Expansion operators in spherically symmetric loop quantum gravity

TL;DR

This work quantizes the ingoing and outgoing null expansions associated with a 2-sphere in a spherically symmetric loop quantum gravity model. It constructs a graph-based kinematical Hilbert space and defines self-adjoint expansion operators using a $ar{\mu}$-regularization for $K_\varphi$, with explicit action on spin-network states. Spectral analysis reveals a common essential spectrum forming a finite band, plus sector-dependent isolated eigenvalues outside the band; zero lies within the continuous part, suggesting a quantum horizon structure and potential singularity avoidance. The results establish a promising route toward a quantum notion of horizons in LQG, while highlighting that alternative quantization schemes could modify the operator properties and semiclassical behavior.

Abstract

The ingoing and outgoing null expansions associated to a spatial 2-sphere are quantized in the spherically symmetric model of loop quantum gravity. It is shown that the resulting expansion operators are self-adjoint in the kinematical Hilbert space with generalized eigenstates. It turns out that the outgoing and ingoing expansion operators share the common continuous part of their spectra but have different additional isolated eigenvalues. These results provide new insights on the avoidance of the singularities in classical general relativity and the establishment of certain notion of quantum horizons.

Figures (5)

• Figure 1: Numerical spectra of the expansion operators obtained from truncated matrix approximations ($n=10000$). (a) Spectrum of $\widehat{\Theta}_{\rm out}$. (b) Spectrum of $\widehat{\Theta}_{\rm in}$. The dashed lines indicate the analytic band edges at $\beta=\pm 2$. Discrete eigenvalues appear only above the band for $\widehat{\Theta}_{\rm out}$ and only below the band for $\widehat{\Theta}_{\rm in}$, while the continuous part fills the band interval.
• Figure 2: The maximum gap between adjacent eigenvalues of $\mathsf{E}$ in the interval $[-2,2]$ for different values of $n$.
• Figure 3: Convergence test for the truncated-matrix approximation of $\widehat{\Theta}_{\rm out}$: Three representative discrete eigenvalues lying above the continuous band are given as the truncation (matrix size) is increased. The observed plateaus indicate the numerical convergence of these discrete levels.
• Figure 4: Representative localized eigenstates $\mathsf{E}_n$ associated with eigenvalues outside the band $[-2,2]$, obtained from truncated matrix approximations with $n=10000$. (a) Eigenstates of $\widehat{\Theta}_{\rm out}$ (top panel). (b) Eigenstates of $\widehat{\Theta}_{\rm in}$ (bottom panel).
• Figure 5: Eigenstate of $\hat{\Theta}_{\rm out}(v_j)$ with eigenvalue $\omega=0$.