Subtleties in non-equilibrium horizon thermodynamics of modified gravity theories

Abstract: Thermodynamic interpretations of gravity often arise from applying the Clausius relation to spacetime horizons. In modified gravity theories with higher-order equations of motion, such as f(R) and scalar-tensor gravity, this relation generally acquires additional entropy-production term. In this context, two distinct formulations have been proposed in literature: the non-equilibrium approach of Eling, Guedens, and Jacobson based on local Rindler horizons, and the thermodynamic formulation of cosmological apparent horizons in FLRW spacetimes. In this article, we present a detailed analysis of these approaches, and show that, even though both employ identical entropy balance relations that resemble non-equilibrium thermodynamics, the exact origin and role of each entropy-production term is fundamentally different. In the Rindler-horizon framework the extra term follows directly from consistency requirements related to the Bianchi identity, whereas in the apparent-horizon approach it is introduced solely to recover the Friedmann equations. Furthermore, we will see that the latter non-equilibrium contribution enters directly into dynamical equations of gravity, while the former does not. Finally, we also highlight the fact that thermodynamic descriptions of horizons in such modified gravity are not unique, and that equilibrium, and non-equilibrium descriptions can arise from different choices of thermodynamic variables. A clear understanding of these distinctions is therefore crucial for establishing a consistent and physically meaningful thermodynamic foundation for gravity beyond general relativity.

• The paper demonstrates that non-equilibrium entropy production emerges as a crucial correction in modified gravity theories.
• It analyzes both EGJ and CAH frameworks, outlining distinct roles for entropy production in reconciling gravitational dynamics.
• It highlights that ambiguity in thermodynamic variable definitions challenges unique derivations of field equations beyond standard GR.

Subtleties in Non-Equilibrium Horizon Thermodynamics of Modified Gravity Theories

Introduction

The thermodynamic interpretation of gravity, first formalized by Jacobson’s derivation of the Einstein equations from the Clausius relation on local Rindler horizons, has deeply influenced the investigation of gravitational dynamics. Central to this approach is the identification of the area of a horizon with entropy and the energy flux as heat. When generalizing to higher-order gravity theories, such as $f(R)$ or scalar-tensor models, the simplicity of this equilibrium picture breaks down due to the appearance of additional degrees of freedom and non-area contributions to the entropy density. The paper "Subtleties in non-equilibrium horizon thermodynamics of modified gravity theories" (2604.04731) offers a systematic and comparative analysis of non-equilibrium thermodynamic formulations for horizons within these extended theories of gravity, focusing on the origins and implications of entropy production terms in different frameworks.

Thermodynamic Derivations of Field Equations

The EGJ (Eling-Guedens-Jacobson) Rindler-Horizon Framework

In the EGJ framework, the derivation of field equations in $f(R)$ gravity from horizon thermodynamics reveals a fundamental non-equilibrium structure. For a local Rindler horizon with entropy density given by the Wald formula, the variation of entropy generically yields terms involving derivatives of $f'(R)$. This prevents the standard Clausius relation, $dS = \delta Q / T$, from directly reproducing the correct gravitational field equations, as additional terms arise that spoil compatibility with the Bianchi identity and local energy-momentum conservation.

The resolution, as established in this approach, is the extension to a non-equilibrium entropy balance law incorporating an explicit entropy production term, $d_i S$, so that the relation becomes $dS = \delta Q / T + d_i S$. The form of $d_i S$ is fixed uniquely by the requirement that the resulting equations respect covariance and local conservation laws. Crucially, the role of $d_i S$ is compensatory: it cancels non-covariant terms arising from the curvature-dependent horizon entropy, and does not enter the final form of the dynamical equations. Instead, it guarantees the internal consistency of the thermodynamic derivation within the modified gravity context.

