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A loop quantization of the marginally bound Lemaître-Tolman-Bondi dust model

Luca Cafaro, Farshid Soltani

Abstract

We present a loop quantization of the marginally bound Lemaître-Tolman-Bondi (LTB) model, describing the gravitational collapse of pressureless dust in spherical symmetry. The full quantum LTB model is constructed as a collection of non-interacting shells, each governed by an individual single-shell loop quantum dynamics. We show that the single-shell evolution is non-singular and that wave packets initially peaked on a collapsing trajectory undergo a bounce at Planckian energy densities and subsequently follow an expanding classical trajectory, resolving the classical central curvature singularity. We also compare the loop quantum theory with the Wheeler-DeWitt quantization of the same model and assess the accuracy of the loop quantum gravity effective theory in reproducing the full quantum dynamics. Specifically, we find that initially collapsing wave packets generically develop an interference pattern at the bounce, which suppresses the accuracy of the effective theory near the center of the dust cloud.

A loop quantization of the marginally bound Lemaître-Tolman-Bondi dust model

Abstract

We present a loop quantization of the marginally bound Lemaître-Tolman-Bondi (LTB) model, describing the gravitational collapse of pressureless dust in spherical symmetry. The full quantum LTB model is constructed as a collection of non-interacting shells, each governed by an individual single-shell loop quantum dynamics. We show that the single-shell evolution is non-singular and that wave packets initially peaked on a collapsing trajectory undergo a bounce at Planckian energy densities and subsequently follow an expanding classical trajectory, resolving the classical central curvature singularity. We also compare the loop quantum theory with the Wheeler-DeWitt quantization of the same model and assess the accuracy of the loop quantum gravity effective theory in reproducing the full quantum dynamics. Specifically, we find that initially collapsing wave packets generically develop an interference pattern at the bounce, which suppresses the accuracy of the effective theory near the center of the dust cloud.
Paper Structure (15 sections, 140 equations, 4 figures, 2 tables)

This paper contains 15 sections, 140 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classical theory
  3. Gauge-fixing and reduction to single shell
  4. Quantum theory
  5. Wheeler-DeWitt quantum theory
  6. Loop quantum theory
  7. Numerical analysis
  8. Multi-shell quantum theory
  9. Summary and Conclusions
  10. LTB conditions
  11. Self-adjointness of the WDW and LQG Hamiltonians
  12. Wheeler-DeWitt Hamiltonian
  13. LQG Hamiltonian
  14. Large v behavior and normalization of the loop eigenfunctions
  15. Semiclassical effective theory

Figures (4)

  • Figure 1: Plots portraying the evolution of $\psi(T, v)$. On the left, a 3D plot showing $\abs{\psi(T, v)}$. On the right, a snapshot of $\abs{\psi}^2$ at the bouncing time $T_b$. Units have been chosen to be $c=\hbar=G=\gamma=1$ and the initial values have been set to $v_0=20000$, $\rho_0=10^{-2} \rho_c$, and $\sigma=0.6$.
  • Figure 2: Expectation values of $\hat{\rho}{}_{{}_{\mathrm G}}$ and $\hat{V}$ with their variance for the state $\psi(T, v)$ in \ref{['fig:wavepacket']} compared with the trajectories predicted by the effective semiclassical theory.
  • Figure 3: A 3x3 table of plots showing $\abs{\psi(T_b, v)}^2$ for different values of $\eta$ and $\sigma$. The initial parameter $v_0$ has been set in such a way that the bouncing volume is the same ($v_b \approx 200$) for all the plots.
  • Figure 4: Probability density at the bounce $\abs{\psi(T_b, v)}^2$ for three different values of $\eta$ and $\sigma=\sigma_{\mathrm{min}}$. Differently from the configurations shown in \ref{['fig:table']}, the volume at which the bounce occurs has been increased to $v_b \approx 2000$.