The 4-dimensional Taub string

TL;DR

This paper constructs a stable, non-singular vacuum solution of Einstein's equations called the 4-dimensional Taub string, formed by joining Taub spaces, and explores its properties, stability, and possible excitations involving new scales.

Contribution

It introduces a novel stable vacuum solution called the Taub string, formed by successive junctions of Taub spaces, and analyzes its stability, dynamics, and potential excitations with additional scales.

Findings

The Taub string is a stable, geodesically complete vacuum solution.

Dynamics of the Taub string imply stability and hierarchy.

Effective stress-energy and torsion emerge from averaging the dynamics.

Abstract

The prototype of a Taub string is formed by successive junctions of copies of Taub's space {\cal T}, joined at their null boundaries \Sigma to create the axially-symmetric Bianchi-type-XI (with compact SL sections of scale {\rm L_o}) vacuum {\cal B}^4_{\rm T}= ...\vee {\cal T}\vee {\cal T}\vee {\cal T}\vee..., which is a {\em proper} one, namely a stable non-singular geodesically complete and globally fit solution of Einstein's vacuum equations without torsion and without a cosmological constant. Each {\cal T} contributes to {\cal B}^4_{\rm T} with its entire life-span as a quantum of time \delta t\sim{\rm L_ o} between two consecutive \Sigma. The latter propagate as shock-wave fronts under string tension of Planck-scale strength \kappa_{\rm o}. The incurring dynamics entails stability and the foundation of hierarchy in {\cal B}^4_{\rm T}. Appropriate averaging of this dynamics generates {\em effective} stress-energy content and torsion in a static \bar{\cal B}^4_{\rm T} vacuum. With the latter as ground state, excitations thereof must involve two new independent scales, \kappa and {\rm L_1}\gg{\rm L_o}, in addition to \kappa_{\rm o}, {\rm L_o}. Elemental finite Taub strings and variant vacua, including the {\rm L_o}\gg\kappa_{\rm o} cosmological case, are also discussed.