Junction conditions in General Relativity with spin sources

TL;DR

This paper derives junction conditions in General Relativity with intrinsic spin sources, revealing how torsion affects the continuity of geometric quantities and applying the results to specific spinning string solutions in three and four dimensions.

Contribution

It introduces a novel framework for junction conditions with spin and torsion, relaxing traditional continuity requirements and linking spin currents to geometric jumps.

Findings

Spin sources induce torsion-related jumps in geometry.

Spinning strings exhibit frame dragging effects that can be shielded.

Solutions extend consistently from three to four dimensions.

Abstract

The junction conditions for General Relativity in the presence of domain walls with intrinsic spin are derived in three and higher dimensions. A stress tensor and a spin current can be defined just by requiring the existence of a well defined volume element instead of an induced metric, so as to allow for generic torsion sources. In general, when the torsion is localized on the domain wall, it is necessary to relax the continuity of the tangential components of the vielbein. In fact it is found that the spin current is proportional to the jump in the vielbein and the stress-energy tensor is proportional to the jump in the spin connection. The consistency of the junction conditions implies a constraint between the direction of flow of energy and the orientation of the spin. As an application, we derive the circularly symmetric solutions for both the rotating string with tension and the spinning dust string in three dimensions. The rotating string with tension generates a rotating truncated cone outside and a flat space-time with inevitable frame dragging inside. In the case of a string made of spinning dust, in opposition to the previous case no frame dragging is present inside, so that in this sense, the dragging effect can be "shielded" by considering spinning instead of rotating sources. Both solutions are consistently lifted as cylinders in the four-dimensional case.