We derive an effective Hamiltonian constraint for the Schwarzschild geometry starting from the full loop quantum gravity Hamiltonian constraint and computing its expectation value on coherent states sharply peaked around a spherically symmetric geometry. We use this effective Hamiltonian to study the interior region of a Schwarzschild black hole, where a homogeneous foliation is available. Descending from the full theory, our effective Hamiltonian preserves all relevant information about the graph structure of quantum space and encapsulates all dominant quantum gravity corrections to spatially homogeneous geometries at the effective level. It carries significant differences from the effective Hamiltonian postulated in the context of minisuperspace loop quantization models in the previous literature. We show how, for several geometrically and physically well motivated choices of coherent states, the classical black hole singularity is replaced by a homogeneous expanding Universe. The resultant geometries have no significant deviations from the classical Schwarzschild geometry in the pre-bounce sub-Planckian curvature regime, evidencing the fact that large quantum effects are avoided in these models. In all cases, we find no evidence of a white hole horizon formation. However, various aspects of the post-bounce effective geometry depend on the choice of quantum states. Finally, we show how a de Sitter Universe extending the classical spacetime past the singularity can be recovered by means of the simplicity constraint.
- Quantum Gravity
- Scientific Series