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

Abstract

A renormalization-group (RG) improved Schwarzschild-like black hole (BH) is studied here, with a lapse that interpolates between a classical Schwarzschild exterior and a quantum-smoothed interior set by a cutoff scale $\hat{I}¾$ and an interpolation parameter ${\hat{I}}^3$. We work out the horizon structure together with the photon sphere and shadow radius $R_{\mathrm{sh}}$, set up the scalar, electromagnetic, and Dirac Regge-Wheeler-Zerilli problems in a single treatment, and compute the fundamental and overtone quasinormal modes by sixth-order WKB, cross-checked against time-domain ringdown. For $\hat{I}¾>0$ and ${\hat{I}}^3>0$ the geometry is regular, with a de Sitter core. Strong Cosmic Censorship (SCC) is examined at the inner Cauchy horizon, which the improved geometry generates without charge or rotation. The quasinormal spectral gap ${\hat{I}}^2=|\mathrm{Im}\ddot{I}‰|/\hat{I}{º}_{-}$ stays multipole-independent at the $6\%$ level and follows ${\hat{I}}^2\simeq \hat{I}{»}_L/(2\hat{I}{º}_{-})$. It remains below the de Sitter Christodoulou bound across the parameter range, and the asymptotically flat late-time tail places the geometry in the SCC-respecting class. A thermodynamic analysis identifies a Davies-type phase transition of the outer horizon, with the Schwarzschild $T_H\propto 1/r_{+}$ decay replaced by a bell-curve profile peaking at $T_H^{\max }\simeq 0.062$. A scan of the $(\hat{I}¾,{\hat{I}}^3)$ plane gathers the joint behavior of the shadow, the scalar barrier, the SCC ratio, and $T_H$. Set against Bardeen, Hayward, and Bonanno-Reuter BHs at matched perturbation scale, the improved Schwarzschild BH is the most Schwarzschild-like of the regular-BH family, its static shadow radius degenerate with Hayward and Bonanno-Reuter at the percent level. The closing analysis takes up the sparsity of the Hawking flux and the energy-emission rate, both tied to the outer-horizon surface gravity through a single auxiliary function.

23 pages, 8 figures, and 9 tables. We are especially grateful to Prof. Roman Konoplya for his valuable comments and suggestions. The manuscript is improved