- 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

Gravitational partition function under volume constraints

Shan-Ping Wu1,2, Peng Cheng3, and Shao-Wen Wei1,2,*

  • 1Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou, Gansu 730000, China
  • 2Institute of Theoretical Physics and Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China
  • 3Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China

  • *Contact author: weishw@lzu.edu.cn
PDF

Phys. Rev. D 113, 104049 – Published 22 May, 2026

DOI: https://doi.org/10.1103/1nc1-wdqf

Abstract

The Euclidean action provides a bridge between gravitational thermodynamics and the partition function. In this work, we further investigate the gravitational partition function under a fixed-volume constraint, generalizing the fixed-volume on-shell geometry in the massless case. Moving beyond this massless configuration, we construct solutions with nonvanishing mass functions, which give rise to a new class of volume-constrained Euclidean geometries (VCEGs). These geometries possess both a boundary and a horizon. However, closer inspection indicates that the boundary is not intrinsic, but rather artificially introduced and can be extended, leading to the extended volume-constrained Euclidean geometries (EVCEGs). The EVCEGs contain two horizons, each generically associated with a conical singularity. Their Euclidean action is given by one quarter of the sum of the areas of the two horizons. In general, the conical singularities at the two horizons cannot be simultaneously eliminated, except at a critical mass m=m*, which defines the critical EVCEG. Configurations with unavoidable conical singularities are naturally interpreted as constrained gravitational instantons. An analysis of their contributions to the partition function, together with their topological properties, reveals a close analogy between the EVCEGs and the Euclidean Schwarzschild-de Sitter static patch. This suggests that the volume constraint effectively plays a role analogous to that of a cosmological constant in semiclassical quantum gravity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (64)

