- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Quantum fluxes and ⟨Φ^2⟩ for a nonminimally coupled scalar field: Ringdown and tail on approaching the polar Kerr inner horizon

Maria Alberti1,2,*, Noa Zilberman3,†, Marc Casals4,5,6,‡, and Adrian C. Ottewill5,§

  • *Contact author: mariaalberti@campus.technion.ac.il
  • †Contact author: nz3745@princeton.edu
  • ‡Contact author: marc.casals@uni-leipzig.de
  • §Contact author: adrian.ottewill@ucd.ie
PDF

Phys. Rev. D 114, 064039 – Published 14 September, 2026

DOI: https://doi.org/10.1103/t6ff-qsdf

Abstract

We compute ⟨Φ^2⟩ren as well as the energy fluxes ⟨T^uu⟩ren and ⟨T^vv⟩ren (where u and v are the standard Eddington-Finkelstein coordinates) associated with a quantum massless real scalar field Φ^, with a general curvature coupling constant ξ, near the inner horizon (IH) of a Kerr black hole, along the axis of rotation. The quantum field is in the Unruh state, corresponding to an evaporating black hole. We renormalize these quantities by the state-subtraction method. We drop the assumption of minimal coupling to the curvature, thereby generalizing the results of [1] for the fluxes at the IH. This requires understanding the asymptotic behavior of ⟨Φ^2⟩ren near the IH. State subtraction allows us to push the computation of ⟨Φ^2⟩ren along the axis of rotation in the Kerr interior in [2] deeper into the near-IH region, exposing their final asymptotic behavior on approaching the IH. For ⟨Φ^2⟩ren (a ξ-independent quantity in the Kerr case), we find that the approach to its finite asymptotic IH value is given, per ℓ-mode, by a ringdown phase (namely exponentially damped oscillations), followed by an inverse-power tail, both in the tortoise coordinate r* (which diverges at the IH). Interestingly, in the regime where the ringing dominates, the ringing’s complex frequencies are (numerically) found to match twice the well-known classical quasinormal-mode frequencies in Kerr, and the inverse-power tails are found to be r*−2ℓ−3 (resembling Price’s law in the classical black hole exterior, upon replacement t→r*). In particular, the sum over ℓ behaves as r*−3 (with a prefactor obtained here analytically for the first time), resembling the known behavior of classical (axially symmetric) scalar perturbations on approaching the IH [3]. These results allow computing the flux components ⟨T^uu⟩ren and ⟨T^vv⟩ren for general ξ at the IH vicinity. We find that the limiting IH values of ⟨T^uu⟩ren and ⟨T^vv⟩ren are independent of the coupling constant ξ. When translated to the Kruskal coordinate V (regular and vanishing at the Cauchy horizon), it implies that the prefactor C characterizing the divergence of the renormalized quantum stress-energy tensor appearing in ⟨T^VV⟩ren≃CV−2, is independent of the coupling to curvature in the case of polar Kerr.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. N. Zilberman, M. Casals, A. Ori, and A. C. Ottewill, Quantum fluxes at the inner horizon of a spinning black hole, Phys. Rev. Lett. 129, 261102 (2022).
  2. N. Zilberman, M. Casals, A. Levi, A. Ori, and A. C. Ottewill, Computation of ⟨Φ2⟩ and quantum fluxes at the polar interior of a spinning black hole, Phys. Rev. D 111, 124054 (2025).
  3. A. Ori, Evolution of scalar field perturbations inside a Kerr black hole, Phys. Rev. D 58, 084016 (1998).
  4. R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, Chicago, 1994).
  5. R. M. Wald, Trace anomaly of a conformally invariant quantum field in curved spacetime, Phys. Rev. D 17, 1477 (1978).
  6. S. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  7. W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976).
  8. F. J. Tipler, Singularities in conformally flat spacetimes, Phys. Lett. A 64, 8 (1977).
  9. A. Ori, Strength of curvature singularities, Phys. Rev. D 61, 064016 (2000).
  10. A. Ori, Structure of the singularity inside a realistic rotating black hole, Phys. Rev. Lett. 68, 2117 (1992).
  11. M. Dafermos and J. Luk, The interior of dynamical vacuum black holes I: The C0-stability of the Kerr Cauchy horizon, Ann. Math. 202, 309 (2025).
  12. S. Gurriaran, Non-linear instability of the Kerr Cauchy horizon near i+, arXiv:2603.17911.
  13. J. Luk and S.-J. Oh, The formation of a weak null singularity in the interior of generic rotating black holes, arXiv:2604.04877.
  14. A. Lanir, A. Ori, N. Zilberman, O. Sela, A. Maline, and A. Levi, Analysis of quantum effects inside spherical charged black holes, Phys. Rev. D 99, 061502 (2019).
  15. N. Zilberman, A. Levi, and A. Ori, Quantum fluxes at the inner horizon of a spherical charged black hole, Phys. Rev. Lett. 124, 171302 (2020).
  16. S. Hollands, C. Klein, and J. Zahn, Quantum stress tensor at the Cauchy horizon of the Reissner–Nordström–de Sitter spacetime, Phys. Rev. D 102, 085004 (2020).
  17. C. Klein, M. Soltani, M. Casals, and S. Hollands, Infinite quantum twisting at the Cauchy horizon of rotating black holes, Phys. Rev. Lett. 132, 121501 (2024).
  18. C. Klein, J. Zahn, and S. Hollands, Quantum (Dis)Charge of Black Hole Interiors, Phys. Rev. Lett. 127, 231301 (2021).
  19. M. Alberti and J. Zahn, Quantum effects in near-extremal charged black hole spacetimes, Phys. Rev. D 111, 104061 (2025).
