Complexity in the interdefinability of timelike, lightlike and spacelike relatedness of Minkowski spacetime

TL;DR

This paper investigates the definability and complexity of relations between timelike, lightlike, and spacelike relatedness in Minkowski spacetime, providing minimal variable definitions and exploring their logical properties across different fields and dimensions.

Contribution

It introduces the simplest possible definitions of these relations using only four variables and analyzes their logical and quantifier complexity across various mathematical settings.

Findings

Definitions work over arbitrary Euclidean fields for n>2

No definitions with only 3 variables exist for these relations

Open problems remain regarding definitions in certain cases

Abstract

Interdefinability of timelike, lightlike and spacelike relatedness of Minkowski spacetime is investigated in detail in the paper, with the aim of finding the simplest definitions. Based on ideas scattered in the literature, definitions are given between any two of these relations that use only 4 variables. All these definitions work over arbitrary Euclidean fields in place of the field of reals, if the dimension n of spacetime is greater than two. If n=2, the definitions work over arbitrary ordered fields except the ones based on lightlike relatedness (where no definition can work by symmetry). None of these relations can be defined from another one using only 3 variables. Our four-variable definitions use only one universal and one existential quantifiers in a specific order. In some of the cases, we show that the order of these quantifiers can be reversed for the price of using twice as many quantifiers. Except two cases, we provide existential/universal definitions using 5 variables or show that no existential/universal definition exists. There are no existential/universal definitions between any two of these relations using only 4 variables. It remains open whether there is an existential (universal) definition of timelike (lightlike) relatedness from spacelike relatedness if $n>2$. Finally, several other open problems related to the quantifier complexity of the simplest possible definitions are given.