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

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-$Λ$ 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.

• The paper demonstrates that treating gravitational fields as a form of matter leads to unified thermodynamic relations across de Sitter, Bonnor-Melvin–Λ, and static Einstein universes.
• It employs a generalized energy density formalism that integrates contributions from vacuum, matter, curvature, and magnetic fields to establish equilibrium conditions.
• The work resolves aspects of the cosmological constant problem and bridges gravitational dynamics with fluid analogies, paving the way for future modifications in gravity theories.

Thermodynamics of Homogeneous Cosmological Solutions: de Sitter, Bonnor-Melvin–$\Lambda$, and Static Einstein Universes

Overview and Motivation

This work formulates and analyzes the thermodynamics of three spatially homogeneous cosmological models — the de Sitter Universe, the Bonnor-Melvin–$\Lambda$ Universe, and the static Einstein Universe — under the paradigm that the gravitational field is a specific manifestation of matter, as advocated by pre-geometric and emergent gravity frameworks. The underlying methodological premise is to treat all contributions, including gravitational, electromagnetic, and vacuum energy, uniformly within a generalized thermodynamic formalism. The study demonstrates that, despite marked differences in matter content and geometric realization, all these cosmological solutions are governed by unified thermodynamic relations involving generalized densities and thermodynamically conjugate pairs.

Generalized Thermodynamic Framework

A central departure from traditional approaches is the identification of the gravitational field with a component of generalized matter, echoing Sakharov’s induced gravity and related pre-geometric theories. The analysis proceeds by expressing the total action as $S = S^{\rm m} + S^{\rm grav}$ and by constructing a total matter Lagrangian $\mathcal{L}^{\rm M}$ that sums the Lagrangians of ordinary and gravitational components. The generalized energy-momentum tensor, obtained by direct (not functional) differentiation of $\mathcal{L}^{\rm M}$ with respect to the metric, is identically zero by the Einstein equations and is thereby automatically conserved.

The generalized energy density $\epsilon^{\rm gen}$ embodies all contributions from ordinary matter, vacuum energy, electromagnetic fields, gravitational curvature, and other possible fields (such as Hawking’s three-form gauge fields), and is structurally of the form

$$\epsilon^{\rm gen} = \epsilon_{\rm vac} + \epsilon_{\rm Matter} + KR = 0,$$

where $K$ is the gravitational “chemical potential” (canonical coupling), $R$ is the Ricci scalar, and $\epsilon_{\rm Matter}$ subsumes all non-vacuum fields.

de Sitter Universe: Two-Component Thermodynamics and Generalized Conjugate Pairs

For de Sitter space ($\Lambda$0), the cosmological constant is interpreted as vacuum energy. The analysis reveals a two-component thermodynamic structure:

• Vacuum component: equation of state parameter $\Lambda$1, $\Lambda$2.
• Thermal (“normal”) component: emerging from gravitational degrees of freedom, with $\Lambda$3 (i.e., Zeldovich stiff-matter), and $\Lambda$4.

The local de Sitter temperature is experienced as the background temperature by comoving observers. Notably, the entropy density is obtained from horizon area and Hubble volume, resulting in $\Lambda$5, correlating the thermal properties of de Sitter with its intrinsic curvature.

A key result is the equivalence and unification, at the thermodynamic level, of vacuum and gravitational contributions, with both components having equal energy densities ($\Lambda$6). The free energy density vanishes in equilibrium, enforcing $\Lambda$7.

Bonnor-Melvin–$\Lambda$8 Universe: Magnetic Contributions and Anisotropy

In the Bonnor-Melvin–$\Lambda$9 Universe, a homogeneous magnetic field is present, made possible by fine-tuning between the magnetic energy density and cosmological constant. The magnetic field replaces the thermal gravitational component of de Sitter. Here, the generalized energy density maintains its structure: $S = S^{\rm m} + S^{\rm grav}$0 where $S = S^{\rm m} + S^{\rm grav}$1. Unlike the de Sitter case, there is no thermal gravitational component ($S = S^{\rm m} + S^{\rm grav}$2) and the universe is fully static.

