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

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 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.