Toward a midisuperspace quantization of LeMaître-Tolman-Bondi collapse models

Abstract

LeMaître-Tolman-Bondi models of spherical dust collapse have been used and continue to be used extensively to study various stellar collapse scenarios. It is by now well known that these models lead to the formation of black holes and naked singularities from regular initial data. The final outcome of the collapse, particularly in the event of naked singularity formation, depends very heavily on quantum effects during the final stages. These quantum effects cannot generally be treated semiclassically as quantum fluctuations of the gravitational field are expected to dominate before the final state is reached. We present a canonical reduction of LeMaître-Tolman-Bondi space-times describing the marginally bound collapse of inhomogeneous dust, in which the physical radius R, the proper time of the collapsing dust , and the mass function F are the canonical coordinates R(r), (r) and F(r) on the phase space. Dirac's constraint quantization leads to a simple functional ͑Wheeler-DeWitt͒ equation. The equation is solved and the solution can be employed to study some of the effects of quantum gravity during gravitational collapse with different initial conditions.

Quantum general relativity and Hawking radiation

“…Using a version of the canonical transformation developed by Kuchař [10], the configuration variable L is replaced by a new variable F (the mass function). In terms of the new chart (τ, R, F, P τ , P R , P F ), the momentum and the Hamiltonian constraints read [8] …”

“…Thus we must transform from the Euclidean flat metric on the (R * , τ ) plane of Eqn. (8) to the metric in the (R, T ) coordinates. Using the relations (6) and (25) we get…”

“…In [8], we defined it in a natural way by exploiting the fact that the DeWitt super-metric is manifestly flat in the configuration space (τ, R * ),…”

“…In Section 2, we briefly recall key results from our previous paper on quantum dust collapse [8], leading up to the WKB solution of the Wheeler-DeWitt equation. In Section 3, this solution is specialized to the case of matter around a black hole, and shown to describe Hawking radiation.…”

“…A midisuperspace program to study the canonical quantum dynamics of the LTB dust collapse has been developed during the last two years [8], following earlier pioneering work by Kastrup and Thiemann [9] and Kuchař [10] on the quantization of the Schwarzschild geometry. Here one sets up a canonical description of the collapse, using the dust proper time, the area radius and the mass function of the cloud as canonical configuration variables.…”