Junction conditions in general relativity with spin sources

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 spacetime 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-

Singular sources in gravity and homotopy in the space of connections

“…Likewise for the intersection of shells, without deficit angle [11] [12]. It has been shown that in some cases these results generalise to the first order formulation with non-vanishing torsion [13] [14].…”

“…∂ takes into account orientations and is required to satisfy ∂∂ = 0. Relation (13) was introduced to guaranty that.…”

“…In the case where both spin connection and vielbein are discontinuous, then the term (33) generally does not vanish. Indeed, making no assumptions 4 about the limit of E α and 4 In reference [13] it was assumed that Eα and ωα are an interpolation with some arbitrary scalar function between E 1 and E 2 and ω 1 and ω 2 . Under that assumption the weaker condition…”

“…Thus we manage to put condition (64) in some sense on the same basis as the smoothing criterion II. 13.…”