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Jaehong Oh

χPreprint · 2025

Internal Time in Loop Quantum Gravity: Computational Verification of General Relativity Emergence

Jaehong Oh

143 words1 min read

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Abstract

We present a computational framework that demonstrates the emergence of General Relativity from Loop Quantum Gravity using an explicit internal time functional T[γ]. The problem of time in quantum gravity — how to define temporal evolution when the Hamiltonian constraint Ĥ|Ψ⟩ = 0 eliminates time as an external parameter — has remained a central conceptual obstacle for decades. We resolve this by constructing a directed, reparametrization-invariant time functional on spin network configuration space, inspired by Fisher information geometry and thermodynamic entropy growth. Through seven key experiments we achieve: (1) Einstein-equations emergence G_μν = Π_μν verified to machine precision (ε < 10⁻¹⁶), with consistency error Δ_cons < 0.001% between area and scale factor; (2) gauge invariance across clock choices (ρ = 1.000); (3) matter emergence with power law H ∝ a^{-1.984} (1.58% deviation from theoretical H ∝ a⁻²); (4) singularity resolution (a_min = 0.707 ℓ_P > 0); (5) sub-quadratic computational scaling to N = 600.

Overview

The middle paper in the three-part LQG internal-time sequence. It takes the internal-time functional established in the bounce paper and uses it to verify — numerically, to machine precision — that classical General Relativity emerges from LQG at macroscopic scales.

What the paper shows

Five headline results from seven numerical experiments:

  1. Einstein-equation emergence verified to ε < 10⁻¹⁶ machine precision in vacuum, with < 0.001% consistency error between area and scale factor.
  2. Gauge invariance — volume and entropy clock choices yield perfectly correlated geometries (ρ = 1.000), confirming clock choice as pure gauge.
  3. Matter emergence — radiation-dominated scaling emerges naturally with only a 1.58% deviation from the theoretical H ∝ a⁻², suggestive of an intrinsic quantum-geometry feature.
  4. Singularity resolution — confirms the bounce paper's finding, a_min = 0.707 ℓ_P.
  5. Scalability — T_comp ∝ N^1.66 up to N = 600.

Where it sits

Continuation of the bounce paper and direct precursor to the scalable simulations paper.

BibTeX· generated

@misc{oh2025lqg,
  title   = {Internal Time in Loop Quantum Gravity: Computational Verification of General Relativity Emergence},
  author  = {Jaehong Oh},
  year    = {2025},
  url     = {https://jack0682.github.io/papers/lqg-internal-time-gr-emergence/},
}