[Paper Review] Dynamical Regge Calculus as Lattice Quantum Gravity

Hiroyuki Hagura|ArXiv.org|Apr 16, 2003

Black Holes and Theoretical Physics16 references3 citations

TL;DR

This paper proposes Dynamical Regge Calculus (DRC), a hybrid lattice model unifying dynamical triangulations and quantum Regge calculus to describe quantum gravity with enhanced physical degrees of freedom. By allowing both link-lengths and triangulation configurations to vary, DRC enables lattice diffeomorphism invariance and topology change via degenerate simplicial complexes, while numerical simulations in 3D pure gravity reveal a first-order phase transition with fractal spacetime geometry in the strong-coupling phase and a milder, lower-fractal-dimension structure in the critical region.

ABSTRACT

We propose a hybrid model of simplicial quantum gravity by performing at once dynamical triangulations and Regge calculus. A motive for the hybridization is to give a dynamical description of topology-changing processes of Euclidean spacetime. In addition, lattice diffeomorphisms as invariance of the simplicial geometry are generated by certain elementary moves in the model. We attempt also a lattice-theoretic derivation of the black hole entropy using the symmetry. Furthermore, numerical simulations of 3D pure gravity are carried out,exhibiting a large hysteresis between two phases. We also measure geometric properties of Euclidean `time slice' based on a geodesic distance, resulting in a fractal structure in the strong-coupling phase. Our hybrid model not only reproduces numerical results consistent with those of dynamical triangulations and Regge calculus, but also opens a possibility of studying quantum black hole physics on the lattice.

Motivation & Objective

• To develop a unified lattice model of quantum gravity that extends both dynamical triangulations and quantum Regge calculus by incorporating dynamical link-lengths and triangulation changes.
• To provide a constructive framework for describing topology-changing processes in Euclidean spacetime, such as those in spacetime foam, via degenerate simplicial configurations.
• To realize lattice diffeomorphism invariance as a symmetry generated by elementary moves, enabling a lattice-theoretic derivation of black hole entropy.
• To investigate the phase structure of 3D pure gravity using two integration measures, assessing the stability and physical consistency of the model.
• To explore the possibility of studying quantum black hole physics and spacetime confinement on the lattice through numerical simulations.

Proposed method

• The model combines dynamical triangulations (sum over triangulations) and quantum Regge calculus (integration over link-lengths) into a single path integral formulation.
• Elementary moves—generalized (p,q) moves—generate lattice diffeomorphisms and ensure ergodicity in the Monte Carlo simulations.
• The gravitational path integral is defined with two measures: the uniform measure ∏dl_i and the scale-invariant measure ∏dl_i/l_i, to test stability and physical consistency.
• Geometric observables such as average curvature ⟨R⟩ and average link-length ⟨l⟩ are measured to identify phase transitions.
• The fractal dimension of spatial slices is analyzed using geodesic distance, revealing distinct scaling behavior in different phases.
• A lattice-theoretic derivation of black hole entropy is attempted by counting quantum fluctuations near the horizon, constrained by a finite minimal length l_min ~ ℓ_P.

Experimental results

Research questions

• RQ1 Can a hybrid lattice model of quantum gravity be constructed that unifies dynamical triangulations and quantum Regge calculus, allowing both link-length and triangulation dynamics?
• RQ2 How can topology-changing processes in Euclidean spacetime be realized dynamically in a lattice quantum gravity model?
• RQ3 Can lattice diffeomorphism invariance be generated and used to derive the Bekenstein-Hawking entropy of a black hole in a discrete setting?
• RQ4 What is the phase structure of 3D pure gravity in this hybrid model, and how do different integration measures affect the stability and physical interpretation?
• RQ5 Does the model exhibit a crumpled or spiky phase, and what does the fractal dimension of spatial slices reveal about the nature of quantum spacetime?

Key findings

• Numerical simulations with the scale-invariant measure ∏dl_i/l_i show no pathological behavior, while the uniform measure ∏dl_i leads to exponential divergence of the number of configurations, indicating instability.
• A large hysteresis is observed in both ⟨R⟩ and ⟨l⟩ across two phases, signaling a first-order phase transition in 3D pure gravity.
• In the strong-coupling phase, spatial slices exhibit a fractal structure with large fractal dimension, suggesting a crumpled, 'spiky' spacetime with possible confinement of geometry.
• In the critical region, the spatial average distance (SAD) function is smooth, indicating a milder, lower-fractal-dimension geometry resembling a fractal manifold.
• In the weak-coupling phase, spacetime becomes spiky, consistent with results from quantum Regge calculus and dynamical triangulations.
• The model successfully reproduces numerical results from both dynamical triangulations and quantum Regge calculus in their respective regimes, validating its consistency and potential as a unified framework.