Non-singular quantum gravitational dynamics of an LTB dust shell model: the role of quantization prescriptions

Abstract

We study some consequences of the loop quantization of the outermost dust shell in the Lemaître-Tolman-Bondi spacetime with a homogeneous dust density using different quantization strategies motivated by loop quantum gravity. Prior work has dealt with loop quantizing this model by employing holonomies and the triads, following the procedure in standard loop quantum cosmology. In this work we compare this quantization with the one in which holonomies and gauge-covariant fluxes are used. While both of the quantization schemes resolve the central singularity, they lead to different mass gaps at which a trapped surface forms. This trapped surface which is matched to an exterior generalized Vaidya spacetime disappears when the density of the dust shell is in the Planck regime. We find that the quantization based on holonomies and gauge-covariant fluxes generically results in an asymmetric evolution of the dust shell in which the effective mass associated with the white hole as seen by an external observer is 2/π of the one for the black hole. This effective difference in masses results from difference in the classical limits in pre-and post-bounce regimes in the two quantizations. This distinctive feature rules out formation of any black hole-white hole twin in presence of gauge-covariant flux modifications which is in contrast to the quantization using holonomies and triads where the gravitational collapse always leads to a black hole-white hole twins. In another striking difference, for the quantization based on holonomies and gauge-covariant fluxes there can be situations in which during a non-singular collapse only a black hole forms without a white hole.

Does loop quantum $μ_o$ scheme permit a black hole formation?

“…In this section, we briefly review the classical dust shell model of the gravitational collapse in the marginally bound case using connection and triad variables. Our discussion would parallel the one in [15], which we refer the reader for further details.…”

“…As already discussed in detail in [15], in terms of the areal radius R and its conjugate momentum, the gravitational and the matter sectors of the Hamiltonian constraint of the outermost dust shell are given respectively by…”

“…where the angular part in the Hamiltonian constraint has already been integrated and the integration along the radial direction has been performed from the center of the dust cloud x = 0 to its outermost shell x = x b (namely the boundary surface), yielding R b = a(t)x b and F b = F (x b ). In order to loop quantize the dust shell model, it is a prerequisite to express the above Hamiltonian constraint of the dust shell model in terms of the Ashtekar variables which was first analyzed in [34] for an inhomogeneous dust cloud and later adapted to the shell model for a homogeneous dust cloud in [15]. In the dust shell model, one can identify the radial component of the densitized triad with the square of the areal radius, namely…”

“…[13,14,[17][18][19][20][21][22][23][24][25]). Investigations on singularity resolution have been carried out also for dynamical gravitational collapse spacetimes when a matter field is taken into account 2 [15,[28][29][30][31][32][33]. These studies mainly focused on the μ scheme and showed that the central singularity is generically resolved and a black hole can form when its mass is above some threshold value.…”

On consistent gauge fixing conditions in polymerized gravitational systems

“…1 In recent years, LQG techniques have been applied for various isotropic and anisotropic spacetimes in cosmology [19], and static and dynamical spherically symmetric spacetimes in the black hole physics (see for eg. [18,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]). Many of these works use the effective theory (or the polymerization) to understand Planck scale physics from LQG.…”