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Effective Quantum Gravitational Collapse in Metric Variables: The $\barμ$ Scheme

L. Boldorini, G. Montani

TL;DR

This work addresses the singularity problem in gravitational collapse by applying the bar_mu polymerization of Loop Quantum Gravity to the Oppenheimer-Snyder model in metric variables. By discretizing the area variable $A$ with a fixed volume lattice and replacing the momentum dependence with a sine-like term, they derive an effective Hamiltonian that yields a bounce at Planckian densities and a transition from a black hole to a white hole state. The bounce occurs inside the horizon for non-Planckian mass, while Planckian mass allows an outside-horizon bounce; a negative quantum gravitational pressure $P_{pol}$ interprets the quantum corrections as a volume tension. The framework provides a transparent, Planck-scale mechanism for quantum corrections to collapse and clarifies the role of the polymer corrections in driving the bounce.

Abstract

We study, using the metric variables, how an effective theory for the Oppenheimer-Snyder gravitational collapse can be built with the $\barμ$ scheme from Loop Quantum Gravity (LQG). The collapse is analyzed for both the flat and spherical models. In both scenarios the effective theory make possible to avoid the formation of the singularity. The source of this is found in the presence of a negative pressure term inside the stress-energy tensor of the gravitational field. This pressure is analyzed and is concluded that the effective polymer model is the reason why the negative pressure appears. A characterization of the solutions for both models is also carried out, showing that the collapse is altered and avoided in favor of a transition from a black hole state to a white hole one, transition that occurs when the collapse has reached a Planckian regime.

Effective Quantum Gravitational Collapse in Metric Variables: The $\barμ$ Scheme

TL;DR

This work addresses the singularity problem in gravitational collapse by applying the bar_mu polymerization of Loop Quantum Gravity to the Oppenheimer-Snyder model in metric variables. By discretizing the area variable with a fixed volume lattice and replacing the momentum dependence with a sine-like term, they derive an effective Hamiltonian that yields a bounce at Planckian densities and a transition from a black hole to a white hole state. The bounce occurs inside the horizon for non-Planckian mass, while Planckian mass allows an outside-horizon bounce; a negative quantum gravitational pressure interprets the quantum corrections as a volume tension. The framework provides a transparent, Planck-scale mechanism for quantum corrections to collapse and clarifies the role of the polymer corrections in driving the bounce.

Abstract

We study, using the metric variables, how an effective theory for the Oppenheimer-Snyder gravitational collapse can be built with the scheme from Loop Quantum Gravity (LQG). The collapse is analyzed for both the flat and spherical models. In both scenarios the effective theory make possible to avoid the formation of the singularity. The source of this is found in the presence of a negative pressure term inside the stress-energy tensor of the gravitational field. This pressure is analyzed and is concluded that the effective polymer model is the reason why the negative pressure appears. A characterization of the solutions for both models is also carried out, showing that the collapse is altered and avoided in favor of a transition from a black hole state to a white hole one, transition that occurs when the collapse has reached a Planckian regime.
Paper Structure (10 sections, 58 equations, 6 figures)

This paper contains 10 sections, 58 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. THE CLASSICAL OPPENHEIMER-SNYDER MODEL
  3. THE EFFECTIVE MODEL
  4. The $\bar{\mu}$ Effective Model
  5. Area Dynamics
  6. Momentum Dynamics
  7. PHENOMENOLOGY
  8. Minima and Consistency Conditions
  9. The Quantum Gravitational Pressure
  10. CONCLUSIONS

Figures (6)

  • Figure 1: Phase Portraits in Geometrical Units for both the Flat and Spherical models.
  • Figure 2: Collapse and Bounce of the Dust Sphere in the Spherical case. The minimum area is $\mathcal{A}_{min} = 1.61\cdot 10^{-49} \textit{Km}^2$
  • Figure 3: Collapse and Bounce of the Dust Sphere in the Flat case. The minimum area is $\mathcal{A}_{min} = 1.06\cdot 10^{-49} \textit{Km}^2$
  • Figure 4: Plot of the Momentum Trajectory against Surface in the Spherical model for both the In-Falling and Out-Going branches. A Symmetric-Log scale is used to dispaly both branches. The y-axis displays only the magnitude of the momentum, being the Log-scale unsigned.
  • Figure 5: Plot of the Momentum Trajectory against Surface in the Flat model for both the In-Falling and Out-Going branches. A Symmetric-Log scale is used to dispaly both branches. The y-axis displays only the magnitude of the momentum, being the Log-scale unsigned.
  • ...and 1 more figures