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

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 binary relations that use 4 variables, i.e., they use only 2 auxiliary 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 one auxiliary variable. These 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 in two cases, we provide existential/universal definitions using 3 auxiliary variables or show that no existential/universal definition exists. There are no existential/universal definitions between any two of these relations using only 2 auxiliary 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. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

This paper thoroughly investigates the interdefinability of timelike, lightlike, and spacelike relatedness in Minkowski spacetime, aiming to find the simplest definitions. Using 4 variables, definitions are provided between any two of these binary relations, where only 2 auxiliary variables are used. These definitions work over arbitrary Euclidean fields for n > 2. In the case of n = 2, they work over arbitrary ordered fields except for lightlike relatedness. It is not possible to define any of these relations using only one auxiliary variable. The order of universal and existential quantifiers can be reversed in some cases at the cost of using twice as many quantifiers. Existential/universal definitions using 3 auxiliary variables are provided in most cases, or it is shown that no existential/universal definition exists. There are no existential/universal definitions between any two of these relations using only 2 auxiliary variables. It remains unknown whether there is an existential (universal) definition of timelike (lightlike) relatedness from spacelike relatedness if n > 2. Several open problems related to the quantifier complexity of the simplest possible definitions are given. Published by Elsevier B.V.