Effective dynamics of Janis-Newman-Winicour spacetime

Abstract

The effective dynamics of the Janis-Newman-Winicour spacetime inspired by loop quantum gravity is studied. Two different schemes are considered to regularize the Hamiltonian constraint for the quantum dynamics. In the $μ_0$ scheme in which the quantum parameters are treated as constants, the equations of motion generated by the effective Hamiltonian are solved analytically. The resulting quantum-corrected effective spacetime obviously extends the effective spacetime previously obtained in the literature. In the new effective spacetime, the naked singularity and the central singularity presented in the classical JNW spacetime are resolved by a series of quantum bounces. In the scheme of choosing the quantum parameters as Dirac observables, the effective dynamics is also solved in the light of the solution in $μ_0$ scheme. It turns out that the resulting effective spacetime has singularities due to the appearance of the zero points of the time reparametrization functions. Hence, the effective theory in this scheme does not remain valid throughout the full spacetime.

Figures (5)

• Figure 1: The plot of classical evolution: the parameters are chosen as $B = 10^3$, $\nu = 0.5$, $\kappa = 8\pi$, $\gamma=0.2375$ and $L_0=1$. The initial data are chosen at $t_0=-0.000775$. Both $p_b$ and $p_c$ monotonically decrease to zero toward both the past and future directions.
• Figure 2: The plot of the evolution in the effective theory with respect to the function of $b$: the parameters are chosen as $B =10^3$, $\nu = 0.5$, $\kappa = 8\pi$, $\hbar=1$ and $L_0=1$. The blue lines represent our solutions, the red lines represent the solutions in Ref.zhang2020quantum.
• Figure 3: The plot of the Penrose diagram for the effective spacetime metric: the curve AOC corresponds to the transition surface at $b=0$. The curve APC, ARC, AQC, and ASC correspond to the transition surfaces at $b=0$$b=-\frac{\pi}{\delta_b}$, $b=-\frac{2\pi}{\delta_b}$, $b=\frac{\pi}{\delta_b}$ and $b=\frac{2\pi}{\delta_b}$ respectively.
• Figure 4: The plot of $F_b$, $F_c$, $t_1$ and $t_2$ with the parameters $B = 4\times 10^4$, $\nu = 0.5$, $\kappa = 8\pi$, $\hbar=1$, $\gamma=0.2375$, $L_0=1$, $\delta_b=0.034$, $\delta_c=0.0374$: the constants of motion are fixed as $\frac{\partial f_b}{\partial O_1}=2.108 \times 10^{-6} >0$ in (a) and (b) while $\frac{\partial f_b}{\partial O_1}=-2.108 \times 10^{-6} <0$ in (c) and (d), $\frac{\partial f_c}{\partial O_2}=-8.386 \times 10^{-6} <0$ in (e) and (f) while $\frac{\partial f_c}{\partial O_2}=8.386 \times 10^{-6} >0$ in (g) and (h).
• Figure 5: The plot of $b$, $p_b$, $c$, $p_c$, $\dot{p}_c$, $\dot{p}_b$ and $R$ with the parameters $B = 4\times 10^4$, $\nu = 0.5$, $\kappa = 8\pi$, $\hbar=1$, $\gamma=0.2375$, $L_0=1$, $\delta_b=0.034$, $\delta_c=0.0374$: the constants of motion are fixed as $\frac{\partial f_b}{\partial O_1}=2.108 \times 10^{-6}$ and $\frac{\partial f_c}{\partial O_2}=-8.386 \times 10^{-6}$ such that both $F_b$ and $F_c$ have zero points.