Annihilation-to-nothing: a quantum gravitational boundary condition for the Schwarzschild black hole

TL;DR

This paper explores quantum boundary conditions inside Schwarzschild black holes using the Wheeler-DeWitt equation, proposing an annihilation process that could resolve the black hole information paradox.

Contribution

It introduces a novel quantum gravitational boundary condition inside black holes and discusses implications for the black hole information loss paradox.

Findings

Wave function solutions match classical trajectories near the horizon.

Ambiguity in the arrow of time leads to two interpretations of wave function behavior.

Probabilities for black hole histories decay over time, favoring trivial geometries.

Abstract

The interior of a static Schwarzschild metric can be written in terms of two functions, similar to some models of anisotropic cosmology. With a suitable choice of canonical variables, we solve the Wheeler-DeWitt equation (WDW) inside the horizon of a Schwarzschild black hole. By imposing classicality near the horizon, and requiring boundedness of the wave function, we get a rather generic solution of the WDW equation, whose steepest-descent solution, i.e., the ridge of the wave function, coincides nicely with the classical trajectory. However, there is an ambiguity in defining the arrow of time which leads to two possible interpretations - (i) if there is only one arrow of time, one can infer that the steepest-descent of the wave function follows the classical trajectory throughout: coming from the event horizon and going all the way down to the singularity, while (ii) if there are two different arrows of time in two separate regimes, it can be inferred that the steepest-descent of the wave function comes inwards from the event horizon in one region while it moves outwards from the singularity in the other region, and there exists an annihilation process of these two parts of the wave function inside the horizon. Adopting the second interpretation could shed light on the information loss paradox: as time goes on, probabilities for histories that include black holes and singularities decay to zero and eventually only trivial geometries dominate.