Quantizing Gravitational Collapse

It is generally believed that the following is true: the classical gravitational field of a sufficiently massive star will overcome its neutron degeneracy pressure and, barring quantum gravitational effects in the final stages of collapse, the star will form a singularity of space-time. The initial conditions determine the precise nature of the singularity of the collapse: either the singularity will be covered, in which case a black hole is formed, or it will be naked, i.e., not covered by a horizon. Naked singularities lead to violations of causality, which encouraged Penrose to propose the Cosmic Censorship Hypothesis (CCH)[1]. The CCH simply states that physically reasonable initial data cannot produce naked singularities. Nevertheless, to date there is little evidence that it is true in general relativity, even though most of our current emphasis on black hole physics is based on the validity of this hypothesis. The LeMâıtre-Tolman-Bondi (LTB) collapse of spherical, inhomogeneous dust[2] is (arguably) the simplest model that allows for the formation both of black holes and of naked singularities. It is defined by specifying two functions, viz., a mass function, F (ρ), and an energy function, f(ρ), where ρ is a shell label coordinate. The former represents the weighted mass contained within the matter shell labeled by ρ, and the latter is related to the velocity profile within the collapsing cloud at the initial time. The configuration space for all LTB models consists of the dust proper time, τ , and the area radius, R. The classical collapse may end in a black hole or in a naked singularity depending on the behavior of F (ρ) and f(ρ) near the center[3]. When it ends in a black hole, a significant fraction of the star is expected to evaporate via Hawking radiation [4] during the semi-classical phase. However, the back-reaction of the space-time prevents us from making definitive predictions at the final stages. A different result is obtained when the collapse is toward a naked singularity[5]. It is quantum mechanically unstable (the radiated power behaves as P ∼ (U0 −U)−α for some – model dependent – value of α > 0) but, during the validity of the semi-classical approximation (curvatures should be less than Planck scale), the collapsing cloud emits only about one Planck unit of energy[6]. Because the back-reaction does not become important so long as gravity can be treated classically, the future evolution of the star is governed exclusively by quantum gravity and it is impossible to say, from the semi-classical approximation, whether the star radiates away its energy on a short time scale or settles down into a black hole state. Quantum gravitational effects are therefore expected to modify the very