On the Generalized Lemaitre Tolman Bondi Metric: Classical Sensitivities and Quantum Einstein-Vaz Shells

Abstract

In this paper, in the classical framework, the lower bounds for the sensitivities of the generalized Lemaitre Tolman Bondi metric are evaluated. The calculated lower bounds via the linear dynamical systems $L_{\frac{\partial }{\partial \theta }}$, $L_{\frac{\partial }{\partial r}}$, and $L_{\frac{\partial }{\partial \varphi }}$ are $-\ln 2+\ln |{\left(\dot{R}B\right)}^2-{\left(R^{\prime }\right)}^2|-2\ln |B|$, $2\ln |\dot{B}|-\ln 2$ and $-\ln 2-2\ln \left|B\right|+\ln |({\dot{R}}^2{B}^2-R^{\prime 2}){\sin }^2\theta -B^2{\cos }^2\theta |$ respectively. The sensitivities and the lower sensitivities via $L_{\frac{\partial }{\partial t}}$ are zero are also shown. In the quantum framework, the properties of the Einstein-Vaz shells which are the final result of the quantum gravitational collapse arising from the Lemaitre Tolman Bondi discussed by Vaz in 2014 are analyzed. In fact, Vaz showed that continued collapse to a singularity can only be obtained if one combines two independent and entire solutions of the Wheeler-DeWitt equation. Forbidding such a combination leads naturally to matter condensing on the Schwarzschild surface during quantum collapse. In that way, an entirely new framework for black holes (BHs) has emerged. The approach of Vaz was also consistent with Einstein's idea in 1939 of the localization of the collapsing particles within a thin spherical shell. Here, following an approach of oned of us (CC), we derive the BH mass and energy spectra via a Schrodinger-like approach, by further supporting Vaz's conclusions that instead of a spacetime singularity covered by an event horizon, the final result of the gravitational collapse is an essentially quantum object, an extremely compact “dark star”. This “gravitational atom” is held up not by any degeneracy pressure but by quantum gravity in the same way that ordinary atoms are sustained by quantum mechanics. Finally, the time evolution of the Einstein-Vaz shells is discussed.