- 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

Canonical ensemble of a self-gravitating matter thin shell in AdS space

Tiago V. Fernandes*, Francisco J. Gandum†, and José P. S. Lemos‡

  • *Contact author: tiago.vasques.fernandes@tecnico.ulisboa.pt
  • †Contact author: francisco.gandum@tecnico.ulisboa.pt
  • ‡Contact author: joselemos@ist.utl.pt
PDF

Phys. Rev. D 112, 064065 – Published 22 September, 2025

DOI: https://doi.org/10.1103/ftpk-r7mk

Abstract

We build the canonical ensemble of a hot self-gravitating matter thin shell in anti–de Sitter (AdS) space by finding its partition function through the Euclidean path integral approach with fixed temperature at the conformal boundary. We obtain the reduced action of the system by restricting the path integral to spherically symmetric metrics with given boundary conditions and with the Hamiltonian constraint satisfied. The stationary conditions, i.e., the mechanical equilibrium and the thermodynamic equilibrium, are obtained from minimizing the reduced action. Evaluating the perturbed reduced action at the stationary points yields the mechanical stability condition and the thermodynamic stability condition. The reduced action calculated at the stationary points gives the partition function in the zero-loop approximation, and from it, the thermodynamic properties of the system are acquired. Within thermodynamics alone, the only stability condition that one can establish is thermodynamic stability, which follows from the computation of the heat capacity. For given specific pressure and temperature equations of state for the shell, we obtain the solutions of the ensemble. There are four different thin shell solutions, one of them is fully stable, i.e., is stable mechanically and thermodynamically. For the equations of state given, we find a first order phase transition from the matter thermodynamic phase to the Hawking-Page black hole phase. Moreover, there is a maximum temperature above which the shell ceases to exist, presumably at these high temperatures the shell inevitably collapses to a black hole.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (36)

  1. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  2. J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws of black hole mechanics, Commun. Math Phys. 31, 161 (1973).
  3. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  4. G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).
  5. S. W. Hawking, Black holes and thermodynamics, Phys. Rev. D 13, 191 (1976).
  6. D. J. Gross, M. J. Perry, and L. G. Yaffe, Instability of flat space at finite temperature, Phys. Rev. D 25, 330 (1982).
  7. S. W. Hawking and D. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87, 577 (1983).
  8. B. Allen, Euclidean Schwarzschild negative mode, Phys. Rev. D 30, 1153 (1984).
  9. J. W. York, Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33, 2092 (1986).
  10. B. F. Whiting and J. W. York, Action principle and partition function for the gravitational field in black-hole topologies, Phys. Rev. D 61, 1336 (1988).
  11. E. A. Martinez and J. W. York, Additivity of the entropies of black holes and matter in equilibrium, Phys. Rev. D 40, 2124 (1989).
  12. O. B. Zaslavskii, Canonical ensemble for arbitrary configurations of self-gravitating systems, Phys. Lett. A 152, 463 (1990).
  13. J. D. Brown, G. L. Comer, E. A. Martinez, J. Melmed, B. F. Whiting, and J. W. York, Thermodynamic ensembles and gravitation, Classical Quantum Gravity 7, 1433 (1990).
  14. R. André and J. P. S. Lemos, Thermodynamics of five-dimensional Schwarzschild black holes in the canonical ensemble, Phys. Rev. D 102, 024006 (2020).
  15. R. André and J. P. S. Lemos, Thermodynamics of d-dimensional Schwarzschild black holes in the canonical ensemble, Phys. Rev. D 103, 064069 (2021).
  16. H. W. Braden, J. D. Brown, B. F. Whiting, and J. W. York, Charged black hole in a grand canonical ensemble, Phys. Rev. D 42, 3376 (1990).
  17. J. P. S. Lemos, Thermodynamics of the two-dimensional black hole in the Teitelboim-Jackiw theory, Phys. Rev. D 54, 6206 (1996).
  18. C. S. Peça and J. P. S. Lemos, Thermodynamics of Reissner-Nordström-anti-de Sitter black holes in the grand canonical ensemble, Phys. Rev. D 59, 124007 (1999).
  19. C. S. Peça and J. P. S. Lemos, Thermodynamics of toroidal black holes, J. Math. Phys. (N.Y.) 41, 4783 (2000).
  20. M. M. Akbar, Schwarzschild-anti-de Sitter black holes within isothermal cavity: Thermodynamics, phase transitions, and the Dirichlet problem, Phys. Rev. D 82, 064001 (2010).
  21. J. X. Lu, S. Roy, and Z. Xiao, Phase transitions and critical behavior of black branes in canonical ensemble, J. High Energy Phys. 01 (2011) 133.
  22. T. V. Fernandes and J. P. S. Lemos, Grand canonical ensemble of a d-dimensional Reissner-Nordström black hole in a cavity, Phys. Rev. D 108, 084053 (2023).
  23. J. P. S. Lemos and O. B. Zaslavskii, Black holes and hot shells in the Euclidean path integral approach to quantum gravity, Classical Quantum Gravity 40, 235012 (2023).
  24. J. P. S. Lemos and O. B. Zaslavskii, Hot spaces with positive cosmological constant in the canonical ensemble: de Sitter solution, Schwarzschild-de Sitter black hole and Nariai universe, Phys. Rev. D 109, 084016 (2024).
  25. T. V. Fernandes and J. P. S. Lemos, Gibbons-Hawking action for electrically charged black holes in the canonical ensemble and Davies’ thermodynamic theory of black holes, Proc. R. Soc. A 481, 20240648 (2025).
  26. E. A. Martinez, Fundamental thermodynamical equation of a self-gravitating system, Phys. Rev. D 53, 7062 (1996).
  27. J. P. S. Lemos, G. M. Quinta, and O. B. Zaslavskii, Entropy of a self-gravitating electrically charged thin shell and the black hole limit, Phys. Rev. D 91, 104027 (2015).
  28. R. André, J. P. S. Lemos, and G. M. Quinta, Thermodynamics and entropy of self-gravitating matter shells and black holes in d dimensions, Phys. Rev. D 99, 125013 (2019).
  29. T. V. Fernandes and J. P. S. Lemos, Electrically charged spherical matter shells in higher dimensions: Entropy, thermodynamic stability, and the black hole limit, Phys. Rev. D 106, 104008 (2022).
  30. J. P. S. Lemos and O. B. Zaslavskii, Entropy of quasiblack holes, Phys. Rev. D 81, 064012 (2010).
  31. J. P. S. Lemos and O. B. Zaslavskii, Entropy of extremal black holes from entropy of quasiblack holes, Phys. Lett. B 695, 37 (2011).
  32. J. P. S. Lemos and O. B. Zaslavskii, Compact objects in general relativity: From Buchdahl stars to quasiblack holes, Int. J. Mod. Phys. D 29, 2041019 (2020).
  33. A. Ashtekar and A. Magnon, Asymptotically anti-de Sitter space-times, Classical Quantum Gravity 1, L39 (1984).
  34. M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter spaces, Commun. Math. Phys. 98, 391 (1985).
  35. D. N. Page and K. C. Phillips, Self-gravitating radiation in anti-de Sitter space, Gen. Relativ. Gravit. 17, 1029 (1985).
  36. T. V. Fernandes, F. J. Gandum and J. P. S. Lemos, Notebook and plots for “Canonical ensemble of a self-gravitating matter thin shell in AdS space” (2025), https://zenodo.org/records/17037882.

Outline

Information

Phys. Rev. D 112, 064065– Published 22 September, 2025
  • Received 10 January 2025
  • Accepted 22 August 2025
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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation