beyond: A review, Rep. Prog. Phys. 82, 086901 (2019).
  41. P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, Dilatonic black holes in higher curvature string gravity, Phys. Rev. D 54, 5049 (1996).
  42. M. Rinaldi, Black holes with non-minimal derivative coupling, Phys. Rev. D 86, 084048 (2012).
  43. E. Babichev and C. Charmousis, Dressing a black hole with a time-dependent Galileon, J. High Energy Phys. 08 (2014) 106.
  44. T. P. Sotiriou and S.-Y. Zhou, Black hole hair in generalized scalar-tensor gravity: An explicit example, Phys. Rev. D 90, 124063 (2014).
  45. H. Lu and Y. Pang, Horndeski gravity as D→4 limit of Gauss-Bonnet, Phys. Lett. B 809, 135717 (2020).
  46. S. E. P. Bergliaffa, R. Maier, and N. d. O. Silvano, Hairy black holes from Horndeski theory, arXiv:2107.07839.
  47. J. B. Hartle and D. H. Sharp, Variational principle for the equilibrium of a relativistic, rotating star, Astrophys. J. 147, 317 (1967).
  48. D. Langlois, K. Noui, and H. Roussille, Black hole perturbations in modified gravity, Phys. Rev. D 104, 124044 (2021).
  49. D. Langlois, K. Noui, and H. Roussille, On the effective metric of axial black hole perturbations in DHOST gravity, J. Cosmol. Astropart. Phys. 08 (2022) 040.
  50. G. W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10, 363 (1974).
  51. T. Kobayashi, H. Motohashi, and T. Suyama, Black hole perturbation in the most general scalar-tensor theory with second-order field equations I: The odd-parity sector, Phys. Rev. D 85, 084025 (2012).
  52. K. Takahashi, H. Motohashi, and M. Minamitsuji, Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations, Phys. Rev. D 100, 024041 (2019).
  53. A. Cisterna, M. Cruz, T. Delsate, and J. Saavedra, Nonminimal derivative coupling scalar-tensor theories: Odd-parity perturbations and black hole stability, Phys. Rev. D 92, 104018 (2015).
  54. E. Babichev, C. Charmousis, and M. Hassaine, Charged Galileon black holes, J. Cosmol. Astropart. Phys. 05 (2015) 031.
  55. J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, Healthy theories beyond Horndeski, Phys. Rev. Lett. 114, 211101 (2015).
  56. P. Pani and V. Cardoso, Are black holes in alternative theories serious astrophysical candidates? The Case for Einstein-Dilaton-Gauss-Bonnet black holes, Phys. Rev. D 79, 084031 (2009).
  57. G. Teschl, Ordinary Differential Equations and Dynamical Systems, vol. 140 of Graduate Studies in Mathematics, (American Mathematical Society, Providence, 2012).
  58. K. Aoki, M. A. Gorji, and S. Mukohyama, A consistent theory of D→4 Einstein-Gauss-Bonnet gravity, Phys. Lett. B 810, 135843 (2020).
  59. D. Langlois, K. Noui, and H. Roussille, Quadratic degenerate higher-order scalar-tensor theories revisited, Phys. Rev. D 103, 084022 (2021).

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Phys. Rev. D 114, 024086– Published 29 July, 2026
  • Received 11 February 2026
  • Accepted 7 July 2026
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