Renormalization-group improved Schwarzschild black hole: shadow, ringdown, and strong cosmic censorship

TL;DR

This paper investigates a quantum-corrected Schwarzschild black hole model, analyzing its horizon, shadow, perturbations, and thermodynamics, revealing novel features like phase transitions and near-extremal behavior, with implications for cosmic censorship.

Contribution

It introduces a renormalization-group improved Schwarzschild black hole model with detailed analysis of its structure, perturbations, thermodynamics, and comparison with other regular black holes, highlighting new quantum effects.

Findings

Derived horizon structure, photon sphere, and shadow radius.

Computed quasinormal modes and ringdown profiles.

Identified a phase transition and near-extremal behavior affecting stability.

Abstract

A renormalization-group (RG) improved Schwarzschild-like black hole is investigated, whose lapse function interpolates between the classical Schwarzschild exterior and a quantum-smoothed interior governed by the cutoff scale $\xi$ and interpolation parameter $\gamma$. The horizon structure, photon sphere, and shadow radius $R_{\mathrm{sh}}$ are derived, while scalar, electromagnetic, and Dirac Regge--Wheeler--Zerilli perturbations are treated in a unified framework. Fundamental and overtone quasinormal modes are obtained through sixth-order WKB calculations and checked against time-domain ringdown profiles. Strong Cosmic Censorship (SCC) is tested at the inner Cauchy horizon, generated here without charge or rotation, and the ratio $\beta=|\mathrm{Im}\,\omega|/\kappa_-$ is found to be multipole-independent at the $6\%$ level, following $\beta\simeq\lambda_L/\kappa_-$. Thermodynamic analysis reveals a Davies-type phase transition at the outer horizon and a nontrivial Weinhold--Ruppeiner geometry on the $(S,\gamma)$ slice. The Schwarzschild decay law $T_H\propto 1/r_+$ is replaced by a bell-shaped profile with a maximum $T_H^{\max}\simeq 0.062$. A scan over the $(\xi,\gamma)$ plane captures the joint behavior of the shadow, scalar barrier, SCC ratio, and Hawking temperature, revealing a thin crescent near the extremal boundary where Christodoulou-SCC is marginally violated. Comparison with Bardeen, Hayward, and Bonanno--Reuter black holes shows that the present solution is the most Schwarzschild-like member of the regular-black-hole family at matched perturbation scale, while remaining shadow-degenerate with Hayward and Bonanno--Reuter geometries at the $1\%$ level. Finally, the sparsity of the Hawking flux and the energy-emission rate are analyzed, both depending on $(\xi,\gamma)$ through a single auxiliary function tied to the outer-horizon surface gravity.