pith. sign in

arxiv: 2605.21047 · v1 · pith:67XM4SK2new · submitted 2026-05-20 · 🌀 gr-qc · cond-mat.stat-mech

Thermodynamics of homogeneous Universes: de Sitter, Bonnor-Melvin and static Einstein

G.E. Volovik This is my paper

Pith reviewed 2026-05-21 03:49 UTC · model grok-4.3

classification 🌀 gr-qc cond-mat.stat-mech
keywords thermodynamicsde Sitter universeBonnor-Melvin universestatic Einstein universecosmological constanthomogeneous cosmologyemergent gravity
0
0 comments X

The pith

Three homogeneous universes with different matter contents have the same thermodynamic properties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that three homogeneous universes—the de Sitter, Bonnor-Melvin-Λ, and static Einstein universes—have identical thermodynamic properties even though they involve different matter fields such as ordinary matter, magnetic fields, gravitational fields, and vacuum energy. Their energy densities follow the same equation that accounts for these densities along with pairs of thermodynamically conjugate variables. In the Minkowski vacuum without any of these fields, the approach yields a zero cosmological constant. A sympathetic reader would care because this provides a unified thermodynamic framework for diverse cosmological models and naturally explains the absence of a cosmological constant in empty space.

Core claim

Although these three Universes have different types of matter fields (ordinary matter, magnetic field, gravitational field and vacuum energy), they have the same thermodynamic properties. Their energy densities obey the same equation, which contains the corresponding matter densities and the pairs of the thermodynamically conjugate variables. In Minkowski vacuum, where the ordinary matter and magnetic and gravitational fields are absent, this thermodynamic approach automatically leads to zero cosmological constant.

What carries the argument

The common thermodynamic equation for energy density incorporating matter densities and conjugate variable pairs, which unifies the three universes.

Load-bearing premise

The gravitational field emerges from underlying matter fields and can be treated as part of matter for thermodynamic purposes.

What would settle it

Computing the energy density equation for the Bonnor-Melvin-Λ universe and finding it does not match the shared form with the other two would disprove the result.

read the original abstract

In the theories, in which dynamic gravitational field emerges from the underlying matter fields, the gravitational field can be considered as a part of matter. Using this approach, we construct the thermodynamics of the homogeneous Universes -- the de Sitter Universe, the Bonnor-Melvin-$\Lambda$ Universe and the static Einstein Universe. It is demonstrated that although these three Universes have different types of matter fields (ordinary matter, magnetic field, gravitational field and vacuum energy), they have the same thermodynamic properties. Their energy densities obey the same equation, which contains the corresponding matter densities and the pairs of the thermodynamically conjugate variables. In Minkowski vacuum, where the ordinary matter and magnetic and gravitational fields are absent, this thermodynamic approach automatically leads to zero cosmological constant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that in theories where the dynamic gravitational field emerges from underlying matter fields, gravity can be treated as part of matter. It constructs a thermodynamic description for three homogeneous spacetimes—the de Sitter universe, the Bonnor-Melvin-Λ universe, and the static Einstein universe—showing that their energy densities obey the same equation involving the respective matter densities (ordinary matter, magnetic field, gravitational field, vacuum energy) and pairs of thermodynamically conjugate variables, despite differing field contents. The framework also implies that the cosmological constant vanishes in the Minkowski vacuum limit.

Significance. If the identification of gravitational energy density holds, the result supplies a uniform thermodynamic relation across these symmetric cases that follows algebraically from the Einstein equations once gravity is included as matter. This offers a concrete illustration of how the thermodynamic identity emerges without additional free parameters and automatically enforces Λ=0 in vacuum, which may inform discussions of the cosmological constant problem and thermodynamic interpretations of gravity in homogeneous settings.

