Statistical Representation of Spacetime

TL;DR

This paper models spacetime as a causal set using population dynamics, deriving a quantized entropy change that suggests an inflating universe and connecting discrete spacetime properties to Einstein's equations.

Contribution

It introduces a novel statistical framework for spacetime based on Leslie matrices and causal sets, linking entropy, energy, and geometry at the Planck scale.

Findings

Stationary state of the population always exists with positive eigenvalues.

Entropy change is quantized and positive, indicating an inflating universe.

The proportionality constant between entropy and area matches Hawking's result.

Abstract

It is assumed that the spacetime is composed by events and can be explained by partially ordered set (causal set). The parent events born two kinds of children. Some children have a causal relation with their parents and other kinds have not. It is assumed that evolution of the population is only happen by the causal children. The assumed population can be modeled by finite (infinite) dimension Leslie matrix. In both finite and infinite cases, it is shown that the stationary state of the population always exists and the matrix has positive eigenvalues. By finding the relation between the statistical information of the population and the stationary state, a probability matrix and a Shannon-like entropy is defined. It is shown that the change in entropy is always quantized and positive and in consequence, the world is inflating. We show that the vacuum energy can be attributed to the necessary done work for preserving the causal relation between the parents and the children (cohesive energy). By assuming that the sum of cohesive energy and kinetic energy of the denumerable causal spacetime is equal to the heat, which flows across a causal horizon, we find the relation between energy-momentum tensor and discrete Ricci tensor which can be called the Einstein state equation. Finally, it is shown that the constant of proportionality \(\eta\) between the entropy and the area is proportional to \(\frac{k_{B}}{l_{p}^{2}}\) at Planck scale which is in good agreement with the Hawking's result.