Shearfree Cylindrical Gravitational Collapse

TL;DR

This paper analyzes shearfree cylindrical gravitational collapse, deriving explicit matching conditions between interior and exterior spacetimes, and explores the limitations on static and dynamic solutions, including energy considerations at the boundary.

Contribution

It provides explicit solutions for boundary matching conditions in shearfree cylindrical collapse and clarifies the restrictions on static and dynamic configurations, extending previous work on energy and asymptotic flatness.

Findings

Static exterior matches only with static interior in shearfree collapse.

Collapse evolution is generally described by elliptic functions of time.

Energy analysis shows the exterior is not radiative at the boundary.

Abstract

We consider diagonal cylindrically symmetric metrics, with an interior representing a general non-rotating fluid with anisotropic pressures. An exterior vacuum Einstein-Rosen spacetime is matched to this using Darmois matching conditions. We show that the matching conditions can be explicitly solved for the boundary values of metric components and their derivatives, either for the interior or exterior. Specializing to shearfree interiors, a static exterior can only be matched to a static interior, and the evolution in the non-static case is found to be given in general by an elliptic function of time. For a collapsing shearfree isotropic fluid, only a Robertson-Walker dust interior is possible, and we show that all such cases were included in Cocke's discussion. For these metrics, Nolan and Nolan have shown that the matching breaks down before collapse is complete and Tod and Mena have shown that the spacetime is not asymptotically flat in the sense of Berger et al. The issues about energy that then arise are revisited and it is shown that the exterior is not in an intrinsic gravitational or superenergy radiative state at the boundary.