Gravitomagnetism in the Lewis cylindrical metrics

TL;DR

This paper clarifies the differences between rotating and static cylindrical spacetimes in general relativity, showing that gravitomagnetic effects arise from the vector potential, similar to electromagnetic phenomena, and provides a canonical form for the Lewis metrics.

Contribution

It introduces a canonical form of the Lewis cylindrical metric that isolates the gravitomagnetic effects, clarifying the global differences between rotating and static cylinders in GR.

Findings

The canonical form depends on mass, angular momentum, and angle deficit.

Rotating and static solutions differ only in the gravitomagnetic vector potential.

Gravitomagnetic effects manifest in Sagnac and clock effects, analogous to electromagnetic Aharonov-Bohm phenomena.

Abstract

The Lewis solutions describe the exterior gravitational field produced by infinitely long rotating cylinders, and are useful models for global gravitational effects. When the metric parameters are real (Weyl class), the exterior metrics of rotating and static cylinders are locally indistinguishable, but known to globally differ. The significance of this difference, both in terms of physical effects (gravitomagnetism) and of the mathematical invariants that detect the rotation, remain open problems in the literature. In this work we show that, by a rigid coordinate rotation, the Weyl class metric can be put into a "canonical" form where the Killing vector field $\partial_{t}$ is time-like everywhere, and which depends explicitly only on three parameters with a clear physical significance: the Komar mass and angular momentum per unit length, plus the angle deficit. This new form of the metric reveals that the two settings differ only at the level of the gravitomagnetic vector potential which, for a rotating cylinder, cannot be eliminated by any global coordinate transformation. It manifests itself in the Sagnac and gravitomagnetic clock effects. The situation is seen to mirror the electromagnetic field of a rotating charged cylinder, which likewise differs from the static case only in the vector potential, responsible for the Aharonov-Bohm effect, formally analogous to the Sagnac effect. The geometrical distinction between the two solutions is also discussed, and the notions of local and global staticity revisited. The matching in canonical form to the van Stockum interior cylinder is also addressed.