Topological and optical signatures of modified black-hole entropies

TL;DR

This paper explores how different modified black-hole entropies affect spacetime geometry, thermodynamics, and optical properties, providing observational bounds and insights into deviations from classical entropy models.

Contribution

It introduces a method to analyze the topological and optical signatures of various modified entropies using effective metrics and observational data.

Findings

Barrow and Rènyi entropies produce a single unstable topological sector.

Logarithmic and Kaniadakis corrections create canceling topological defects.

Modified entropies induce characteristic shifts in photon-sphere and shadow size.

Abstract

We investigate how deviations from the Bekenstein-Hawking entropy modify black-hole spacetimes through the recently proposed entropy-geometry correspondence. For four representative modified entropies, namely Barrow, R\'enyi, Kaniadakis, and logarithmic, we derive the corresponding effective metrics and analyze their thermodynamic and topological classification using the off-shell free energy and winding numbers. We show that Barrow and R\'enyi entropies yield a single unstable sector with global charge $W=-1$, while logarithmic and Kaniadakis corrections produce canceling defects with $W=0$, revealing topological structures absent in the Schwarzschild case. Using the modified metrics, we further calculate the photon-sphere radius and shadow size, showing that each modified entropy relation induces characteristic optical shifts. Thus, by comparing with Event Horizon Telescope observations of Sgr A$^\ast$, we extract new bounds on all entropy-deformation parameters. Our results demonstrate that thermodynamic topology, together with photon-sphere phenomenology, offers a viable way to test generalized entropy frameworks and probe departures from the Bekenstein-Hawking area law.