On the emergence of the Lorentzian metric structure of space-time in general relativity

TL;DR

The paper argues that a Lorentzian metric naturally emerges as the macroscopic structure of space-time in general relativity due to the statistical patterns of physical events, despite potential microscopic inaccuracies.

Contribution

It introduces a probabilistic interpretation and a principle from statistics to justify the emergence of Lorentzian geometry in macroscopic space-time.

Findings

Large collections of physical events exhibit two fundamental patterns.

A Lorentzian metric is necessary to describe matter-filled macroscopic regions.

Statistical principles support the emergence of space-time structure.

Abstract

In this short note we argue that, even if, as sometimes remarked, a Lorentzian manifold does not model correctly the structure of the spatio-temporal continuum as it is, yet a Lorentzian manifold should describe its macroscopic structure as we experience it. More precisely, theoretically motivated by von Weizs\"acker's chronological relative frequency interpretation of probability, and taking the Diaconis--Mosteller principle (also called the law of truly large numbers) as an empirical evidence in the macroscopic world, we argue that large collections of physical events appear in a composition of two fundamentally different patterns, termed as a progression and a sample here, making it unavoidable to use a Lorentzian-type metric on a manifold to describe matter-filled macroscopic regions of the spatio-temporal continuum.