[Paper Review] Emergent spacetime from purely random structures

Ioannis Kleftogiannis, Ilias Amanatidis|arXiv (Cornell University)|Oct 3, 2022

Advanced Mathematical Theories and Applications6 citations

TL;DR

This paper proposes that spacetime emerges from purely random, uniform random graphs with minimal constraints, where geometric properties like dimensionality (D≈3) and curvature arise from connectivity. Through an evolution mechanism removing one edge per time step, the model exhibits exponential expansion and yields emergent flat 3D-like geometry, along with derived quantum and relativistic effects such as time dilation and the Heisenberg uncertainty principle from statistical fluctuations.

ABSTRACT

We examine the fundamental question whether a random discrete structure with the minimal number of restrictions can converge to continuous metric space. We study the geometrical properties such as the dimensionality and the curvature emerging out of the connectivity properties of uniform random graphs. In addition we introduce a simple evolution mechanism for the graph by removing one edge per a fundamental quantum of time from an initially complete graph. We show an exponential growth of the radius of the graph, that ends up in a random structure with emergent average spatial dimension $D=3$ and zero curvature $K=0$, resembling a flat 3D manifold, that could describe the observed space in our universe and some of its geometrical properties. In addition, we introduce a generalized action for graphs based on physical quantities on different subgraph structures that helps to recover the well known properties of spacetime as described in general relativity, like time dilation due to gravity. Also, we show how various quantum mechanical concepts such as generalized uncertainty principles based on the statistical fluctuations can emerge from random discrete models. Moreover, our approach leads to a unification of space and matter-energy, for which we propose a mass-energy-space equivalence that leads to a way to transform between empty space and matter-energy via the cosmological constant.

Motivation & Objective

• To investigate whether a purely random discrete structure can give rise to continuous spacetime with emergent geometric properties.
• To explore how fundamental physical concepts—such as dimensionality, curvature, and time evolution—can emerge from connectivity in uniform random graphs.
• To unify space, matter, and energy via a proposed mass-energy-space equivalence, enabling transformation between vacuum and particles.
• To derive quantum mechanical principles like the uncertainty principle from statistical fluctuations in discrete random networks.
• To recover relativistic effects such as time dilation from a generalized graph action based on subgraph physical quantities.

Proposed method

• Model spacetime as a uniform random graph G=(V,E) with n vertices and m randomly distributed edges, maximizing entropy under uniform probability over all configurations.
• Introduce a time evolution mechanism where one edge is removed per fundamental time step from an initially complete graph, simulating cosmic expansion.
• Define a generalized graph action S_gr = ΣΣ a_ij f_i(V_j) over subgraphs V_j and physical functionals f_i, enabling derivation of spacetime dynamics.
• Use the degree distribution and its fluctuations as proxies for geometric and physical quantities, linking them to energy, dimension, and volume fluctuations.
• Derive the Heisenberg uncertainty principle by showing ΔEΔt ≥ ħ/2 through statistical variance of local energy over time intervals in the thermodynamic limit.
• Propose a mass-energy-space equivalence where fluctuations in spatial dimension D lead to effective mass via ΔW ∝ exp((ΔD ln L)^2), enabling particle emergence from vacuum fluctuations.

Experimental results

Research questions

• RQ1 Can a purely random discrete structure with minimal constraints converge to a continuous metric space with emergent spatial geometry?
• RQ2 How do dimensionality D≈3 and zero curvature arise from the connectivity properties of uniform random graphs?
• RQ3 Can relativistic effects such as time dilation due to gravity emerge from a graph-based action principle?
• RQ4 To what extent can quantum mechanical principles like the uncertainty principle be derived from statistical fluctuations in discrete random networks?
• RQ5 How can particles and the mass-energy-space equivalence emerge from fluctuations in the dimension and volume of a random graph vacuum?

Key findings

• The model exhibits exponential growth of graph radius under edge-removal evolution, leading to a large-scale structure resembling a flat 3D manifold with D≈3 and zero curvature.
• The entropy of the uniform random graph peaks at R = (n−1)/4, indicating a critical connectivity regime where maximal disorder corresponds to emergent geometric order.
• The Heisenberg uncertainty principle ΔEΔt ≥ ħ/2 is derived naturally from statistical fluctuations in local energy over time intervals, with the inequality saturated in the thermodynamic limit.
• Dimensional fluctuations ΔD lead to log-normal volume fluctuations ΔW, suggesting that dimensional uncertainty could be detectable at high energies and short timescales.
• Persistent localized 0D and 1D subgraph structures emerge, resembling point-like and string-like particles, with linear energy dispersion E = ħv_nt k, mimicking relativistic massless particles.
• A mass-energy-space equivalence is proposed, where fluctuations in spatial dimension D transform into effective mass, enabling the emergence of matter from vacuum via dimensional fluctuations.