New Improved Schwarzschild Black Hole and Its Thermodynamics and Topological Classification

Abstract

We construct a renormalization-group improved Schwarzschild-like black hole geometry using the exact new scheme running for the Newton coupling. The scale identification is implemented via a standard interpolating proper-distance function that smoothly connects the ultraviolet and infrared regimes. We present the resulting coordinate-dependent coupling and the improved metric function, analyzing its asymptotic expansions. The large-distance limit is shown to recover the classical Schwarzschild solution, while the short-distance behavior exhibits a regular de Sitter-like core, demonstrating the regularization of the central singularity. We also analyze the thermodynamic properties of the solution, showing that quantum corrections significantly modify the small-radius behavior, leading to a remnant configuration and a nontrivial phase structure. Finally, we perform a topological classification of the thermodynamic phase space and demonstrate that asymptotically safe effects shift the critical point while preserving the global topological number of the Schwarzschild solution.

Figures (6)

• Figure 1: Metric potential as a function of the radial coordinate, for $M=1.0$. Left panel: Varying the interpolation parameter $\gamma$, with $\xi=0.6$. Right panel: Varying the cutoff scale $\xi$, with $\gamma=0.7$. Note that $\xi=0.0$ curve represents the Schwarzschild black hole.
• Figure 2: Parameter space $(M,\xi)$ of the new improved Schwarzschild black hole. The shaded region indicates the values of the parameters yielding black hole solutions with event horizons, bounded by the critical line where the horizon degenerates. Beyond this boundary, the spacetime describes no black hole. The interpolation parameter is set to $\gamma=0.6$.
• Figure 3: Left panel: Kretschmann scalar as a function of $r$, varying the interpolation parameter $\gamma$, with $\xi=0.6$. Right panel: The same quantity, now varying the cutoff scale $\xi$, with $\gamma=0.3$. It was considered $M=1.0$.
• Figure 4: Top panel:Hawking temperature as a function of the horizon radius $r_+$, varying the interpolation parameter $\gamma$, with $\xi=0.6$. Bottom panel: The same quantity, now varying the cutoff scale $\xi$, with $\gamma=0.3$.
• Figure 5: Approximate black hole entropy obtained from a perturbative expansion up to second order in $\xi$, for different values of the parameter $\xi$. In all curves we fix $\gamma = 0.3$.