The Emergence of the Minkowski Metric from Causal Sets via Thermodynamic Action Penalties

Target Venue: Entropy

Abstract

Deriving the Minkowski metric from discrete graphs requires overcoming the Kleitman-Rothschild (KR) order collapse. A generic causal set overwhelmingly favors non-manifold KR-orders (e.g., three-layer structures). We formulate the Intellecton Lattice as a directed causal set and introduce a thermodynamic partition function governed by the Benincasa-Dowker action augmented with a local volume penalty. This partition function explicitly suppresses KR-orders, inducing a thermodynamic phase transition that heavily favors manifold-like geometries in the continuum limit. Consequently, the pseudo-Riemannian metric SO(1, D-1) and the Poincaré algebra are shown to rigorously emerge as the macroscopic thermodynamic ground state of discrete causal interactions.

1. Introduction

A simple unweighted graph Laplacian yields a positive-definite Riemannian metric. To recover Lorentz invariance, we use a Causal Set. However, Causal Sets generically collapse into non-manifold structures.

2. The Partition Function and KR-Order Suppression

Let the network be a causal set C representing the partial ordering of agent updates. To extract the continuous metric signature, we evaluate the system statistically using the partition function:


Z = \sum_{C} e^{-S_{BD}(C) - \beta V(C)}

where S_{BD}(C) is the discrete Benincasa-Dowker action, and V(C) is a non-local volume penalty that counts the number of localized intervals. The parameter \beta acts as an inverse topological temperature.

3. The Emergence of the Minkowski Metric

At low topological temperatures (high \beta), the volume penalty \beta V(C) thermodynamically suppresses the highly entropic, non-manifold Kleitman-Rothschild orders. The system undergoes a phase transition into a manifold-like phase where the continuum limit of the Benincasa-Dowker action yields the Einstein-Hilbert action over a pseudo-Riemannian manifold. Because the surviving geometries rigorously preserve the causal precedence of the directed graph, the continuum limit metric tensor g_{\mu\nu} natively possesses the minus sign required for Lorentz invariance. The Poincaré symmetry group SO(1, D-1) is therefore derived as the thermodynamic limit of the augmented causal set.

4. Conclusion

Relativistic spacetime and the Minkowski metric emerge neither from classical graphs nor generic causal sets, but specifically from the thermodynamic ground state of causal graphs governed by volume-penalized discrete actions.

References

Benincasa, D. M. T., & Dowker, F. (2010). The Scalar Curvature of a Causal Set. Physical Review Letters.
Surya, S. (2019). The causal set approach to quantum gravity. Living Reviews in Relativity.

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