Quantum Collapse of a Thin Shell Revisited

TL;DR

This paper compares two quantum descriptions of a collapsing thin shell using different time parameters, revealing how the shell's mass relative to the Planck mass influences the existence of stationary states.

Contribution

It provides exact solutions to the Wheeler-DeWitt equation for a thin shell in two different time frameworks, highlighting the role of the Planck mass in quantum collapse.

Findings

Stationary states exist only if shell mass is less than the Planck mass in interior time.

Stationary states exist only if shell mass exceeds the Planck mass in proper time.

In coordinate time, both scattering and bound states are present, with a well-defined energy spectrum.

Abstract

There are several possible choices of the time parameter for the canonical description of a self-gravitating thin shell, but quantum thories built on different time parameters lead to unitarily inequivalent descriptions. We compare the quantum collapse of a thin dust shell in two different times {\it viz.,} the time coordinate in the interior of the shell (originally addressed in \cite{hajicek92a}) and the time coordinate of the comoving observer (proper time). In each case, we obtain exact solutions to the Wheeler-DeWitt equation requiring only a finite and well behaved $U(1)$ current. The two quantum theories are complementary and each highlights the role played by the Planck mass: stationary states of positive energy in interior time exist only if the shell rest mass in smaller than the Planck mass. In proper time they exist only when the shell rest mass is {\it greater} than the Planck mass. In coordinate time there are both scattering states and bound states with a well defined energy spectrum. In the proper time description there are only bound states, whose spectrum we determine.