Effective matter sectors from modified entropies

TL;DR

The paper develops a general mapping from modified horizon entropies $S(r)$ to a spacetime metric $f(r)=1-\frac{4\pi M}{S'(r)}$ by promoting the horizon relation to a global function $g(r)=4\pi/S'(r)$, effectively encoding entropy corrections as geometric backreaction. This leads to an emergent anisotropic stress–energy tensor $T^\mu_{\ \nu}=\mathrm{diag}(-\rho,-\rho,p_t,p_t)$ with $p_r=-\rho$, representing an entropic matter sector without ordinary fields. Applying the framework to Barrow, Tsallis–Cirto, Rényi, Kaniadakis, logarithmic, LQG, and exponential entropies yields explicit $f(r)$ forms, their Einstein tensors, gravitational forces, and the corresponding $\rho,p_r,p_t$, along with energy-condition analyses. The results show how entropy deformations can regularize black-hole interiors via a de Sitter-like core and provide a unified way to compare entropic corrections across different formalisms, with potential extensions to cosmology and astrophysical tests.

Abstract

We present a general formalism linking modified entropy functions directly to a modified spacetime metric and, subsequently, to an effective matter sector of entropic origin. In particular, within the framework of general relativity, starting from the first law of black-hole thermodynamics we establish an explicit correspondence between the entropy derivative and the metric function, which naturally leads to an emergent stress-energy tensor representing an anisotropic effective fluid. This backreaction effect of horizon entropy may resolve possible inconsistencies recently identified in black hole physics with modified entropies. As specific examples, we apply this procedure to a wide class of modified entropies, such as Barrow, Tsallis-Cirto, Renyi, Kaniadakis, logarithmic, power-law, loop-quantum-gravity, and exponential modifications, and we derive the associated effective matter sectors, analyzing their physical properties and energy conditions.