Gravity theory of the generalized mass to horizon entropy The Iyer Wald approach

Abstract: In this letter, our aim is to study the gravitational origin of the \textit{generalized mass-to-horizon entropy} (GMHE), described by an entropic exponent index $n$ and a multiplicative parameter $γ$. We employ Iyer-Wald's approach within the modified $\mathfrak{f(R)}$ gravity scenario in order to assess the horizon entropy of the black hole (BH) by considering solutions with constant curvature for the spherically symmetric vacuum field equations. In this process, we explicitly show that the GMHE can effectively be reconstructed from a generalized Lagrangian $\mathfrak{L}\propto \mathfrak{R}^{1+ε}$, where $ε\equiv\frac{n-1}{2}$ measures tiny departures from the standard formulation of general relativity (GR). Finally, we discuss the physical implications of our results in relation to cosmology and the prospects for alleviating the thermodynamic instability of Schwarzschild BHs.

• The paper establishes the GMHE framework, extending the mass-to-horizon relation beyond the traditional Bekenstein-Hawking entropy.
• It utilizes the Iyer-Wald formalism in modified f(R) gravity to derive horizon entropy as a Noether charge and modify the Einstein-Hilbert action.
• The analysis reveals logarithmic corrections and stability conditions that align with quantum gravity models and cosmological constraints.

Gravity Theory of the Generalized Mass-to-Horizon Entropy: Iyer-Wald Approach

Context and Motivation

This paper presents a rigorous investigation into the gravitational roots of the Generalized Mass-to-Horizon Entropy (GMHE) using the Iyer-Wald formalism within modified $\mathfrak{f(R)}$ gravity, focusing on spherically symmetric, vacuum black hole (BH) solutions with constant spacetime curvature. The research addresses persistent inadequacies of the Boltzmann-Gibbs statistical mechanics and the Bekenstein-Hawking entropy in characterizing the thermodynamics of black holes with non-standard, nonextensive entropy forms. Central to the analysis is the GMHE, parameterized by an entropic exponent $n$ and a multiplicative factor $\gamma$, which generalizes the conventional mass-to-horizon area relationship to accommodate nonadditive and nonextensive entropy models.

Generalized Mass-to-Horizon Entropy Framework

The GMHE approach establishes a fundamental entropy-horizon relation for black holes that supersedes the Bekenstein-Hawking area law. The entropy expression,

$$S_{GS} = \gamma\,\frac{2\pi}{G}\, \frac{n}{n+1} \left(\frac{\mathcal{A}_{hor}}{4\pi}\right)^{\frac{n+1}{2}},$$

features $n$ as a real, dimensionless entropic exponent and $\gamma$ as a multiplicative scale factor. This form allows for the recovery of various nonextensive entropy theories (e.g., Tsallis-Cirto, Zamora-Tsallis, Barrow) by suitable choices of $n$. Importantly, the framework resolves the thermodynamic inconsistencies observed when nonextensive entropies are combined with the Hawking temperature and standard mass-energy relations. It has been demonstrated that these relations can be reconciled by adopting a generalized mass-to-horizon power law, rather than the linear scaling inherent to Bekenstein-Hawking entropy.

Iyer-Wald Formalism in Modified Gravity

The Iyer-Wald approach enables the derivation of horizon entropy as a Noether charge for any diffeomorphism-invariant gravity theory. For a generic $\mathfrak{f(R)}$ Lagrangian, the entropy formula at the horizon is given by

$$S = \frac{\mathcal{A}_{hor}}{4G}\left[\mathfrak{f}'(R)\right]_{\Sigma}.$$

By imposing that the entropy matches the GMHE form and analyzing perturbations around general relativity (GR), the authors reconstruct an effective Lagrangian:

$\mathfrak{L} \propto \mathfrak{R}^{1+\epsilon},$

where $n$0 quantifies deviations from the Einstein-Hilbert action. This result bridges entropic gravity scenarios—especially those aligned with nonextensive entropy models—and geometric modifications to GR, fortifying the theoretical basis for GMHE in black hole thermodynamics.

Black Hole Solutions and Entropic Corrections

The paper examines spherically symmetric black hole metrics in $n$1 gravity, focusing on constant curvature vacuum solutions. The event horizon is analytically expanded in the presence of a small background curvature, yielding corrections to the Schwarzschild radius and event horizon area. Using the derived $n$2, the entropy expression is shown to display logarithmic corrections in the perturbative regime:

$n$3

Such corrections are ubiquitous in quantum gravity models (e.g., loop quantum gravity, string theory), reinforcing the structural similarity between GMHE and quantum-inspired black hole thermodynamics.

Physical Implications

Cosmological Constraints

The effective Lagrangian $n$4 aligns with prior studies in cosmology, where observational bounds on $n$5 from light element synthesis constrain the entropic exponent: $n$6. This result is consistent with cosmological analyses using DESI DR2 BAO data and baryogenesis models, suggesting that GMHE-inspired modifications are observationally viable over a range compatible with the standard $n$7CDM paradigm.

Thermodynamic Stability of Black Holes

Instability of Schwarzschild black holes in GR arises from negative heat capacity. In the GMHE framework, the analytic condition for thermodynamic stability is derived via a perturbative expansion:

$n$8

for BH masses $n$9. For $\gamma$0 set to the Planck length, stability can be achieved for $\gamma$1 below a threshold—compatible with cosmological bounds—thereby offering a mechanism for resolving the instability via entropic modification.

Theoretical Significance and Future Directions

The study solidifies the gravitational origin of GMHE, showing that a minor deformation of the Hilbert-Einstein action naturally embeds nonextensive entropy models in black hole physics. The logarithmic corrections to the horizon entropy, mirrored in quantum gravity theories, suggest deep connections between classical gravitational thermodynamics, quantum statistical mechanics, and information theory. The formalism paves the way for further exploration of quantum gravity influences on black hole entropy and stability, as well as applications to cosmological structure formation and phase transitions.

Conclusion

This work rigorously establishes the gravitational foundation for GMHE using the Iyer-Wald formalism in $\gamma$2 gravity. It demonstrates that the GMHE, parameterized by entropic exponent $\gamma$3 and scale $\gamma$4, can be dynamically realized through a modified Lagrangian $\gamma$5, thereby connecting black hole thermodynamics, nonextensive statistical mechanics, and modified gravity. The results yield nontrivial constraints on entropic parameters from cosmology and provide analytic criteria for black hole stability. The appearance of logarithmic corrections highlights synergy with quantum gravity theories, suggesting far-reaching implications for the understanding of gravitational entropy, information-theoretic aspects of spacetime, and thermodynamic phases of black holes.