Dynamics of Rotating Cylindrical Shells in General Relativity

TL;DR

This paper analyzes rotating cylindrical spacetimes in general relativity, deriving the physical meaning of Riemann tensor components, and studies rotating shells that generate Lewis solutions, showing they can satisfy energy conditions.

Contribution

It introduces a method to relate the surface energy-momentum tensor of rotating shells to metric discontinuities and applies it to Lewis solutions with flat interior.

Findings

The shell can satisfy all energy conditions with proper parameters.

Derived general expressions for the surface energy-momentum tensor.

Connected shell parameters to physical quantities like mass per unit length.

Abstract

Cylindrical spacetimes with rotation are studied using the Newmann-Penrose formulas. By studying null geodesic deviations the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuation of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that $ 0 \le \sigma \le 1$, where $\sigma$ is related to the mass per unit length of the shell.