Thermodynamic Gravity with Non-Extensive Horizon Entropy and Topological Calibration

TL;DR

This paper extends Jacobson's thermodynamic derivation of gravity to non-extensive horizon entropies, linking entropy models with effective gravitational couplings and topology, and exploring implications for cosmology.

Contribution

It introduces a topological calibration principle and models horizon entropy with a power law, connecting non-extensive thermodynamics to modified gravity and topology-dependent couplings.

Findings

Reproduces Einstein's equations with effective coupling $G_{eff}$ from entropy slope.

Derives $f(R)$ gravity equations with curvature-dependent entropy densities.

Establishes a topology-dependent effective gravitational coupling influenced by horizon entropy models.

Abstract

We revisit Jacobson's thermodynamic derivation of gravitational dynamics in the presence of generalized, non-extensive horizon entropies. Working within a local Rindler-wedge framework, we formulate the Clausius relation as the stationarity condition of a Massieu functional at fixed Unruh temperature, which identifies the entropy slope as the parameter controlling the effective gravitational coupling. For area-type entropies with constant slope, the construction reproduces Einstein's equations with $G_{eff} = 1/(4s_0)$, while curvature-dependent entropy densities supplemented by an internal entropy-production term yield the field equations of $f(R)$ gravity. Motivated by group-entropic considerations and long-range correlations, we model the entropy of horizon cross sections by a power law $S(A) = \eta (A/4G)^\delta$ and analyze its local and global implications. To fix the otherwise arbitrary coarse-graining scale entering the entropy slope, we introduce a Topological Calibration Principle that ties the reference area to intrinsic geometric data through the Gauss-Bonnet theorem. For compact two-dimensional sections, this selects a canonical calibration area and leads to a topology-dependent effective coupling $G_{eff}(\chi) \propto |\chi|^{1-\delta}$ where $\chi$ represents the Euler characteristic. Consistency across scales and topologies yields logarithmic bounds on $|1-\delta|$, while the associated scale dependence induces a characteristic modulation of the gravitational coupling in cosmology. The framework thus provides a controlled route to confront non-extensive horizon thermodynamics with both theoretical consistency requirements and observational constraints.