Quantum black holes: inside and outside

TL;DR

The paper investigates unitary evolution in evaporating black holes within canonical quantum gravity by studying the Wheeler-DeWitt equation on time slices that may cross the horizon. Employing the Page-Wootters clock formalism, it argues that coherent-state slicings must remain outside the event horizon to preserve unitarity, while the interior becomes a horizon-scale superposition of coherent states, implying a highly quantum macroscopic interior. A concrete Schwarzschild quantization is analyzed, revealing an annihilation-to-nothing behavior at the horizon—interpreted not as literal disappearance but as a superposition of geometries with horizon-scale uncertainty, consistent with a horizon-scale GUP-like picture. The work further connects these results to the De Sitter example, infalling-observer issues, and nonperturbative topology-change channels, ultimately suggesting a unitary evolution for external observers with interior quantum complexity and posing questions for deeper microscopic understanding and potential experimental analogs.

Abstract

For a unitary description of an evaporating black hole, one usually chooses the time slices that cover only outside of the event horizon, which is mostly problem-free because the event horizon is not encountered. However, is there any justification for avoiding time slices that cover inside the event horizon? To answer the question, we investigate the Wheeler-DeWitt equation, where the time slices can cover both inside and outside the event horizon. We find that one can reasonably construct a wave packet that covers outside, but the wave function must be annihilated near the event horizon. This observation strongly suggests that we cannot choose a coherent state for a spacelike hypersurface that crosses the event horizon. To explain the unitary time evolution, we must keep the slices as coherent states; hence, they must always be outside the event horizon. In contrast, inside the horizon, we cannot have a single coherent state of a classical spacetime. Hence, the interior must be a superposition of several coherent states, which implies that there exists a horizon-scale uncertainty and a black hole should be viewed as a highly quantum macroscopic object. We provide a synthetic approach to understanding the information loss paradox from this perspective.

Figures (11)

• Figure 1: Typical time slices of a semi-classical black hole. Which slices are self-consistent in terms of canonical quantum gravity?
• Figure 2: Classical trajectory of $a$ and $b$ space (black) for $M = 100$. The green-colored line denotes $r = 2M$. The left end corresponds to infinity, while the right end corresponds to a singularity.
• Figure 3: Several cases of potential boundaries. Left: the early time limit ($M = 100$, $t_{0} = 0.0001$). Middle: the late time limit ($M = 100$, $t_{0} = 5$). Right: the small mass limit ($M = 0.1$, $t = 0.001$). The red and violet dashed curves are potential wells, while the blue dashed curve is the potential barrier.
• Figure 4: A numerical solution for a wave packet starting from outside the event horizon. Here, we give $a_{m} = -100$, $a_{0} = -50$, $A = 1$, $\sigma = 1$, $M = 100$, and $a_{M} = -8.1$. One can provide a vanishing boundary condition at the potential well. The blue contour is the classical trajectory.
• Figure 5: Left: If we consider a wave packet inside the horizon for early time, there must exist its mirror image that may correspond to a sub-Planckian mass limit geometry. Right: The early Kruskal slicing does not touch the singularity. Hence, it is not surprising that there is no consistent classical wave packet near the singularity for early time slicings.