CAH (Cosmological Apparent Horizon) Framework

A distinct route involves the apparent (or Hubble) horizon of an FLRW universe. Applying the equilibrium Clausius relation with the Wald entropy and standard energy flux yields dynamical equations for the expansion rate. However, in extended gravity models, essential terms present in the field equations are missing, while spurious terms appear. To reproduce the modified Friedmann equations, the method commonly adopted is to ad hoc introduce a non-equilibrium entropy production term, $dS_p$, and recast the horizon thermodynamics as $dS + dS_p = \delta Q / T$.

Unlike the EGJ case, the entropy production term in the CAH framework directly contributes to the modified dynamical equations (e.g., Friedmann equations) and is not fixed by independent physical principles. Instead, its specific form is adjusted post hoc to ensure the correct structure of the field equations. This leads to a certain degree of circularity: the entropy production term is chosen such that the non-equilibrium Clausius relation is mathematically equivalent to the set of field equations previously derived via variational principles.

Physical Origin and Interpretation of Entropy Production Terms

The formal similarity between entropy balance relations in the two frameworks masks substantial physical differences. In the EGJ approach, the entropy production term arises from the structure of the underlying theory (specifically, from the variation of curvature-dependent entropy) and serves to ensure the validity of conservation laws. It acts as a non-dynamical correction. In contrast, in the CAH framework, the non-equilibrium term is introduced as a mathematical device to retrofit the Clausius relation to the known cosmological dynamics. Therefore, there is no unique prescription for $f(R)$0 independent of the choice of background metric or gravity theory.

This discrepancy highlights that thermodynamic descriptions of horizons in modified gravity are non-unique. Depending on the definition of heat flux, quasi-local energy, and horizon entropy, one can construct either an equilibrium or non-equilibrium reformulation of the same gravitational dynamics. In FLRW cosmology, it is also possible to absorb terms usually associated with non-equilibrium entropy production into redefinitions of the Misner-Sharp mass or the energy flux, thereby restoring an equilibrium-like first-law structure.

Ambiguity and Physical Consistency

The analysis clarifies that the distinction between equilibrium and non-equilibrium horizon thermodynamics in higher-order gravity is primarily a consequence of the ambiguity in thermodynamic variable definitions, not a fundamental physical feature of gravitational dynamics. The freedom to reinterpret additional curvature-dependent terms as either entropy production or as part of the heat flow underscores the absence of a direct, unique mapping between thermodynamic and geometric variables once one moves beyond general relativity. As such, the mere introduction of an entropy production term does not necessarily signal genuinely irreversible processes.

Implications and Future Directions

Pragmatically, this non-uniqueness implies that thermodynamic derivations alone cannot decisively constrain the form of modified gravity field equations or distinguish between genuinely equilibrium and non-equilibrium processes at the horizon. The flexibility in defining effective variables highlights the limits of the thermodynamic analogy without a deeper statistical or microphysical interpretation of horizon entropy in these theories. The paper’s critical comparison suggests the need for a more fundamental principle to underpin the specific partitioning between heat, work, and entropy production in gravitational systems beyond GR.

Looking forward, future work could aim to:

• Identify horizon thermodynamic variables with genuine statistical ensembles or microstates in extended gravity theories.
• Develop covariant non-equilibrium thermodynamic formalisms that do not depend on background-specific ansätze for entropy production.
• Investigate the implications of these ambiguities for semiclassical and quantum gravitational phenomena, particularly in cosmology and black hole contexts.

Conclusion

"Subtleties in non-equilibrium horizon thermodynamics of modified gravity theories" (2604.04731) delivers a rigorous dissection of the conceptual and mathematical underpinnings of entropy production terms within horizon thermodynamics for higher-order gravity. By exposing the distinct physical origins and the context-dependent roles of non-equilibrium terms in the EGJ and CAH frameworks, the authors clarify the source of ambiguity and its implications for the thermodynamic interpretation of gravity. The main takeaway is that the division between equilibrium and non-equilibrium descriptions in horizon thermodynamics is not absolute, but reflects freedom in the definition of effective variables—a fact that both constrains and challenges future attempts to ground gravity on thermodynamic principles.