Investigation of the gravitational dust collapse of the LQG-inspired effective asymmetric bounce model

Abstract

We investigate gravitational dust collapse within an effective loop quantum gravity (LQG)-inspired model exhibiting an asymmetric bounce in the marginally bound case. This work extends previous studies, which have predominantly focused on models with either symmetric bounces or asymmetric bounces restricted to homogeneous dust configurations. Our analysis emphasises the phenomenological implications of the model through a combination of analytical and numerical investigations, with particular attention to singularity resolution and the formation of trapped surfaces. As in symmetric bounce models, the central curvature singularity inside the collapsing dust cloud is resolved. However, in contrast to the symmetric case, we find that a singularity emerges in the polymerised vacuum region during the bounce phase. This singularity can be identified as a shell-crossing singularity and exhibits the expected power-law behaviour of curvature scalars. Furthermore, likewise to the symmetric bounce models, we find a critical mass threshold governing the formation of inner and outer horizons in the pre-bounce phase. No analogous critical mass restriction arises for the formation of the inner horizon in the post-bounce phase, highlighting a qualitative difference between the pre- and post-bounce dynamics.

Figures (6)

• Figure 1: The evolution of the areal radius (left panel) for the mass $M(x) = 5$ in the symmetric (red) and asymmetric (blue) bounce model and the parametric relation between the time coordinate $t$ and the parameter $\eta$ (right panel).
• Figure 2: The signs of the expansion parameters $\theta_+$ (blue) and $\theta_-$ (red) plotted over $s(x) - t$ for the mass $M(x) = 5$ (left panel) and the time derivative of the solution $R(t, x)$ plotted over $s(x) - t$ for the masses $M(x) = 0.2, 0.747326, 2$ (orange, red, blue) indicating the existence of a critical mass for the formation of trapped regions (right panel). The inner diagram in the left panel shows an enlarged view of a small region around $s(x) - t = 0$.
• Figure 3: Inhomogeneous profiles (blue) for the integration constant $s(x)$ (left panel) and the mass function $M(x)$ with a maximum value of $M = 5$ in the polymerised vacuum region (right panel) based on the homogeneous reduction to the Oppenheimer-Snyder model (orange dashed).
• Figure 4: The Ricci scalar (left figure) and the Kretschmann scalar (right figure) shown over the $t - x$ plane. The red line indicates the spacetime location of the shell-crossing singularity and the white dashed lines mark the apparent horizons. The vertical lines at $x = 1$ and $x = 4$ display the position where the cross-sections for the diagrams provided in Fig. \ref{['fig:curvature1D']} were taken.
• Figure 5: The top left and right panels show the time evolution of the Ricci and Kretschmann scalar for fixed radial coordinates $x = 1$ (outer diagram) and $x = 4$ (inner diagram). The bottom diagram displays the curvature scalars at $x = 4$ plotted near the singularity on a double logarithmic scale where $\widetilde{\eta} := \eta - \eta_0$.