Semigroup extensions of isometry groups of compactified spacetimes

TL;DR

This paper explores conditions under which the isometry group of a compactified spacetime can be extended to a semigroup, providing explicit examples and establishing when such extensions are possible or obstructed.

Contribution

It introduces criteria for semigroup extensions of isometry groups in compactified spacetimes and constructs explicit examples involving Lorentzian metrics with lightlike vectors.

Findings

Semigroup extensions are impossible if the restricted metric is Euclidean.

An explicit semigroup extension is constructed for a Lorentzian spacetime with a lightlike basis vector.

Extensions are obstructed when the covering spacetime's metric is Euclidean.

Abstract

We investigate the possibility of semigroup extensions of the isometry group of an identification space, in particular, of a compactified spacetime arising from an identification map $p: \RR^n_t \to \RR^n_t / \Gamma$, where $\RR^n_t$ is a flat pseudo-Euclidean covering space and $\Gamma$ is a discrete group of primitive lattice translations on this space. We show that the conditions under which such an extension is possible are related to the index of the metric on the subvector space spanned by the lattice vectors: If this restricted metric is Euclidean, no extensions are possible. Furthermore, we provide an explicit example of a semigroup extension of the isometry group of the identification space obtained by compactifying a Lorentzian spacetime over a lattice which contains a lightlike basis vector. The extension of the isometry group is shown to be isomorphic to the semigroup $(\ZZ^{\times},\cdot)$, i.e. the set of nonzero integers with multiplication as composition and 1 as unit element. A theorem is proven which illustrates that such an extension is obstructed whenever the metric on the covering spacetime is Euclidean.