Scalar$-$Tensor Gravity as a Probe of Generalized Black Hole Entropy

Abstract: We develop a geometric realization of a broad class of generalized black hole entropy functionals by establishing their direct correspondence with the Misner$-$Sharp quasilocal mass and the Wald Noether$-$charge entropy in scalar$-$tensor theories of gravity. The resulting models feature a scale-dependent effective gravitational coupling, whose functional dependence is determined by the underlying entropy parameters. Within this framework, we derive explicit Einstein-frame scalar potentials: for Barrow entropy, a steep exponential potential; for Tsallis$-$Cirto entropy, an exponential potential governed by the nonextensivity parameter; and for quantum-gravity and entanglement$-$induced corrections, an approximately linear potential. These distinct potentials generate characteristic cosmological phenomenology, with implications for inflationary dynamics, late-time dark-energy behavior, and non-singular bouncing cosmologies. The framework is compatible with current constraints from solar-system tests, big-bang nucleosynthesis, and pulsar-timing observations, and it yields predictions that can be probed by forthcoming observational surveys. In this way, the analysis establishes a unified and geometrically grounded connection between information$-$theoretic entropy proposals and gravitational field theory.

• The paper establishes the mass-to-horizon relation that connects generalized entropy functionals with horizon-scale gravitational properties in a scalar-tensor framework.
• The paper employs the Misner-Sharp mass and Wald entropy formalism to derive explicit scalar potentials for Barrow, Tsallis, and quantum-gravity entropy corrections.
• The paper highlights cosmological implications by showing how scale-dependent gravitational couplings and inflationary dynamics align with observational constraints.

Geometric Realization of Generalized Black Hole Entropy in Scalar-Tensor Gravity

Introduction

The manuscript "Scalar-Tensor Gravity as a Probe of Generalized Black Hole Entropy" (2605.18551) rigorously addresses the geometric foundation and cosmological implications of generalized black hole entropy proposals. The author establishes a direct correspondence between information-theoretic entropy functionals and scalar-tensor gravity, mapping entropy parameters to the functional form of the Misner-Sharp quasilocal mass and Wald Noether-charge entropy. This theoretical framework yields scale-dependent modifications to the effective gravitational coupling, which manifest as distinct scalar potentials with cosmological relevance.

Generalized Entropy and Mass-to-Horizon Relation

A central contribution is the formalization of the mass-to-horizon relation (MHR), which connects generalized entropy functionals to horizon-scale gravitational properties. The entropy functional is shown to be thermodynamically consistent with the standard Hawking or Cai-Kim temperature, rectifying previous inconsistencies arising from ad hoc temperature generalizations. The MHR is parameterized by $(m, \beta, \alpha)$, enabling systematic construction of entropy functionals that include quantum-gravity, entanglement, Tsallis-Cirto, and Barrow forms. The formalism unifies disparate entropy proposals by encoding them as specific parameter choices within the generalized MHR.

The entropy functional

$$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$

captures leading and subleading corrections, including logarithmic quantum-gravity terms, power-law Tsallis nonextensivity, and Barrow fractal corrections. This is achieved without modifying the Hawking temperature, preserving consistency with quantum field theory in curved spacetime.

Scalar-Tensor Realization: Misner-Sharp Mass and Wald Entropy

The paper utilizes the Misner-Sharp quasilocal mass formalism to geometrize entropy modifications. In scalar-tensor theories, the effective gravitational coupling is $G_{\mathrm{eff}} = G_0^{-1} F(\phi) f'(R)$, with $F(\phi)$ and $f(R)$ encoding non-minimal couplings and curvature corrections. The scalar field profile and resulting dynamics are directly connected to the entropy parameters via matching the generalized MHR to the Misner-Sharp mass.

For each entropy prescription, the author derives explicit scalar potentials in both Jordan and Einstein frames:

• Barrow entropy: yields a steep exponential Einstein-frame potential ($V_E \sim \exp(-\lambda \varphi)$), forbidding slow-roll inflation (minimum slow-roll parameter $\epsilon_v \geq 3$).
• Tsallis-Cirto entropy: results in an exponential $V_E$, with slope set by the nonextensivity parameter $\delta$, supporting exact power-law inflation and quintessence.
• Quantum-gravity and entanglement corrections: produce an approximately linear $V_E$ ($$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$0), with subleading corrections that reconcile predictions for tensor-to-scalar ratio $$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$1 with BICEP/Keck constraints.

