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

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 Weizsäcker’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.