Critical collapse of a massive scalar field in semi-classical loop quantum gravity

TL;DR

This work tests whether semi-classical loop quantum gravity corrections alter the well-known critical phenomena in gravitational collapse of a massive scalar field. Using two polymerization schemes, it shows that the two canonical regimes—type II behavior with universal echoing period $\Delta$ near $3.4$ and exponent $\gamma$ near $0.37$, and type I behavior with a finite black-hole mass gap at large $\mu$—persist when quantum corrections are included. The results, robust to the polymerization parameter $k$ and to the two covariant formalisms, indicate that the semi-classical corrections have negligible impact on critical collapse dynamics, preserving the classical universality and scaling. This supports the view that semi-classical LQG effects do not impose new thresholds for black-hole formation in these regimes, reinforcing the universality of critical phenomena in gravity.

Abstract

We investigate critical phenomena during the gravitational collapse of a massive scalar field under two distinct semi-classical loop quantum gravity (LQG) approaches within spherical symmetry. Numerical simulations reveal that the massive scalar field in both semi-classical frameworks exhibits two distinct types of critical behavior, consistent with the classical scenario. When the scalar field's mass parameter is small, type II critical phenomena emerge, with the resulting echoing periods and critical exponents precisely matching those obtained in general relativity. In contrast, a large mass parameter triggers type I critical phenomena, where the resulting black holes possess a finite minimum mass. These findings suggest that semi-classical corrections from LQG have a negligible impact on the dynamics of critical collapse.

Figures (4)

• Figure 1: Critical behavior of a massive scalar field with potential $V\left(\phi\right) = \mu^{2}\phi^{2}/2$ for $\mu=1.0$. (a) Evolution of the scalar field $\varphi$ at the origin $(r = 0)$ as a function of $-\ln{(T^{*}-T})$, where $T^{*}$ denotes the time of naked singularity formation. (b) Power-law scaling of the black hole mass in the supercritical regime. The initial profile parameters in Equation (\ref{['initial_psi']}) are $r_{0}=1.0$, and $\sigma=0.2$. The blue solid line and the magenta dashed line correspond to the cases $k=0$ and $k=1$, respectively.
• Figure 2: Relationship between the black hole mass and the initial amplitude in the supercritical regime for a scalar field with potential $V \left(\phi \right) = \mu^{2}\phi^{2}/2$ and mass parameter $\mu = 8.0$. The blue stars and magenta circles correspond to the cases $k=0$ and $k=1$, respectively.
• Figure 3: Critical behavior of a massive scalar field with potential $V\left(\phi\right) = \mu^{2}\phi^{2}/2$ for $\mu=1.0$. (a) Evolution of the scalar field $\varphi$ at the origin $(x = 0)$ as a function of $-\ln{(T^{*}-T)}$, where $T^{*}$ denotes the time of naked singularity formation. (b) Power-law scaling of the black hole mass in the supercritical regime. The initial profile parameters in Equation (\ref{['initial_psi2']}) are $x_{0}=1.0$, and $\sigma=0.2$. The blue solid line and the magenta dashed line correspond to the cases $k=0$ and $k=1$, respectively.
• Figure 4: Relationship between the black hole mass and the initial amplitude in the supercritical regime for a scalar field with potential $V \left(\phi \right) = \mu^{2}\phi^{2}/2$ and mass parameter $\mu = 8.0$. The blue stars and magenta circles correspond to the cases $k=0$ and $k=1$, respectively.