Decreasing "circumference" for increasing "radius" in axially symmetric gravitating systems

TL;DR

This paper explores a unique cylindrically symmetric solution in Einstein's general relativity where the circumference of circles diminishes as the radius increases, analyzing its geometric, field boundary, and particle trajectory implications.

Contribution

It introduces a novel interpretation of this solution's geometry, examines boundary conditions for scalar and vector fields, and studies particle trajectories and fermionic condensates in this background.

Findings

Circumference tends to zero as radius increases in this solution.

Unbroken gauge symmetry is required for non-increasing energy density.

The geometry supports a normalizable fermionic condensate but not a current.

Abstract

Apart from the flat space with an angular deficit, Einstein general relativity possesses another cylindrically symmetric solution. Because this configuration displays circles whose "circumferences" tend to zero when their "radius" go to infinity, it has not received much attention in the past. We propose a geometric interpretation of this feature and find that it implies field boundary conditions different from the ones found in the literature if one considers a source consisting of the scalar and the vector fields of a U(1) system . To obtain a non increasing energy density the gauge symmetry must be unbroken . For the Higgs potential this is achieved only with a vanishing vacuum expectation value but then the solution has a null scalar field. A non trivial scalar behaviour is exhibited for a potential of sixth order. The trajectories of test particles in this geometry are studied, its causal structure discussed. We find that this bosonic background can support a normalizable fermionic condensate but not such a current.