On isotropic cylindrically symmetric stellar models

TL;DR

This paper proves that isotropic cylindrically symmetric stars cannot exist due to incompatible boundary and interior conditions, using matching conditions, polarization, and null geodesic analysis.

Contribution

It demonstrates the impossibility of isotropic cylindrically symmetric stellar models by analyzing boundary conditions and null geodesic expansions, extending previous static symmetry results.

Findings

Cylindrical vacuum must be polarized.

No trapped cylinders can evolve from initial conditions.

Isotropic cylindrically symmetric stars cannot exist.

Abstract

We attempt to match the most general cylindrically symmetric vacuum space-time with a Robertson-Walker interior. The matching conditions show that the interior must be dust filled and that the boundary must be comoving. Further, we show that the vacuum region must be polarized. Imposing the condition that there are no trapped cylinders on an initial time slice, we can apply a result of Thorne's and show that trapped cylinders never evolve. This results in a simplified line element which we prove to be incompatible with the dust interior. This result demonstrates the impossibility of the existence of an isotropic cylindrically symmetric star (or even a star which has a cylindrically symmetric portion). We investigate the problem from a different perspective by looking at the expansion scalars of invariant null geodesic congruences and, applying to the cylindrical case, the result that the product of the signs of the expansion scalars must be continuous across the boundary. The result may also be understood in relation to recent results about the impossibility of the static axially symmetric analogue of the Einstein-Straus model.