Quantum collapse of a small dust shell

Abstract

The full quantum mechanical collapse of a small relativistic dust shell is studied analytically, asymptotically and numerically starting from the exact finite dimensional classical reduced Hamiltonian recently derived by Hájíček and Kuchař. The formulation of the quantum mechanics encounters two problems. The first is the multivalued nature of the Hamiltonian and the second is the construction of an appropriate self adjoint momentum operator in the space of the shell motion which is confined to a half line. The first problem is solved by identifying and neglecting orbits of small action in order to obtain a single valued Hamiltonian. The second problem is solved by introducing an appropriate lapse function. The resulting quantum mechanics is then studied by means of analytical and numerical techniques. We find that the region of total collapse has very small probability. We also find that the solution concentrates around the classical Schwarzschild radius. The present work obtains from first principles a quantum mechanics for the shell and provides numerical solutions, whose behavior is explained by a detailed WKB analysis for a wide class of collapsing shells.

Quantum collapse of dust shells in 2 + 1 gravity

“…[5], [9], [8], especially the the existence of the interference fringes that were noticed in Refs. [5] and [9]. The results for horizon formation are at least consistent with those of previous work.…”

“…Once this definition has been decided upon, one must try to find out if some collapse process will result in the formation of such a horizon, with the result being a probability of horizon formation. Theories of quantum gravity in their present state are far from being able to give us this result, so in shell collapse some articles [5], [9], [8] have tried to give an estimate of horizon formation by finding out if a sharply peaked wave packet, during its collapse toward R = 0, will fall, in some sense, below the classical horizon radius, R H . In some sense, because the packet will usually spread and basically will never lie entirely below R = R H .…”

“…Now, in order to make contact with previous work, we would like to study the evolution of a wave packet sharply peaked around a value of y = y 0 , some distance outside the point where a classical horizon would form if the radius of the shell were to fall below R H = ℓ √ 2M 0 − 1 and follow its movement as the packet falls toward R = 0. In previous work it was necessary to solve this problem numerically, since in the 3 + 1 case analytic solutions to the Schrödinger equation did not exist [5] [9], and in the Newtonian case, [8], while analytic solutions existed, it was impossible to sum the expansions for the relevant wave functions to give an analytic expression for the collapsing wave packet. In the 2 + 1 case, however, we can give an exact analytic expression for the wave packet as a coherent harmonic-oscillator state.…”

“…Hájíček [4] has given a sufficient but not necessary condition that shows that we might expect problems above a couple of Planck masses. In [5], a qualitative argument was presented that used Compton wavelength considerations to give a similar bound.…”

Minisuperspace, WKB and Quantum States of General Relativistic Extended Objects

“…Starting from these toeholds many authors have then tackled the problem of finding some hints about the properties of the still undiscovered quantum theory of gravitational phenomena using shells as convenient models. In this context, just as examples of what can be found in the literature, we quote the seminal works of Berezin [5] and Visser [6], that date back to the early nineties, or the more recent [7,8] and references therein.What we are going to shortly discuss in the present contribution is set in this last perspective and suggests a semiclassical approach to define WKB quantum states for spherically symmetric shells. This method has already been used in [4].…”