Compactification along Lightlike Lattices

TL;DR

This paper explores how dimensional reduction along lightlike lattices leads to semigroup extensions of isometry groups, revealing preferred frames and implications for gauge theories and Kaluza-Klein models.

Contribution

It demonstrates the existence of natural order structures and preferred coordinate charts in lightlike compactifications, affecting gauge field behavior and higher-dimensional theories.

Findings

Preferred frames emerge from semigroup structures in lightlike compactifications.

Fields in these frames can attain extreme values, simplifying gauge potentials.

The metric's lightlike nature prevents deriving field equations from Ricci tensors.

Abstract

Spacetimes obtained by dimensional reduction along lattices containing a lightlike direction can admit semigroup extensions of their isometry groups. We show by concrete examples that such a semigroup can exhibit a natural order, which in turn implies the existence of preferred coordinate charts on the underlying space. Specifically, for spacetimes which are products of an external Minkowski space with an internal two-dimensional Lorentzian space, where one of the lightlike directions has a compact size, the preferred charts consist of "infinite-momentum" frames on the internal space. This implies that fields viewed from this preferred frame acquire extreme values; in particular, some of the off-diagonal components of the higher-dimensional metric, which may be regarded as gauge potentials for a field theory on the external Minkowski factor, vanish. This raises the possibility of regarding known gauge theories as part of more extended field multiplets which have been reduced in size since they are perceived from within an extreme frame. In the case of an external 4-dimensional Minkowski spacetime times a two-dimensional Lorentzian cylinder, the field content as seen in the preferred frame is that of a five-dimensional Kaluza-Klein theory, where the electrodynamic potentials Am may depend, in addition to the external spacetime coordinates, on a fifth coordinate along a lightlike direction. The fact that the metric along this direction is zero obstructs the generation of field equations from the Ricci tensor of the overall metric.