major comments (1)
  1. [Sections deriving the thermodynamic relations for each universe (following the premise statement)] The central thermodynamic equation is obtained by defining the gravitational contribution so that it satisfies the Einstein equations by construction for these homogeneous, static or de Sitter cases; this makes the common relation a direct algebraic consequence of the adopted identification rather than an independent derivation. A concrete test of robustness would be to verify whether the same equation persists when the gravitational energy density is computed from an explicit matter-field Lagrangian without presupposing the Einstein equations (e.g., in a perturbative or non-homogeneous extension).
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction state the emergence premise clearly but refer to prior publications for its justification; a short self-contained paragraph recalling the definition of gravitational energy density and conjugate variables would improve readability.
  2. [Thermodynamic equation presentation] Notation for the conjugate pairs (e.g., pressure-like terms) should be defined explicitly at first use to avoid ambiguity when comparing the three cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. We address the major point below.

read point-by-point responses

  1. Referee: [Sections deriving the thermodynamic relations for each universe (following the premise statement)] The central thermodynamic equation is obtained by defining the gravitational contribution so that it satisfies the Einstein equations by construction for these homogeneous, static or de Sitter cases; this makes the common relation a direct algebraic consequence of the adopted identification rather than an independent derivation. A concrete test of robustness would be to verify whether the same equation persists when the gravitational energy density is computed from an explicit matter-field Lagrangian without presupposing the Einstein equations (e.g., in a perturbative or non-homogeneous extension).

    Authors: We appreciate the referee's observation that the thermodynamic relation follows from identifying the gravitational energy density to satisfy the Einstein equations in these homogeneous cases. This identification is deliberate and follows directly from the manuscript's premise that, in theories where the dynamic gravitational field emerges from underlying matter fields, gravity can be treated as part of matter. For the de Sitter, Bonnor-Melvin-Λ and static Einstein universes, this yields a uniform thermodynamic equation involving the respective matter densities and conjugate variables, despite the different field contents. We regard the algebraic consistency as a strength of the framework, since it demonstrates that the same thermodynamic structure emerges once gravity is included in the matter sector, and it automatically enforces a vanishing cosmological constant in the Minkowski vacuum. The suggested robustness test—deriving the gravitational energy density from an explicit matter-field Lagrangian without presupposing the Einstein equations, for instance in perturbative or non-homogeneous settings—is a natural direction for further work. However, the present manuscript is restricted to the homogeneous spacetimes where the identification is direct from the field equations. We have added a short paragraph in the conclusions acknowledging this scope limitation and noting the potential for such extensions in future research. revision: partial

Circularity Check

1 steps flagged

Thermodynamic equivalence follows by construction from treating gravity as emergent matter

specific steps
  1. self definitional [Abstract]
    "In the theories, in which dynamic gravitational field emerges from the underlying matter fields, the gravitational field can be considered as a part of matter. Using this approach, we construct the thermodynamics of the homogeneous Universes"

    The demonstration that the three universes share the same thermodynamic equation is obtained by defining the gravitational field as one more matter component inside the adopted framework; once this definition is imposed, the algebraic relations from the Einstein equations force the energy-density expressions to take identical form for all cases, so the equivalence is true by construction of the uniform treatment.

full rationale

The paper adopts the premise that dynamic gravity emerges from underlying matter fields and therefore can be treated as an additional matter component. It then constructs a single thermodynamic framework in which energy densities for ordinary matter, magnetic field, gravitational field, and vacuum energy all obey the same equation involving conjugate pairs. Because the gravitational contribution is defined to satisfy the Einstein equations in these homogeneous spacetimes exactly as the other components do, the claimed identity of thermodynamic properties across the three universes reduces to the uniform application of the input identification rather than an independent derivation. The automatic vanishing of Lambda in the Minkowski limit is likewise a direct algebraic consequence of the same construction once all fields are absent.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The entire construction rests on the single domain assumption that gravity emerges from matter fields; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Dynamic gravitational field emerges from the underlying matter fields, allowing gravity to be treated as part of matter.
    Explicitly stated as the starting point for constructing the thermodynamics of the three universes.

pith-pipeline@v0.9.0 · 5657 in / 1136 out tokens · 33245 ms · 2026-05-21T03:49:15.443295+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · 14 internal anchors

  1. [1]

    Akama, An Attempt at Pregeometry: Gravity with Composite Metric, Progress of Theoretical Physics, 60 , 1900--1909 (1978)

    K. Akama, An Attempt at Pregeometry: Gravity with Composite Metric, Progress of Theoretical Physics, 60 , 1900--1909 (1978)

    work page 1900

  2. [2]

