Spherically Symmetric Gravity on a Graph I: Theoretical Foundations

TL;DR

This paper establishes the theoretical foundations for analyzing spherically symmetric solutions in Loop Quantum Gravity, focusing on symmetry-adapted lattice structures, and sets the stage for future studies on cosmological and black hole models.

Contribution

It introduces a symmetry-invariant lattice framework and derives the symplectic structure and scalar constraint for spherically symmetric effective dynamics in Loop Quantum Gravity.

Findings

Defined a symmetry-adapted lattice for spherical symmetry

Computed the symplectic structure on the reduced phase space

Derived the scalar constraint for the symmetry-restricted system

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.