  20. C. Klein and J. Zahn, Renormalized charged scalar current in the Reissner-Nordström-de Sitter spacetime, Phys. Rev. D 104, 025009 (2021).
  21. T. McMaken, Backreaction from quantum fluxes at the Kerr inner horizon, Phys. Rev. D 110, 045019 (2024).
  22. W. A. Hiscock, Stress energy tensor near a charged, rotating, evaporating black hole, Phys. Rev. D 15, 3054 (1977).
  23. N. Birrell and P. Davies, On falling through a black hole into another universe, Nature (London) 272, 35 (1978).
  24. W. A. Hiscock, Quantum-mechanical instability of the Kerr-Newman black-hole interior, Phys. Rev. D 21, 2057 (1980).
  25. A. C. Ottewill and E. Winstanley, The renormalized stress tensor in Kerr space-time: General results, Phys. Rev. D 62, 084018 (2000).
  26. G. Arad, Numerical investigation of the interior geometry of semiclassical evaporating spherical charged black holes (2025).
  27. N. Zilberman and A. Ori, Semiclassical backreaction inside a non-evaporating four-dimensional spherical charged black hole (to be published).
  28. P. Candelas and B. P. Jensen, The Feynman Green function inside a Schwarzschild black hole, Phys. Rev. D 33, 1596 (1986).
  29. J. B. Hartle and S. W. Hawking, Path-integral derivation of black-hole radiance, Phys. Rev. D 13, 2188 (1976).
  30. B. S. DeWitt, Quantum field theory in curved spacetime, Phys. Rep. 19, 295 (1975).
  31. S. M. Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point separation method, Phys. Rev. D 14, 2490 (1976).
  32. A. Lanir, A. Levi, and A. Ori, Mode-sum renormalization of ⟨Φ2⟩ for a quantum scalar field inside a Schwarzschild black hole, Phys. Rev. D 98, 084017 (2018).
  33. A. Levi and A. Ori, Pragmatic mode-sum regularization method for semiclassical black-hole spacetimes, Phys. Rev. D 91, 104028 (2015).
  34. B. S. Kay and R. M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon, Phys. Rep. 207, 49 (1991).
  35. D. Häfner and C. K. M. Klein, The Unruh state for bosonic Teukolsky fields on subextreme Kerr spacetimes, arXiv:2602.09796.
  36. R. H. Price, Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations, Phys. Rev. D 5, 2419 (1972).
  37. A. Ori, Evolution of perturbations inside a charged black hole: Linear scalar field, Phys. Rev. D 55, 4860 (1997).
  38. N. Zilberman, M. Casals, A. Ori, and A. C. Ottewill, Two-point function of a quantum scalar field in the interior region of a Kerr black hole, Phys. Rev. D 106, 125011 (2022).
  39. S. A. Teukolsky, Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations, Phys. Rev. Lett. 29, 1114 (1972).
  40. E. Berti, V. Cardoso, and M. Casals, Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions, Phys. Rev. D 73, 024013 (2006).
  41. Wolfram Research, Inc., Mathematica, Version 14.1 (2024).
  42. C. Iuliano and J. Zahn, Canonical quantization of Teukolsky fields on Kerr backgrounds, Phys. Rev. D 108, 125017 (2023).
  43. A. Starobinskii, Amplification of waves during reflection from a rotating black hole, Zh. Eksp. Teor. Fiz. 64, 48 (1973).
  44. Y. B. Zeldovich, Generation of waves by a rotating body, Sov. J. Exp. Theor. Phys. Lett. 14, 180 (1971).
  45. S. Hollands and R. M. Wald, Local Wick polynomials and time ordered products of quantum fields in curved space-time, Commun. Math. Phys. 223, 289 (2001).
  46. M. Alberti and A. Ori (to be published).
  47. M. Alberti and N. Zilberman, Supplement on the analytical derivation of the leading inverse-power tail, Zenodo (2026), 10.5281/zenodo.20808262.
  48. E. Berti, Ringdown, https://pages.jh.edu/eberti2/ringdown/.
  49. S. R. Dolan, The quasinormal mode spectrum of a Kerr black hole in the Eikonal limit, Phys. Rev. D 82, 104003 (2010).
  50. M. Casals, C. Kavanagh, and A. C. Ottewill, High-order late-time tail in a Kerr spacetime, Phys. Rev. D 94, 124053 (2016).
  51. A. Levi and A. Ori, Mode-sum regularization of ⟨ϕ2⟩ in the angular-splitting method, Phys. Rev. D 94, 044054 (2016).
  52. A. Levi and A. Ori, Versatile method for renormalized stress-energy computation in black-hole spacetimes, Phys. Rev. Lett. 117, 231101 (2016).
  53. M. Alberti, N. Zilberman, and A. Ori (to be published).
  54. E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009).
  55. V. P. Frolov and I. D. Novikov, Black Hole Physics. Basic Concepts and New Developments (Kluwer Academic Publishers, Dordrecht/Boston/London, 1998).
  56. A. Lanir, A. Levi, A. Ori, and O. Sela, Two-point function of a quantum scalar field in the interior region of a Reissner-Nordstrom black hole, Phys. Rev. D 97, 024033 (2018).
  57. S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational, electromagnetic and neutrino-field perturbations, Astrophys. J. 185, 635 (1973).
  58. M. Sasaki and H. Tagoshi, Analytic black hole perturbation approach to gravitational radiation, Living Rev. Relativity 6, 6 (2003).
  59. T. McMaken and A. J. S. Hamilton, Hawking radiation inside a rotating black hole, Phys. Rev. D 109, 065023 (2024).

Outline

Information

Phys. Rev. D 114, 064039– Published 14 September, 2026
  • Received 9 July 2026
  • Accepted 19 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation