- Physical Review Journals
All JournalsPhysics Magazine

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

Vacuum polarization and renormalized stress-energy tensor of spherical thin shells

Julio Arrechea*

Cormac Breen†

Adrian Ottewill‡

Lorenzo Pisani§ and Peter Taylor∥

  • Center for Astrophysics and Relativity, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

  • *Contact author: julio.arrechea@sissa.it
  • †Contact author: cormac.breen@tudublin.ie
  • ‡Contact author: adrian.ottewill@ucd.ie
  • §Contact author: lorenzo.pisani2@mail.dcu.ie
  • ∥Contact author: peter.taylor@dcu.ie
PDF

Phys. Rev. D 114, 085016 – Published 20 October, 2026

DOI: https://doi.org/10.1103/vslc-4dhv

Abstract

We provide a thorough study of the properties of the Boulware vacuum in the spacetime of a spherical, static thin shell with a Minkowski interior. To this end, we calculate the renormalized vacuum polarization and stress-energy tensor of massless scalar fields via the extended-coordinate prescription, paying particular attention to their scaling as the shell approaches the black hole limit. Near the surface of the thin shell, we obtain the expected leading-order singular behavior of both quantities via two independent methods: a high-frequency approximation for the modes and a weak-field approximation. At the center of the shell we find nonlocal, Casimir-like contributions that remain finite in the black hole limit, and whose backreaction effects we compute via the semiclassical Einstein equations. Away from these regions amenable to analytic treatment, we obtain numerical results for a wide range of shell compactnesses and field couplings. In the black hole limit, we show that the vacuum polarization and renormalized stress-energy tensor outside the shell quickly approach the ones generated by a Schwarzschild black hole, suggesting a possible universality in the vacuum outside highly compact horizonless objects. This work addresses the conceptual and technical aspects necessary for computing renormalized expectation values in matter configurations, laying the foundations for future explorations on the subject.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (70)

  1. L. Parker, Phys. Rev. Lett. 21, 562 (1968).
  2. S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
  3. S. M. Christensen and S. A. Fulling, Phys. Rev. D 15, 2088 (1977).
  4. S. A. Fulling, Phys. Rev. D 15, 2411 (1977).
  5. P. Candelas, Phys. Rev. D 21, 2185 (1980).
  6. K. W. Howard and P. Candelas, Phys. Rev. Lett. 53, 403 (1984).
  7. P. R. Anderson, W. A. Hiscock, and D. A. Samuel, Phys. Rev. D 51, 4337 (1995).
  8. A. Levi, E. Eilon, A. Ori, and M. van de Meent, Phys. Rev. Lett. 118, 141102 (2017).
  9. J. B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188 (1976).
  10. W. G. Unruh, Phys. Rev. D 14, 870 (1976).
  11. D. G. Boulware, Phys. Rev. D 11, 1404 (1975).
  12. J. W. York, Phys. Rev. D 31, 775 (1985).
  13. W. A. Hiscock, Phys. Rev. D 37, 2142 (1988).
  14. K. Numajiri, K. Okabayashi, and S. Mukohyama, Phys. Rev. D 111, 085024 (2025).
  15. P. M. Saffin and Q.-X. Xie, Phys. Rev. D 113, 125025 (2026).
  16. M. Visser, Phys. Rev. D 54, 5116 (1996).
  17. C. Barceló, S. Liberati, S. Sonego, and M. Visser, Sci. Am. 301, 38 (2009).
  18. J. Arrechea and C. Barceló, Int. J. Mod. Phys. D 32, 2342006 (2023).
  19. J. Arrechea, C. Barceló, R. Carballo-Rubio, and L. J. Garay, Phys. Rev. D 109, 104056 (2024).
  20. I. A. Reyes and G. M. Tomaselli, Phys. Rev. D 108, 065006 (2023).
  21. P. Taylor and C. Breen, Phys. Rev. D 96, 105020 (2017).
  22. J. Arrechea, C. Breen, A. Ottewill, L. Pisani, and P. Taylor, Phys. Rev. D 111, 085009 (2025).
  23. D. Deutsch and P. Candelas, Phys. Rev. D 20, 3063 (1979).
  24. G. Kennedy, R. Critchley, and J. S. Dowker, Ann. Phys. (N.Y.) 125, 346 (1980).
  25. S. M. Christensen, Phys. Rev. D 14, 2490 (1976).
  26. J. Dowker and G. Kennedy, J. Phys. A 11, 895 (1978).
  27. J. Zahn, Phys. Rev. D 97, 066028 (2018).
  28. K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916, 424 (1916), https://ui.adsabs.harvard.edu/abs/1916skpa.conf..424S/abstract.
  29. J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).
  30. P. O. Mazur and E. Mottola, Universe 9, 88 (2023).
  31. X. Calmet and M. Sebastianutti, Eur. Phys. J. C 86, 512 (2026).
  32. L. H. Ford and N. F. Svaiter, Phys. Rev. D 58, 065007 (1998).
  33. M. R. Brown and A. C. Ottewill, Phys. Rev. D 34, 1776 (1986).
  34. M. Bordag, U. Mohideen, and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001).
  35. S. A. Fulling, J. Phys. A 36, 6857 (2003).
  36. K. Milton and A. A. Saharian, Phys. Rev. D 85, 064005 (2012).
  37. P. Taylor and C. Breen, Phys. Rev. D 94, 125024 (2016).
  38. P. Taylor, C. Breen, and A. Ottewill, Phys. Rev. D 106, 065023 (2022).
  39. P. Candelas and D. J. Raine, Phys. Rev. D 15, 1494 (1977).
  40. B. S. Kay and R. M. Wald, Phys. Rep. 207, 49 (1991).
  41. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
  42. P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959).
  43. R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (The University of Chicago Press, Chicago, 1994).
  44. W. Israel, Nuovo Cimento B 44S10, 1 (1966); 48, 463(E) (1967).
  45. E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, Cambridge, England, 2009).
  46. G. Darmois, Les équations de la gravitation einsteinienne, Mémorial des sciences mathématiques No. 25 (Gauthier-Villars, 1927), https://www.numdam.org/item/MSM_1927__25__1_0/ (1927).
  47. N. Sen, Ann. Phys. (Berlin) 378, 365 (1924).
  48. K. Lanczos, Ann. Phys. (Berlin) 379, 518 (1924).
  49. A. Urbano and H. Veermäe, J. Cosmol. Astropart. Phys. 04 (2019) 011.
  50. H. A. Buchdahl, Phys. Rev. 116, 1027 (1959).
  51. D. A. R. Dalvit and F. D. Mazzitelli, Phys. Rev. D 50, 1001 (1994).
  52. A. Satz, F. D. Mazzitelli, and E. Alvarez, Phys. Rev. D 71, 064001 (2005).
  53. A. Boasso and F. D. Mazzitelli, Phys. Rev. D 112, 105016 (2025).
  54. A. Barvinsky and G. Vilkovisky, Nucl. Phys. B282, 163 (1987).
  55. A. Barvinsky and G. Vilkovisky, Nucl. Phys. B333, 471 (1990).
  56. I. G. Avramidi, Nucl. Phys. B355, 712 (1991).
  57. K. Sanders, Ann. Henri Poincare 23, 1321 (2022).
  58. J. T. Jebsen, Gen. Relativ. Gravit. 37, 2253 (2005).
  59. X. Calmet, R. Casadio, S. D. H. Hsu, and F. Kuipers, Phys. Rev. Lett. 128, 111301 (2022).
  60. I. Perrucci, F. Kuipers, and R. Casadio, J. Cosmol. Astropart. Phys. 03 (2025) 005.
  61. P. R. Anderson and A. Fabbri, Phys. Rev. D 75, 044015 (2007).
  62. J. Percival, Classical and quantum bosonic fields on the spacetimes of black holes and stars, Ph.D thesis, University of Sheffield, 2022.
  63. E. D. Carlson, P. R. Anderson, A. Fabbri, S. Fagnocchi, W. H. Hirsch, and S. Klyap, Phys. Rev. D 82, 124070 (2010).
  64. J. Arrechea, C. Barceló, G. García-Moreno, and J. Polo-Gómez, Phys. Rev. D 111, 124017 (2025).
  65. C. Cattoen, T. Faber, and M. Visser, Classical Quantum Gravity 22, 4189 (2005).
  66. P. R. Anderson, W. A. Hiscock, and D. J. Loranz, Phys. Rev. Lett. 74, 4365 (1995).
  67. Y. Matsuo, arXiv:2603.25683.
  68. A. Ori and N. Zilberman, Phys. Rev. Lett. 135, 021401 (2025).
  69. See Supplemental Material at http://link.aps.org/supplemental/10.1103/vslc-4dhv for a Wolfram Mathematica notebook containing the expressions for the Hadamard coefficients and for the regularization parameters.
  70. DLMF, NIST Digital Library of Mathematical Functions, edited by f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, https://dlmf.nist.gov/ (Release 1.2.2 of 2024-09-15).

Outline

Information

Phys. Rev. D 114, 085016– Published 20 October, 2026
  • Received 8 July 2026
  • Accepted 17 September 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation