Thermodynamic Gravity with Non-Extensive Horizon Entropy and Topological Calibration

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 $S(A) = η(A/4G)^δ$ 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 $G_{eff}(χ) \propto |χ|^{1-δ}$ where $χ$ represents the Euler characteristic. Consistency across scales and topologies yields logarithmic bounds on $|1-δ|$, 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.

Figures (2)

• Figure 1: Local Rindler-wedge geometry near a spacetime point $p$ (with $p\in\Sigma$). The boundary $\mathcal{H}$ is a null hypersurface generated by an affinely parametrized null congruence $k^a$, emanating from the spacelike local bifurcation two-surface $\Sigma$. An auxiliary null vector $\ell^a$ fixes the transverse direction, and $dA$ denotes the induced area element on the spacelike two-dimensional cuts of $\mathcal{H}$.
• Figure 2: TCP-induced running of the effective coupling (fixed topology) and its impact on the growth observable $f\sigma_8(z)$. The reference area is identified with the cosmological apparent-horizon area, $A_*\equiv A_H=4\pi R_A^2$, where $R_A=(H^2+k/a^2)^{-1/2}$; hence $A_H=4\pi/H^2$ for $k=0$Hayward1998CaiKim2005. Using the entropy--slope relation \ref{['eq:Geff-Astar']}, one obtains the dimensionless modulation $\mu(a)\equiv G_{\rm eff}(a)/G_{\rm eff}(1)=[H(a)/H_0]^{2(\delta-1)}$, cf. \ref{['eq:mu-a']}. The curves show $f\sigma_8(z)$ computed by integrating \ref{['eq:growth-mu']} on a fixed flat $\Lambda$CDM background with $(\Omega_{m0},\Omega_{\Lambda 0})=(0.3,0.7)$ (neglecting radiation at the redshifts of interest), growth normalization $D(1)=1$, and $\sigma_8=0.8$, for $\delta=1$ (GR) and for $|1-\delta|\in\{1/500,\;1/100,\;1/10\}$. Points with $1\sigma$ uncertainties are the compilation of $f\sigma_8$ measurements from NesserisPantazisPerivolaropoulos2017, shown for comparison only (no fit is performed).