Lemaitre-Tolman-Bondi collapse from the perspective of loop quantum gravity

Lemaitre-Tolman-Bondi models as specific spherically symmetric solutions of general relativity simplify in their reduced form some of the mathematical ingredients of black hole or cosmological applications. The conditions imposed in addition to spherical symmetry turn out to take a simple form at the kinematical level of loop quantum gravity, which allows a discussion of their implications at the quantum level. Moreover, the spherically symmetric setting of inhomogeneity illustrates several nontrivial properties of lattice refinements of discrete quantum gravity. Nevertheless, the situation at the dynamical level is quite nontrivial and thus provides insights to the anomaly problem. At an effective level, consistent versions of the dynamics are presented which implement the conditions together with the dynamical constraints of gravity in an anomaly-free manner. These are then used for analytical as well as numerical investigations of the fate of classical singularities, including nonspacelike ones, as they generically develop in these models. None of the corrections used here resolve those singularities by regular effective geometries. However, there are numerical indications that the collapse ends in a tamer shell-crossing singularity prior to the formation of central singularities for mass functions giving a regular conserved mass density. Moreover, we find quantum gravitational obstructions to the existence of exactly homogeneous solutions within this class of models. This indicates that homogeneous models must be seen in a wider context of inhomogeneous solutions and their reduction in order to provide reliable dynamical conclusions.

Figure 1

The correction functions $\alpha (R)$ (solid line) and $f(R)$ (dashed line) where $R$ is taken relative to $R_{*}≔\sqrt{\gamma /2}{\ell }_{\mathrm{P}}$.

Figure 2

Convergence test for time integration.

Figure 3

The collapse of a homogeneous dust ball with $R_{\mathrm{s}}=0.1$ and $M=0.01$ in classical general relativity: the snapshots at $t=0$, 0.088 79, 0.1251, 0.1395, 0.1453, and 0.1476 are plotted.

Figure 4

The collapse of an inhomogeneous dust ball with $R_{\mathrm{s}}=0.1$ and $M=0.01$ in classical general relativity: the snapshots at $t=0$, 0.073 65, 0.089 84, 0.093 32, 0.094 07, and 0.094 24 are plotted.

Figure 5

The collapse of a homogeneous dust ball with $R_{\mathrm{s}}=1$ and $M=0.01$ in the first version of loop quantum gravity: the snapshots at $t=0$, 3.069, 4.046, 4.393, 4.519, and 4.564 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.

Figure 6

The collapse of an inhomogeneous dust ball with $R_{\mathrm{s}}=1$ and $M=0.01$ in the first version of loop quantum gravity: the snapshots at $t=0$, 2.008, 2.658, 2.889, 2.973, and 3.003 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.

Figure 7

The collapse of a homogeneous dust ball with $R_{\mathrm{s}}=0.1$ and $M=0.01$ in the first version of loop quantum gravity: the snapshots at $t=0$, 0.1051, 0.1394, 0.1511, 0.1552, and 0.1567 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.

Figure 8

The collapse of an inhomogeneous dust ball $R_{\mathrm{s}}=0.1$ and $M=0.01$ in the first version of loop quantum gravity: the snapshots at $t=0$, 0.1349, 0.1845, 0.2024, 0.2089, and 0.2113 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.

Figure 9

The collapse of a homogeneous dust ball with $R_{\mathrm{s}}=1$ and $M=0.01$ in the second version of loop quantum gravity: the snapshots at $t=0$, 1.433, 2.347, 2.954, 3.364, and 3.391 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.

Figure 10

The collapse of an inhomogeneous dust ball with $R_{\mathrm{s}}=1$ and $M=0.01$ in the second version of loop quantum gravity: the snapshots at $t=0$, 0.9379, 1.557, 2.004, 2.245, and 2.246 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.

Figure 11

The collapse of a homogeneous dust ball with $R_{\mathrm{s}}=0.1$ and $M=0.01$ in the second version formulation of loop quantum gravity: the snapshots at $t=0$, 0.5509, 1.173, 1.836, and 2.395 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.

Figure 12

The collapse of an inhomogeneous dust ball with $R_{\mathrm{s}}=0.1$ and $M=0.01$ in the second version formulation of loop quantum gravity: the snapshots at $t=0$, 1.501, 3.239, 5.121, 6.444 are plotted. In (a), the solid and dashed lines denote the effective density and the conserved mass density, respectively. In (c), the solid and dashed lines denote the Misner-Sharp mass and the conserved mass, respectively.