The Wald entropy functional, applicable to stationary horizons, is shown to be determined solely by the effective coupling, $$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$2, providing a geometric interpretation for the entropy corrections.

Dynamics of Scale-Dependent Gravitational Coupling

The scalar-tensor mapping gives rise to a scale-dependent gravitational coupling, $$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$3, with logarithmic and power-law running determined by the entropy parameters. In cosmological settings, this translates to a slowly varying Newton constant, $$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$4, consistent with pulsar-timing, solar-system ($$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$5), and nucleosynthesis ($$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$6) constraints. For power-law corrections with $$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$7, deviations are negligible at low energies but can dominate near the Planck scale, potentially giving rise to bounce cosmologies and a smooth transition beyond classical singularities.

Crucially, the author identifies a critical scale $$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$8 where $$S_G = \text{(const.)} + \text{(parameter-dependent powers and logarithms of horizon scale)}$$9 vanishes. This signals a breakdown of the effective field theory regime and invites further analysis within loop quantum cosmology or Wheeler-DeWitt frameworks.

Cosmological Implications

The derived potentials support a variety of cosmological scenarios:

• Barrow entropy: precludes conventional slow-roll inflation but may realize power-law or quintessence-like expansion.
• Linear potentials from quantum-gravity/entanglement: predict $G_{\mathrm{eff}} = G_0^{-1} F(\phi) f'(R)$0, $G_{\mathrm{eff}} = G_0^{-1} F(\phi) f'(R)$1 for $G_{\mathrm{eff}} = G_0^{-1} F(\phi) f'(R)$2 e-folds, with corrections enabling compliance with $G_{\mathrm{eff}} = G_0^{-1} F(\phi) f'(R)$3 (BICEP/Keck upper bound).
• Tsallis-Cirto entropy: enables exact power-law inflation for moderate nonextensivity, and mimics a cosmological constant for $G_{\mathrm{eff}} = G_0^{-1} F(\phi) f'(R)$4.
• Generalized entanglement scenarios: allow adjustable inflation or dark energy dynamics by tuning shape parameters, offering a flexible model-building platform.

This unified framework is experimentally viable, with cosmological predictions that can be systematically falsified using future Stage IV surveys (Euclid, Rubin Observatory).

Theoretical Significance and Future Directions

The paper resolves the longstanding ambiguity between phenomenological entropy proposals and their geometric gravitational origin. Each entropy functional now corresponds to a uniquely determined scalar-tensor theory: the mapping is explicit between entropy parameters and the gravitational action, scalar profile, and cosmological dynamics.

This geometrization places generalized entropy on the same foundational level as Bekenstein-Hawking, enabling extensions to other entropy forms (Rényi, Kaniadakis), which were previously disconnected from gravitational dynamics. The approach facilitates the creation of a "dictionary" linking statistical mechanics and gravitational field theory, with wide-ranging implications for the interpretation of horizon thermodynamics.

Outstanding theoretical tasks include:

• Full dynamical system analysis beyond the quasi-static, scalar-dominated regime;
• Generalization to spatially curved and anisotropic geometries and rotating horizons;
• Non-perturbative treatment near critical scales;
• Quantitative confrontation with observational datasets (Planck, BICEP/Keck, future cosmic surveys).

Conclusion

The work establishes a rigorous geometric correspondence between generalized black hole entropy proposals and scalar-tensor theories of gravity, by deploying the mass-to-horizon relation as an organizing principle. The mapping is thermodynamically consistent, observationally viable, and theoretically explicit, linking entropy parameters to gravitational coupling and cosmological observables. This framework elevates phenomenological entropy constructs to geometric foundations, promises systematic extensions to new entropy forms, and provides a tractable pathway for confronting quantum-gravitational corrections with high-precision data. Future research should focus on non-perturbative regimes, non-spherical generalizations, and detailed phenomenological constraints to fully realize the implications of this correspondence for gravitational and cosmological physics.