    Wetterich, Gravity from spinors, Phys

    C. Wetterich, Gravity from spinors, Phys. Rev. D 70 , 105004 (2004)

    work page 2004

  3. [3]

    Wetterich, Pregeometry and spontaneous time-space asymmetry, JHEP 06 (2022) 069

    C. Wetterich, Pregeometry and spontaneous time-space asymmetry, JHEP 06 (2022) 069

    work page 2022

  4. [4]

    Towards lattice-regularized Quantum Gravity

    D. Diakonov, Towards lattice-regularized Quantum Gravity, arXiv:1109.0091

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Frank Wilczek, Riemann-Einstein Structure from Volume and Gauge Symmetry, Phys. Rev. Lett. 80 4851--4854 (1998), arXiv:hep-th/9801184

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  6. [6]

    Addazi, S

    A. Addazi, S. Capozziello, A. Marciano and G. Meluccio, Gravity from Pre-geometry, Class. Quantum Grav. 42 , 045012 (2025), https://doi.org/10.1088/1361-6382/ada767

    work page doi:10.1088/1361-6382/ada767 2025

  7. [7]

    Andrea Addazi, Salvatore Capozziello and Giuseppe Meluccio, Holographic Naturalness and Pre-Geometric Gravity, Physics 8 , 2 (2025)

    work page 2025

  8. [8]

    Giuseppe Meluccio, Pre-geometric Einstein–Cartan Field Equations and Emergent Cosmology, arXiv:2505.02925

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Sakharov, Vacuum Quantum Fluctuations In Curved Space And The Theory Of Gravitation, Sov

    A.D. Sakharov, Vacuum Quantum Fluctuations In Curved Space And The Theory Of Gravitation, Sov. Phys. Dokl. 12 , 1040 (1968); Reprinted in Gen. Rel. Grav. 32 , 365--367 (2000)

    work page 1968

  10. [10]

    Zel'dovich, Interpretation of Electrodynamics as a Consequence of Quantum Theory, JETP Lett

    Ya.B. Zel'dovich, Interpretation of Electrodynamics as a Consequence of Quantum Theory, JETP Lett. 6 , 345--347 (1967)

    work page 1967

  11. [11]

    Volovik, The Universe in a Helium Droplet , Clarendon Press, Oxford (2003),

    G.E. Volovik, The Universe in a Helium Droplet , Clarendon Press, Oxford (2003),

    work page 2003

  12. [12]

    York, Role of conformal three-geometry in the dynamics of gravitation, Phys

    J.W. York, Role of conformal three-geometry in the dynamics of gravitation, Phys. Rev- Lett. 28 ,1082 (1972)

    work page 1972

  13. [13]

    Original Contributions, Essays, and Recollections in Honor of Alexei Starobinsky,

    G.E. Volovik, Proton decay in de Sitter environment, tbp in: Open Issues in Gravitation and Cosmology, "Original Contributions, Essays, and Recollections in Honor of Alexei Starobinsky," Editors: Andrei Barvinsky, Alexander Kamenshchik, Part of the book series: Fundamental Theories of Physics, FTPH 225 , Springer 2026, arXiv:2411.01892

    work page arXiv 2026

  14. [14]

    Maxfield and Z

    H. Maxfield and Z. Zahraee, Holographic solar systems and hydrogen atoms: non-relativistic physics in AdS and its CFT dual, JHEP 11 , 093 (2022)

    work page 2022

  15. [16]

    Zel'dovich, The equation of state at ultrahigh densities and its relativistic limitations, JETP 14 , 1143 (1962)

    Ya.B. Zel'dovich, The equation of state at ultrahigh densities and its relativistic limitations, JETP 14 , 1143 (1962)

    work page 1962

  16. [17]

    Volovik, First law of de Sitter thermodynamics, JETP Letters 121 , 766--770 (2025), arXiv:2504.05763

    G.E. Volovik, First law of de Sitter thermodynamics, JETP Letters 121 , 766--770 (2025), arXiv:2504.05763

    work page arXiv 2025

  17. [18]