A distinctive feature is the inherent anisotropy of the energy-momentum tensor, reflected in the pressure components orthogonal and parallel to the magnetic field direction. The pressure balance is restored by curvature and cosmological contributions.

Static Einstein Universe: Matter Dominance and Universality

The static Einstein Universe, defined on $S = S^{\rm m} + S^{\rm grav}$3, contains ordinary matter and spatial curvature. The same overarching thermodynamic relationships are preserved: $S = S^{\rm m} + S^{\rm grav}$4 with the matter component's equation of state parameter $S = S^{\rm m} + S^{\rm grav}$5 explicitly entering the relations for energy densities. For radiation ($S = S^{\rm m} + S^{\rm grav}$6), the energies of ordinary matter and vacuum are again equal, paralleling the results for de Sitter and Bonnor-Melvin universes.

Generalization and Dynamical Vacuum Energy

The formalism is extended to cases with dynamical vacuum contributions, specifically via Hawking's three-form gauge field. This adds a new thermodynamically conjugate pair ($S = S^{\rm m} + S^{\rm grav}$7, $S = S^{\rm m} + S^{\rm grav}$8) to the generalized energy density: $S = S^{\rm m} + S^{\rm grav}$9 This allows addressing the cosmological constant problem: in the Minkowski vacuum (absence of all matter fields except vacuum), the formulation automatically yields $\mathcal{L}^{\rm M}$0 without fine-tuning, as the Planck-scale contributions from $\mathcal{L}^{\rm M}$1 are precisely cancelled by the conjugate term $\mathcal{L}^{\rm M}$2.

Massless Gravitons and Two-Fluid Analogy

Gravitational excitations in de Sitter can be viewed analogously to second sound in Landau two-fluid hydrodynamics. The propagation speed of these graviton modes coincides with that of light ($\mathcal{L}^{\rm M}$3) as a direct thermodynamic consequence of the stiff-matter equation of state ($\mathcal{L}^{\rm M}$4). This underscores the universality of thermodynamic reasoning across disparate physical systems, from superfluids to cosmological spacetimes.

The paper notes ongoing debate regarding the existence and dynamics of massless graviton modes in de Sitter, referencing various approaches in quantum and classical field theory. Nevertheless, the thermodynamic origin — via the Kronecker anomaly and conjugate variables — is emphasized as robust.

Implications and Future Prospects

This unified thermodynamic treatment has several broad implications:

• Theoretical Universality: The structure $\mathcal{L}^{\rm M}$5 underpins the energetics of a wide range of homogeneous spacetimes, including those with emergent gravitational dynamics.
• Resolution of the Cosmological Constant Problem: The formalism provides a concrete mechanism for the self-adjustment of $\mathcal{L}^{\rm M}$6 to zero in empty spacetimes, without recourse to unspecified ultraviolet physics.
• Extension to Dynamical and Inhomogeneous Cases: While currently limited to homogeneous settings, the approach may be adaptable to dynamical or inhomogeneous scenarios (e.g., black holes, anisotropic universes) with appropriate identification of conjugate pairs and correction for boundary contributions.
• Bridges to Condensed Matter Analogs: The analogy to two-fluid hydrodynamics in superfluids suggests potential cross-fertilization between relativistic cosmology and condensed matter systems, especially for studying collective excitations and nonequilibrium dynamics.

Prospective developments may involve rigorous derivation of the graviton spectrum in de Sitter backgrounds, generalization to $\mathcal{L}^{\rm M}$7 and other modified gravity theories, and formal elucidation of the Kronecker anomaly’s consequences for gravitational thermodynamics.

Conclusion

By treating the gravitational field as a component of matter and identifying thermodynamically conjugate variables linking curvature and coupling, this work constructs a comprehensive thermodynamic theory for spatially homogeneous universes. The resulting formalism reveals deep structural similarities among de Sitter, Bonnor-Melvin–$\mathcal{L}^{\rm M}$8, and static Einstein universes, and naturally resolves the cosmological constant problem in the thermodynamic limit. The approach paves the way for a more unified understanding of gravitational dynamics and thermodynamics in both cosmological and more general settings.