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

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.

💡 Research Summary

This paper investigates how scale‑dependent gravitational couplings, as predicted by the asymptotic safety program, affect the Wheeler‑DeWitt (WDW) equation for the interior of black holes. Starting from the functional renormalization group (FRG), the authors obtain running Newton’s constant G(k) and cosmological constant Λ(k) that approach a non‑Gaussian ultraviolet (UV) fixed point. By identifying the RG scale k with a spacetime‑dependent quantity k(x) (e.g., a proper distance or a curvature invariant), these couplings become external fields G(x) and Λ(x). The modified Einstein‑Hilbert action, \