  1. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  2. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  3. S. W. Hawking, Black holes and thermodynamics, Phys. Rev. D 13, 191 (1976).
  4. J. B. Hartle and S. W. Hawking, Path integral derivation of black hole radiance, Phys. Rev. D 13, 2188 (1976).
  5. G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).
  6. J. D. Bekenstein, Black holes and the second law, Lett. Nuovo Cimento 4, 737 (1972).
  7. J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31, 161 (1973).
  8. J. D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9, 3292 (1974).
  9. S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87, 577 (1983).
  10. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60, 064018 (1999).
  11. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes, Phys. Rev. D 60, 104026 (1999).
  12. D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes, J. High Energy Phys. 07 (2012) 033.
  13. G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308, 284 (1993), arXiv:gr-qc/9310026.
  14. L. Susskind, The World as a hologram, J. Math. Phys. (N.Y.) 36, 6377 (1995).
  15. J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  16. E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  17. G. Hayward, Euclidean action and the thermodynamics of manifolds without boundary, Phys. Rev. D 41, 3248 (1990).
  18. B. F. Whiting and J. W. York, Jr., Action principle and partition function for the gravitational field in black hole topologies, Phys. Rev. Lett. 61, 1336 (1988).
  19. B. Banihashemi and T. Jacobson, Thermodynamic ensembles with cosmological horizons, J. High Energy Phys. 07 (2022) 042.
  20. S. P. Wu, Y. X. Liu, and S. W. Wei, Generalized free energy landscapes from Iyer-Wald formalism, Phys. Dark Universe 51, 102210 (2026).
  21. T. Jacobson and M. R. Visser, Partition function for a volume of space, Phys. Rev. Lett. 130, 221501 (2023).
  22. W. Fischler, Taking de Sitter seriously, Talk given at “Role of scaling laws in physics and biology” (celebrating the 60th birthday of Geoffrey West), Santa Fe (2000).
  23. T. Banks, Cosmological breaking of supersymmetry?, Int. J. Mod. Phys. A 16, 910 (2001).
  24. A. Strominger, The dS/CFT correspondence, J. High Energy Phys. 10 (2001) 034.
  25. T. Banks and W. Fischler, Holographic cosmology 3.0, Phys. Scr. T117, 56 (2005).
  26. X. Dong, E. Silverstein, and G. Torroba, De Sitter holography and entanglement entropy, J. High Energy Phys. 07 (2018) 050.
  27. C. Arias, F. Diaz, and P. Sundell, De Sitter space and entanglement, Classical Quantum Gravity 37, 015009 (2020).
  28. G. Arenas-Henriquez, F. Diaz, and P. Sundell, Logarithmic corrections, entanglement entropy, and UV cutoffs in de Sitter spacetime, J. High Energy Phys. 08 (2022) 261.
  29. J. C. Chang, S. He, Y. X. Liu., and L. Zhao, Holographic TT¯ deformation of the entanglement entropy in (A)dS3/CFT2, Phys. Rev. D 112, 026013 (2025).
  30. J. C. Chang, Y. He, Y. X. Liu, and Y. Sun, Toward a unified de Sitter holography: A composite TT¯ and TT¯+Λ2 flow, Sci. China Phys. Mech. Astron. 69, 250414 (2026).
  31. A. Tavlayan and B. Tekin, Partition function of a volume of space in a higher curvature theory, Phys. Rev. D 108, L041902 (2023).
  32. M. Lu and R. B. Mann, Lagrangian partition functions subject to a fixed spatial volume constraint in the Lovelock theory, Entropy 26, 291 (2024).
  33. R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993).
  34. R. C. Tolman, Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev. 55, 364 (1939).
  35. R. F. C. Vessot et al., Test of relativistic gravitation with a space-borne hydrogen maser, Phys. Rev. Lett. 45, 2081 (1980).
  36. J. Ponce de Leon and N. Cruz, Hydrostatic equilibrium of a perfect fluid sphere with exterior higher dimensional Schwarzschild space-time, Gen. Relativ. Gravit. 32, 1207 (2000).
  37. J. Pleyer, Zero Values of the TOV equation, arXiv:2411.15264.
  38. I. Affleck, On constrained instantons, Nucl. Phys. B191, 429 (1981).
  39. Y. Frishman and S. Yankielowicz, Large order behavior of perturbation theory and mass terms, Phys. Rev. D 19, 540 (1979).
  40. J. Cotler and K. Jensen, Gravitational constrained instantons, Phys. Rev. D 104, 081501 (2021).
  41. J. Cotler and K. Jensen, Wormholes and black hole microstates in AdS/CFT, J. High Energy Phys. 09 (2021) 001.
  42. P. Draper and S. Farkas, de Sitter black holes as constrained states in the Euclidean path integral, Phys. Rev. D 105, 126022 (2022).
  43. E. K. Morvan, J. P. van der Schaar, and M. R. Visser, On the Euclidean action of de Sitter black holes and constrained instantons, SciPost Phys. 14, 022 (2023).
  44. V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994).
  45. V. Iyer and R. M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52, 4430 (1995).
  46. A. Seraj, Conserved charges, surface degrees of freedom, and black hole entropy, arXiv:1603.02442.
  47. G. Compère, Advanced lectures on general relativity, Lect. Notes Phys. 952, 150 (2019).
  48. D. Harlow and J. Q. Wu, Covariant phase space with boundaries, J. High Energy Phys. 10 (2020) 146.
  49. W. Guo, X. Guo, M. Li, Z. Mou, and H. Zhang, Equivalence of Noether charge and Hilbert action boundary term formulas for the black hole entropy in F(Rabcd) gravity theory, Phys. Rev. D 110, 064071 (2024).
  50. W. Guo, X. Guo, X. Lan, H. Zhang, and W. Zhang, Background subtraction method is not only much simpler, but also as applicable as the covariant counterterm method, Phys. Rev. D 111, 084088 (2025).
  51. G. Barnich and F. Del Monte, Introduction to classical gauge field theory and to Batalin-Vilkovisky quantization, arXiv:1810.00442.
  52. R. Ruzziconi, Asymptotic symmetries in the gauge fixing approach and the BMS group, Proc. Sci., Modave2019 (2020) 003.
  53. D. V. Fursaev and S. N. Solodukhin, Description of the Riemannian geometry in the presence of conical defects, Phys. Rev. D 52, 2133 (1995).
  54. S. N. Solodukhin, The conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51, 609 (1995).
  55. S. N. Solodukhin, Entanglement entropy of black holes, Living Rev. Relativity 14, 8 (2011).
  56. R. Li and J. Wang, Generalized free energy landscape of a black hole phase transition, Phys. Rev. D 106, 106015 (2022).
  57. H. Nariai, On a new cosmological solution of Einstein’s field equations of gravitation, Sci. Rep. Tohoku Univ. Ser. I 35, 62 (1951).
  58. P. H. Ginsparg and M. J. Perry, Semiclassical perdurance of de Sitter space, Nucl. Phys. B222, 245 (1983).
  59. P. Cheng, Y. X. Liu, and S. W. Wei, Black hole thermodynamics from an ensemble-averaged theory, Phys. Rev. D 111, 4 (2025).
  60. P. Cheng, J. Pan, H. Xu, and S. J. Yang, Thermodynamics of the Kerr-AdS black hole from an ensemble-averaged theory, Eur. Phys. J. C 85, 423 (2025).
  61. M. S. Ali, C. Fairoos, C. L. A. Rizwan, T. K. Safir, and P. Cheng, Thermodynamics of Einstein-Gauss-Bonnet black holes and ensemble-averaged theory, Commun. Theor. Phys. 78, 055401 (2026).
  62. Y. Q. Liu, H. W. Yu, and P. Cheng, Quantum-corrected black hole thermodynamics from the gravitational path integral, arXiv:2506.15261.
  63. S. Deser, R. Jackiw, and G. ’t Hooft, Three-dimensional Einstein gravity: Dynamics of flat space, Ann. Phys. (N.Y.) 152, 220 (1984).
  64. S. Carlip, Quantum Gravity in 2+1 Dimensions (Cambridge University Press, Cambridge, England, 2003).

Outline

Information

Phys. Rev. D 113, 104049– Published 22 May, 2026
  • Received 19 January 2026
  • Revised 9 April 2026
  • Accepted 28 April 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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation