intellecton/papers/Relativistic_Latency_in_Markovian_Networks.md

The Thermodynamic Bias Toward Manifolds in Causal Sets: Prerequisites for Lorentz Invariance

Target Venue: Entropy

Abstract

The extraction of the Minkowski metric from discrete causal graphs is complicated by the Kleitman-Rothschild (KR) order collapse. A generic causal set overwhelmingly favors non-manifold KR-orders. We introduce a thermodynamic partition function governed by the Benincasa-Dowker action augmented with a non-local volume penalty. This partition function explicitly suppresses KR-orders. While this does not constitute a full derivation of 4D Einstein equations—which remains an open problem in quantum gravity—it successfully induces a thermodynamic phase transition that heavily biases the causal set toward manifold-like geometries in the continuum limit. This thermodynamic bias is a necessary prerequisite for the emergence of the pseudo-Riemannian metric SO(1, D-1) and macroscopic Lorentz invariance.

1. Introduction

A simple graph Laplacian yields a Riemannian metric. Recovering the pseudo-Riemannian metric of relativity requires Causal Sets. However, Causal Sets generically collapse into non-manifold three-layer structures.

2. The Partition Function and Topological Temperature

Let the network be a causal set C representing a discrete partial ordering. To extract continuous manifold properties, 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 volume penalty counting the number of localized intervals. The parameter \beta acts as an inverse topological temperature.

3. Biasing Toward Manifolds

At low topological temperatures (high \beta), the volume penalty \beta V(C) thermodynamically suppresses the highly entropic, non-manifold KR-orders. The system undergoes a phase transition into a phase where the continuum limit strongly favors manifold-like structures. In this manifold phase, the causal precedence preserved by the directed graph inherently generates a continuum limit metric tensor g_{\mu\nu} with a Lorentzian signature. Thus, a volume-penalized thermodynamic action is a strict prerequisite for the emergence of relativistic spacetime.

4. Conclusion

Macroscopic Lorentz invariance requires the thermodynamic suppression of non-manifold causal set structures via volume-penalized discrete actions.

References

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