Wave Functions for Quantum Black Hole Formation in Scalar Field Collapse

TL;DR

This paper investigates quantum wave functions for scalar field collapse leading to black hole formation, revealing quantum effects such as outgoing flux and slight horizon deviations, and compares quantum and classical results.

Contribution

It introduces quantum wave functions for self-similar black hole formation, analyzing quantum effects and their relation to classical evolution in scalar field collapse.

Findings

Quantum wave functions exhibit both incoming and outgoing flux.

Quantum corrections cause only slight deviations in the apparent horizon.

The quantum results align with semiclassical tunneling rates in the subcritical case.

Abstract

We study quantum mechanically the self-similar black hole formation by collapsing scalar field and find the wave functions that give the correct semiclassical limit. In contrast to classical theory, the wave functions for the black hole formation even in the supercritical case have not only incoming flux but also outgoing flux. From this result we compute the rate for the black hole formation. In the subcritical case our result agrees with the semiclassical tunneling rate. Furthermore, we show how to recover the classical evolution of black hole formation from the wave function by defining the Hamilton-Jacobi characteristic function as $W = \hbar {\rm Im} \ln \psi$. We find that the quantum corrected apparent horizon deviates from the classical value only slightly without any qualitative change even in the critical case.