Space-Time Defects and Group Momentum Space

Michele Arzano,Tomasz Trześniewski

doi:10.1155/2017/4731050

Michele Arzano, Tomasz Trześniewski

Open Access

https://doi.org/10.1155/2017/4731050

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Abstract

We study massive and massless conical defects in Minkowski and de Sitter spaces in various space-time dimensions. The energy momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its space-time metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects, respectively. In particular, if we fix the direction of propagation of a massless defect in n+1-dimensional Minkowski space, then its space of holonomies is a maximal Abelian subgroup of the AN(n-1) group, which corresponds to the well known momentum space associated with the n-dimensional κ-Minkowski noncommutative space-time and κ-deformed Poincaré algebra. We also conjecture that massless defects in n-dimensional de Sitter space can be analogously characterized by holonomies belonging to the same subgroup. This shows how group-valued momenta related to four-dimensional deformations of relativistic symmetries can arise in the description of motion of space-time defects.

Highlights

Conical space-time defects were first introduced by Staruszkiewicz [1] as point particles coupled to gravity in 2 + 1 space-time dimensions
We look here at a generalization of the derivation of a massless conical defect as it was done in 2 + 1-dimensional de Sitter space [32]. (In Appendix B, we have included a similar discussion for defects in anti-de Sitter space.) We begin with a static massive conical defect in 3 + 1 dimensions [41, 42], analogous to a point particle in 2 + 1 dimensions [3], whose metric in static de Sitter coordinates has the form ds2 = − (1 − λr2) dτ2 + (1 − λr2)−1 dr2 (22)
We have provided an exploration of the relation between the holonomies of conical defects in more than three space-time dimensions and group-valued momenta, which appear in scenarios of deformed relativistic symmetries

Summary

Introduction

Conical space-time defects were first introduced by Staruszkiewicz [1] as point particles coupled to gravity in 2 + 1 space-time dimensions. In 3 + 1 dimensions, a conical defect can be obtained by replacing the point-like singularity with a singular one-dimensional object Such linear defects are known under the name of cosmic strings and first turned out to be possibly generated during a spontaneous gauge symmetry breaking in the early universe [4]. In order to achieve a nontrivial limit, one usually applies the boost following a prescription of Aichelburg and Sexl, which was first proposed to derive the gravitational field of a photon from the Schwarzschild metric [26] This method was subsequently extended to other singular null objects, in particular massless cosmic strings [27,28,29].

Conical Defects in Minkowski Space

Conical Defects in de Sitter Space

Summary

Conical Defects in Anti-de Sitter Space