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Drawing from the optimal transport theory adapted to the Lorentzian setting, we propose and study the extension of the Sorkin–Woolgar causal relation

One of the vital fibers of mathematical relativity is causality theory; i.e., the study of the causal properties of spacetimes. The fundamental role in this theory is played by certain binary relations on a given spacetime, designed to describe which events (e.g., points of spacetime) are causally connected, and which can never constitute links of a cause–effect chain. The two most important such relations—called

Motivated by the approach to Lorentzian noncommutative geometry developed in [

Let us emphasize that the quantity

This definition rigorously encapsulates the following common intuition: infinitesimal portions of probability distributions can propagate in spacetime only along future-directed causal curves. The thus-obtained formalism can be successfully used to model the causal time-evolution of spatially distributed physical quantities [

Even though

In his work [

From now on, the term “measure” will always stand for “Borel probability measure”.

Analogously as for the relations

As announced in the Introduction, let us put forward the following definition of the Sorkin–Woolgar relation between measures.

As a first result, let us observe that the defining properties of the Sorkin–Woolgar relation—closedness and transitivity—still hold after extending it onto

Reflexivity and transitivity can be shown exactly as in the proof of ([

As for the closedness, take any sequences

What we need to prove is that

It now remains to show that

We now provide several characterizations of the relation

Observe that conditions

All above conditions (with the exception of

We will also need the following lemma, the proof of which is a straightforward adaptation of that of ([

To this end, observe that the first inequality in ii) follows directly from

For any Borel subset

The second inequality in condition ii) of Theorem 3 follows from

To this end, let

The functions

We go along the lines of the suitably modified proof of ([

Using the dominant convergence theorem twice, first for taking

It is now crucial to observe that the map

By the above observation, the integrands in (

To begin with, let

Notice now that

The converse implication holds trivially, because every bounded

We now move to proving that replacing “time” in

To this end, assume that

To move between conditions (

In this work, we have defined the natural extension of the Sorkin–Woolgar relation

Let us remark that all the obtained results would also hold if one, instead of

There are still some natural questions concerning the extension of

The author wishes to thank Stefan Suhr for his clarifying comments on Theorem 3.

The author declares no conflict of interest.

For the reader’s convenience, below we outline some topological and measure-theoretic notions and results used in the paper.

Polish spaces are completely metrizable spaces possessing a countable dense subset. Their basic examples include (second countable) manifolds as well as certain spaces of continuous functions ([

The space

It can be shown that

Two seminal results of the probability theory on Polish spaces used in this paper are the Portmanteau theorem and the Prokhorov theorem ([

Prokhorov’s theorem, on the other hand, characterizes the relatively compact subsets of

This in particular implies that any Borel probability measure on a Polish space is inner regular, i.e., for any

^{0}Lorentzian metrics: Proof of compactness of the space of causal curves