Wheeler-DeWitt Equation for Black Hole Interiors in Asymptotically Safe Gravity

TL;DR

This paper analyzes the Wheeler–DeWitt equation for black hole interiors within asymptotically safe gravity by embedding scale-dependent couplings into a Hamiltonian framework derived from a renormalization-group–improved Einstein–Hilbert action. In the canonical formulation, closure of the constraint algebra enforces a Gaussian normal foliation, leading to classical vacuum solutions that are largely insensitive to the running of Newton’s constant $G(k)$. In the quantum regime, ultraviolet running of $G$ and the cosmological constant $ ext{Λ}(k)$ introduces an effective potential in minisuperspace that drives the WDW wave function to decay exponentially near the classical singularity, irrespective of how the RG scale is tied to spacetime coordinates. This provides a robust mechanism for singularity suppression in black hole interiors driven by asymptotic safety, while highlighting that the precise form of scale identification and the inclusion of higher-curvature terms remain important directions for future work.

Abstract

In this work, we analyze the Wheeler-DeWitt equation with scale-dependent gravitational couplings within the framework of asymptotically safe gravity. In the Hamiltonian formulation based on a renormalization-group improved Einstein-Hilbert action, the consistency of the theory and the Poisson algebra of constraints have been clarified. Within this framework, we show that, despite the explicit scale dependence of Newton's constant, the classical solutions are generically unaffected by the running of the coupling. We then derive the Wheeler-DeWitt equation incorporating the scale dependence of the gravitational couplings and analyze its solutions in the minisuperspace framework. In the classical limit, while the scale dependence of Newton's constant does not affect the classical behavior, the running of the cosmological constant can contribute to the classical solutions. Moreover, we show that the quantum behavior in the ultraviolet regime acts toward suppressing singularity formation in all cases, independently of how the renormalization-group scale is identified with spacetime coordinates and of the relative magnitudes of the ultraviolet fixed points of the running Newton's constant and cosmological constant.