Spherically Symmetric Gravity on a Graph I: Theoretical Foundations

Abstract

This manuscript is the first in a series of instalments that investigate spherically symmetric solutions within the effective dynamics program of Loop Quantum Gravity. The choice of lattice is adapted such that it remains invariant under a set of symmetry transformations maximally mapping spherical symmetry to the discrete setting. The conditions for symmetry restriction of the dynamics are investigated and a subspace is identified to make computations feasible. Afterwards symplectic structure and scalar constraint are explicitly computed on this subspace. This lays the groundwork to target several particular solutions, such $k=1$ cosmology and black holes, which will serve as the subjects of forthcoming follow-up papers.

Figures (6)

• Figure 1: Diagrammatic representation of the effective dynamics algorithm outlined in Sec. \ref{['s4.1:algorithm']}. The top row depicts the symmetry restriction procedure in the continuum theory (see Sec. \ref{['s2.3:symmetry_restriction']}), while the second row illustrates the analogous procedure in the discretized theory (see Sec. \ref{['s3.3:symmetry_restriction']}). In transitioning from the continuum to the discretized theory, one must choose a discretization map $\mathfrak{D}_\gamma$, a regularized Hamiltonian $H_\gamma$, and translate the continuum symmetry group $\Phi^\Psi_\text{AB} < O(3)\rtimes\mathop{\mathrm{Diff}}\nolimits(\Sigma)$ to a discrete analog $\Phi^\Psi_\gamma < SU(2) \rtimes \mathrm{Stab}(\gamma,\gamma^*)$, where $\mathrm{Stab}(\gamma,\gamma^*) < \mathop{\mathrm{Diff}}\nolimits(\Sigma)$ is the set of all graph-preserving diffeomorphisms for the chosen spatial discretization. Once the truncated system is restricted to $(\overline{\mathcal{M}}_\gamma,\overline{\omega}_\gamma,\overline{H}_\gamma)$, one may choose to impose controlled approximations and construct an effective system $(\overline{\mathcal{M}}_\gamma, \overline{\omega}_\text{eff},\overline{H}_\text{eff})$. Lastly, one replaces the unphysical lattice spacing with a physical regulator, $\varepsilon \mapsto \bar{\mu}$, in order to obtain the final effective theory.
• Figure 2: Graphic depiction of the edges (left) and surfaces (right) comprising the building blocks of a spherical graph $\gamma$ and its dual $\gamma^*$. On the left, arrows have been placed on the three positively-oriented edges defined at the vertex $v\in\gamma^v$, and the discretization parameter $n\in\mathbb{N}$ is encoded by the coordinate lengths $\varepsilon_i$ defined in \ref{['edge_length']}. On the right, the three positively-oriented surfaces associated with the vertex $v$ (situated in the centre of the cubic region) are labelled, and we note that each $\mathcal{S}^v_i$ has a coordinate area $\varepsilon_j\varepsilon_k$ with $i\neq j \neq k$.
• Figure 3: Visualization of a spherical graph $\gamma$ (left) and the associated dual cell complex $\gamma^*$ (right) embedded in $\mathbb{R}^3$. Each green point represents a vertex $v\in \gamma^v$, which are connected by edges adapted to the spherical coordinate directions. For each edge, there is exactly one corresponding surface in the dual graph.
• Figure 4: Visual representations of the azimuthal (left) and polar (right) subgraphs defined in Eq. \ref{['subgraph_def']}. In Fig. \ref{['fig:gamma_phi']}, the red and blue lines correspond to the "mirrors" (i.e., axes of reflection) associated with the symmetry transformations $\rho_\varphi$, $\sigma_\varphi$, respectively (see \ref{['refl_phi']}). The dashed lines indicate mirrors associated with all other symmetries of the planar graph, each of which can be obtained from $\rho_\varphi$ or $\sigma_\varphi$ by means of the rotation \ref{['R_phi']}. Fig. \ref{['fig:gamma_pm']} contains a 3-dimensional mirror corresponding to the reflection $\rho_\theta$ (see \ref{['refl_theta']}), which exchanges the two disconnected components of the subgraph.
• Figure 5: Depiction of the various phase spaces that we consider in both the continuum and truncated theories. On the left, the invariant subspace $\overline{\mathcal{M}}_{\text{AB}}\subset\mathcal{M}_{\text{AB}}$ consists of all fixed points under the action of $\Phi^\Psi_{\text{AB}}$ (e.g., the blue point $m$). For points such as $m'\in\mathcal{M}_{\text{AB}}\backslash\overline{\mathcal{M}}_{\text{AB}}$, there exists at least one element $\phi_\psi\in\Phi^\Psi_{\text{AB}}$ such that $\mathcal{A}^R_{\phi_\psi}(m') = m" \neq m'$. On the right, we have the discretized phase space $\mathcal{M}_\gamma = \mathfrak{D}_\gamma(\mathcal{M}_{\text{AB}})$, where $\mathfrak{D}_\gamma = \{\mathfrak{D}_e\}_{e\in\gamma}$ denotes the collection of discretization maps associated with a graph $\gamma$. As in the continuum case, the invariant subspace $\overline{\mathcal{M}}_\gamma \subset \mathcal{M}_\gamma$ is defined with respect to a group $\Phi^\Psi_\gamma$ by $m_\gamma\in\overline{\mathcal{M}}_\gamma \iff \mathcal{A}^R_{\phi_\psi^\gamma}(m_\gamma) = m_\gamma$, $\forall \phi_\psi^\gamma\in\Phi^\Psi_\gamma$. Within $\overline{\mathcal{M}}_\gamma$, we also have the physically-relevant subspace $\overline{M}_{\gamma,\text{sym}} = \mathfrak{D}_\gamma(\overline{\mathcal{M}}_\gamma)$, which completes the tower of truncated phase spaces, $\overline{\mathcal{M}}_{\gamma,\text{sym}} \subset \overline{\mathcal{M}}_\gamma \subset \mathcal{M}_\gamma$.

Theorems & Definitions (3)

• Theorem 2.1: Symmetry Restriction of Dynamics
• proof
• proof