Lessons from gauge fixing and polymerization of loop quantum black holes with a cosmological constant

TL;DR

This work assesses the viability of loop-quantized Schwarzschild spacetimes with a cosmological constant under constant polymerization parameters in the Kantowski-Sachs gauge. Using an effective Hamiltonian with $b o \frac{\sin(\boldsymbol{\delta_b} b)}{\boldsymbol{\delta_b}}$ and $c\to \frac{\sin(\boldsymbol{\delta_c} c)}{\boldsymbol{\delta_c}}$, the authors solve the dynamics across the regimes Λ>0, 0<Λ<Λc, Λ=Λc, and Λ<0, anchored to GR at early times. They find that for Λ>0 a regular transition surface replacing the central singularity always accompanies an additional black-hole-like horizon far from the center, even at low curvatures, a pathology tied to the Kantowski–Sachs gauge with constant polymerization; while Λ<0 yields well-behaved quantum corrections with standard BH/WH structure and a resolved singularity. The results highlight that fixed-parameter schemes struggle to reproduce GR in the infrared for positive Λ and align with μ_o scheme limitations, suggesting the need for δ_b, δ_c that vary with phase space or hybrid approaches to obtain a consistent loop quantization of Schwarzschild–de Sitter spacetimes.

Abstract

Loop quantization of Schwarzschild black holes with a cosmological constant for polymerization parameters which are constant is studied in the effective spacetime description. We show that for the positive cosmological constant there can be an appearance of large quantum effects at small spacetime curvatures. These effects can manifest as an additional black hole horizon. While the central singularity is resolved in all the cases, these limitations demonstrate incompatibility of the Kantowski-Sachs gauge and schemes with fixed polymerization parameters in the presence of a positive cosmological constant. In contrast, the case of a negative cosmological constant is free of such problematic features. Noted limitations are similar to those in the $μ_o$ scheme for the loop quantization of cosmological models.

Figures (14)

• Figure 1: (a) Plot of the function ${\cal{A}}(\tau) \left(\equiv \Lambda\tau^2/3 + 2m/\tau - 1\right)$ for $\Lambda > 0$. The black dot dashed curve represents the $0 < \Lambda < \Lambda_c$ case with two horizons, denoted by $\tau_{\mathrm{BH}}$ and $\tau_{\mathrm{CH}}$ respectively, where $\Lambda_c \equiv 1/(9m^2)$. The red solid curve represents the $\Lambda = \Lambda_c$ case with a degenerate horizon denoted by $\tau_{\mathrm{DH}}$, and the blue dashed curve represents $\Lambda < \Lambda_c$ and has no horizons, so the singularity located at $\tau = 0$ is naked. (b) Plot of ${\cal{A}}(\tau)$ for $\Lambda<0$, in which a black hole horizon always exists. In all the regions where ${\cal{A}}(\tau) > 0$ the spacetime can be written locally in the Kantowski-Sachs form (\ref{['eq2.1']}), and the effective loop quantization is applicable to such regions.
• Figure 2: The effective dynamics of the phase space variables and metric components for $m=10^4$, $\delta_b=0.1172$, $\delta_c=0.0126$ and $\Lambda/\Lambda_c = 1.1$ for the $\Lambda > \Lambda_c$ case, in which there appears classically a naked singularity at the center $p_c^{\mathrm{GR}} = 0$. The normalized Hamiltonian is plotted to monitor the accuracy of our numerical results. Plots of $\sin{\left(\delta_b b\right)}$ and $\Theta_+$ are also presented to understand the existence of horizons and transition surfaces. The transition surface is located at $T_{\cal{T}} \simeq -8.5499$, while the white hole and black hole are located at $T_{\mathrm{WH}} \simeq -22.1844$ and $T_{\mathrm{BH}} \simeq 5.0845$, respectively. The initial conditions used to produce these plots are set at $T_i = 0.5$.
• Figure 3: Curvature invariants such as the Kretschmann scalar $K (\equiv R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$), $R^{ab} R_{ab}$, the Weyl scalar $C (\equiv C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta})$, the energy density $\rho$, and the radial and tangential pressures $p_r$ and $p_{\theta}$ of the effective energy-momentum tensor for $m=10^4$, $\delta_b=0.1172$, $\delta_c=0.0126$ and $\Lambda/\Lambda_c = 1.1$.
• Figure 4: Plots of the phase space variables $b,p_b, c$ and the metric components $N^2, g_{xx}$ show deviations from the classical trajectories near the black hole horizon located at $T_{\mathrm{BH}} \simeq 5.0845$ for $m=10^4$, $\delta_b=0.1172$, $\delta_c=0.0126$ and $\Lambda/\Lambda_c = 1.1$.
• Figure 5: The dynamics of the phase space variables in the black hole interior ($T < T_{\mathrm{BH}}$) for $m=10^4$, $\delta_b=0.1172$, $\delta_c=0.0126$ and $\Lambda/\Lambda_c = 0.9$. The normalized Hamiltonian is plotted to monitor the accuracy of our numerical results, and additional plots are for the lapse function $N^2$, the metric component $g_{xx}$ and the quantities $\sin{\left(\delta_b b\right)}$ and $\Theta_+$. The initial conditions used to produce these plots is set at $T_i = -10^{-3}$, where the classical black hole horizon is located at $T_{\mathrm{BH}} = 0$. A transition surface, a black hole and a white hole horizon are developed at $T_{\cal{T}} \simeq -8.79595$, $T_{\mathrm{BH}} \simeq 0$ and $T_{\mathrm{WH}} \simeq -17.579$, respectively.