    Polyakov and Fedor K

    Alexander M. Polyakov and Fedor K. Popov, Kronecker anomalies and gravitational striction, in: In Dialogues Between Physics and Mathematics, C. N. Yang at 100, 1st ed., Ge, M.L., He, Y.H., Eds.; Springer: Cham, Switzerland, 2022, arXiv:2203.07101 [hep-th]

    work page arXiv 2022

  18. [19]

    Plebanski and S

    J.F. Plebanski and S. Hacyan, Some exceptional electrovac type D metrics with cosmological constant, J. Math. Phys. 20 , 1004 (1979)

    work page 1979

  19. [20]

    Marco Astorino, Charging axisymmetric space-times with cosmological constant, JHEP 06 (2012) 086, arXiv:1205.6998v2 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  20. [21]

    Martin Zofka, Bonnor-Melvin universe with a cosmological constant, Phys. Rev. D 99 , 044058 (2019). arXiv:1903.08563

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  21. [22]

    Faizuddin Ahmed and Abdelmalek Bouzenada, Scalar fields in Bonnor-Melvin-Lambda universe with potential: A study of dynamics of spin-zero particles-antiparticles, Phys. Scr. 99 , 065033 (2024), arXiv:2312.06612

    work page arXiv 2024

  22. [23]

    Castro, Angel E

    Luis B. Castro, Angel E. Obispo and Andrés G. Jirón, Charged scalar bosons in a Bonnor-Melvin- universe at conical approximation, Eur. Phys. J. C 84 , 536 (2024)

    work page 2024

  23. [24]

    Volovik, Thermodynamics of Einstein static Universe with boundary, arXiv:2410.10549

    G.E. Volovik, Thermodynamics of Einstein static Universe with boundary, arXiv:2410.10549

    work page arXiv

  24. [25]

    Hawking, The cosmological constant is probably zero, Phys

    S.W. Hawking, The cosmological constant is probably zero, Phys. Lett. B 134 , 403 (1984)

    work page 1984

  25. [26]

    Duff, The cosmological constant is possibly zero, but the proof is probably wrong, Phys.\ Lett.\ B 226 , 36 (1989); Z.C

    M.J. Duff, The cosmological constant is possibly zero, but the proof is probably wrong, Phys.\ Lett.\ B 226 , 36 (1989); Z.C. Wu, The cosmological constant is probably zero, and a proof is possibly right, Phys. Lett. B 659 , 891 (2008)

    work page 1989

  26. [27]

    Dynamic vacuum variable and equilibrium approach in cosmology

    F.R. Klinkhamer and G.E. Volovik, Dynamic vacuum variable and equilibrium approach in cosmology, Phys. Rev. D 78 , 063528 (2008); arXiv:0806.2805 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  27. [28]

    Self-tuning vacuum variable and cosmological constant

    F.R. Klinkhamer and G.E. Volovik, Self-tuning vacuum variable and cosmological constant, Phys. Rev. D 77 , 085015 (2008); arXiv:0711.3170

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  28. [29]

    f(R) Cosmology from q-Theory

    F.R. Klinkhamer and G.E. Volovik, f(R) cosmology from q --theory, Pis'ma ZhETF 88 , 339--344 (2008); JETP Lett. 88 , 289--294 (2008); arXiv:0807.3896 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  29. [30]

    Massless graviton in de Sitter as second sound in two-fluid hydrodynamics

    G.E. Volovik, Massless graviton in de Sitter as second sound in two-fluid hydrodynamics, arXiv:2601.00639 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv

  30. [31]

    Introduction to superfluidity -- Field-theoretical approach and applications

    Andreas Schmitt, Introduction to superfluidity. Field-theoretical approach and applications, Lecture Notes in Physics volume 888, Springer Cham, 2014, arXiv:1404.1284

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  31. [32]

    What is mass in desitterian physics?

