Quantum Collapse of a Thin Shell Revisited

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 viz., the time coordinate in the interior of the shell (originally addressed in [1]) 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 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.

Configurational entropy of black hole quantum cores

“…This reduction of degrees of freedom is common to canonical quantisation of most general relativistic models[34]. For similar results for thin shells, see Ref [35]. (see also Refs [36][37][38][39]…”

Slowly rotating quantum dust cores and black holes

“…1 For thin shells, see Refs. [16][17][18][19][20]. 2 We shall always use units with c 1 and often write the Planck constant p m p and the Newton constant G N p /m p , where p and m p are the Planck length and mass, respectively.…”