Effective geometrodynamics for renormalization-group improved black-hole spacetimes in spherical symmetry

TL;DR

The paper develops an operational framework to implement renormalization-group (RG) improvements in spherically symmetric spacetimes by embedding scale-dependent gravity into a generally covariant, second-order theory based on two-dimensional Horndeski dynamics. Static RG-improved Schwarzschild spacetimes are shown to arise as vacuum solutions of master field equations derived from 2D Horndeski actions, enabling explicit reconstruction of the corresponding Horndeski functions and clarifying the relation to the spherically reduced Einstein–Hilbert action. The work systematically compares RG-improvement at the level of the action, equations, and solutions, revealing essential discrepancies and providing truncation schemes that preserve second-order dynamics while capturing partial higher-curvature effects. It also extends the framework to dynamical collapse and discusses how 2D Horndeski theories can be connected to 4D covariant actions, with implications for regular black-hole spacetimes and potential links to quasi-topological gravities. The result is a coherent, scalable method to study quantum-gravity corrections in black-hole spacetimes and their dynamical evolution within a covariant, lower-dimensional effective theory.

Abstract

We consider the spherically reduced Einstein-Hilbert action, Einstein field equations and Schwarzschild spacetime modified by a renormalization-group (RG) scale-dependent gravitational Newton coupling, and present a systematic and operational approach to such an RG-improvement. The master field equations for spherically symmetric gravitational fields, recently constructed from two-dimensional Horndeski theory, allow us to retain partial contributions from higher-curvature truncations of the effective action, while preserving the second-order nature of the resulting field equations. Static RG-improved black-hole spacetimes with an effective gravitational coupling depending on the areal radius and the Misner-Sharp mass are derived as vacuum solutions to these master field equations, and are thereby identified as solutions to generally covariant two-dimensional Horndeski theories. We discuss explicitly the embedding of previous key works on RG-improvement into the newly developed formalism to illustrate its broad range of applicability. This formalism moreover allows us to establish explicitly the discrepancies in the outcomes of RG-improvement when implemented at the level of the action, in the field equations, or in the Schwarzschild solution.

Figures (3)

• Figure 1: Effective average action $\Gamma_k$ interpolating between the bare action $S$ for $k \to \infty$ and the quantum effective action $\Gamma$ for $k \to 0$.
• Figure 2: Schematic representation of interrelations between RG-improvement at the level of solutions, field equations and actions in spherical symmetry. From left to right, the figure depicts the sets of most general spherically symmetric geometries, the most general field equations with up to second-order derivatives of the metric, and the most general action functionals for the separate degrees of freedom defining a spherically symmetric geometry, i.e., a two-dimensional metric and a scalar field, that lead to such field equations, i.e., the set of two-dimensional Horndeski theories. The black dots in each of these sets represent, respectively, the Schwarzschild solution, the spherically symmetric Einstein field equations, and the spherically reduced Einstein--Hilbert action. The solid red arrows represent the RG-improved Schwarzschild solution within the space of spherically symmetric geometries, as well as the induced RG-improvements within the space of master field equations and two-dimensional Horndeski actions. The red stars indicate specific examples of RG-improvements of the Schwarzschild solution, e.g., the Bonanno--Reuter black hole, for which we provide in this paper a matching set of spherically symmetric field equations and two-dimensional Horndeski action. The dashed orange arrows represent the RG-improvement implemented at the level of field equations, as well as the induced RG-improvements at the level of geometries and actions. The dotted green arrows indicate the RG-improvement implemented at the level of actions, as well as the induced RG-improvements at the level of geometries and field equations. The outcomes of these different RG-improvement procedures, indicated by the solid red, dashed orange and dotted green arrows, are in general inequivalent. The directions of the arrows connecting the sets of geometries, master field equations and two-dimensional Horndeski theories indicate the qualitative steps needed to evaluate the outcome of RG-improvement at these different levels.
• Figure 3: Gravitational collapse into a regular black-hole spacetime with time-time component of the metric $X_k(v,r)$ given by \ref{['eq:XkSol']} with a linear mass function $m(v) \propto v$, at different constant-$v$ slices. The free parameters are set to $M/m_{Pl}=1$, $G_0/m_{Pl}^{-2}=1$ and $\omega =5$.