    T. Garidi, What is mass in de Sitterian physics? arXiv:hep-th/0309104

    work page internal anchor Pith review Pith/arXiv arXiv

  32. [33]

    Akhmedov, U

    E.T. Akhmedov, U. Moschella, K.E. Pavlenko and F.K. Popov, Infrared dynamics of massive scalars from the complementary series in de Sitter space, Phys. Rev. D 96 , 025002 (2017)

    work page 2017

  33. [34]

    Jean-Pierre Gazeau and Hamed Pejhan, Covariant Quantization of the Partially Massless Graviton Field in de Sitter Spacetime, Phys. Rev. D 108 , 065012 (2023), arXiv:2306.10086 [gr-qc]

    work page arXiv 2023

  34. [35]

    Sadekov, Effective graviton mass in de Sitter space, Phys

    D.I. Sadekov, Effective graviton mass in de Sitter space, Phys. Rev. D 109 , 085001 (2024)

    work page 2024

  35. [36]

    Kurt Hinterbichler and Austin Joyce, A Partially Massless Superconductor, arXiv:2507.15932 [hep-th]

    work page arXiv

  36. [37]

    Drazen Glavan, Graviton propagator in de Sitter space in a simple one-parameter gauge, arXiv:2511.13660

    work page internal anchor Pith review Pith/arXiv arXiv

  37. [38]

    Landau and E.M

    L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, section 96

    work page 1951

  38. [39]

    Volovik, From Landau two-fluid model to de Sitter Universe, Uspekhi Fizicheskikh Nauk 195 , 1137--1156 (2025), Sov

    G.E. Volovik, From Landau two-fluid model to de Sitter Universe, Uspekhi Fizicheskikh Nauk 195 , 1137--1156 (2025), Sov. Phys. Usp. 68 , 1074--1091 (2025), arXiv:2410.04392

    work page arXiv 2025

  39. [40]

    Hawking radiation: black hole vs de Sitter

    G.E. Volovik, Entropy of de Sitter (d+1)-spacetime: modification of the Gibbons-Hawking entropy of cosmological horizon, arXiv:2510.24502 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv

  40. [41]

    Dmitrii Diakonov, De Sitter entropy: on-shell versus off-shell, Physics Letters B 871 , 139967 (2025), doi: https://doi.org/10.1016/j.physletb.2025.139967, arXiv:2504.01942 [hep-th]

    work page doi:10.1016/j.physletb.2025.139967 2025

  41. [42]

    Larkin and S.A

    A.I. Larkin and S.A. Pikin, On phase transitions of the first order resembling those of the second order. JETP 29 , 891--896 (1969)

    work page 1969

  42. [43]

    Anzhong Wang and Qiang Wu, Stability of spin-0 graviton and strong coupling in Horava-Lifshitz theory of gravity, Phys. Rev. D 83 , 044025 (2011)

    work page 2011

  43. [44]

    Particle production in models with helicity-0 graviton ghost in de Sitter spacetime

    Keisuke Izumi and Takahiro Tanaka, Particle production in models with helicity-0 graviton ghost in de Sitter spacetime, Prog.Theor.Phys. 121 , 427--436 (2009), arXiv:0810.4811 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  44. [45]

    Freese, J.A

    K. Freese, J.A. Frieman, A.V. Olinto, Natural inflation with pseudo Nambu-Goldstone bosons, Phys. Rev. Lett. 65 , 3233 (1990)

    work page 1990

  45. [46]

    Noriaki Kitazawa, Inflaton as a pseudo-Nambu-Goldstone boson, Physics Letters B 840 , 137846 (2023)

    work page 2023

  46. [47]

    Marco Merchand, Exploring Ultra-Slow-Roll Inflation in Composite Pseudo-Nambu-Goldstone Boson Models: Implications for Primordial Black Holes and Gravitational Waves, arXiv:2510.15460

    work page arXiv

  47. [48]

    Sloth, Soft gravitons as Goldstone modes of spontaneously broken asymptotic symmetries in de Sitter spacetimes, arXiv:2502.03520

    Martin S. Sloth, Soft gravitons as Goldstone modes of spontaneously broken asymptotic symmetries in de Sitter spacetimes, arXiv:2502.03520

    work page arXiv

  48. [49]

    Juan Maldacena, Non-gaussian features of primordial fluctuations in single field inflationary models, JHEP05(2003)013

    work page 2003

  49. [50]

    Starobinsky, A new type of isotropic cosmological models without singularity, Phys

    A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 , 99 (1980). red The structure, spin dynamics and spin polarization of HQV were investigated in last few decades years

    work page 1980

  50. [51]

    Landau and E.M

    red L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, section 96

    work page 1951