Interpretation of the Weyl tensor

TL;DR

This paper challenges the standard interpretation of the Weyl tensor in cylindrical geometries, showing it fails there but works for spherical and planar cases, and proposes Thorne's local energy as a better diagnostic tool.

Contribution

It demonstrates the limitations of the Weyl tensor interpretation in cylindrical symmetry and introduces Thorne's local energy as an effective alternative diagnostic method.

Findings

Weyl tensor interpretation fails for cylindrical geometries

Standard interpretation works for spherical and planar geometries

Thorne's local energy provides reliable physical interpretation in cylindrical cases

Abstract

According to folklore in general relativity, the Weyl tensor can be decomposed into parts corresponding to Newton-like, incoming and outgoing wavelike field components. It is shown here that this one-to-one correspondence does not hold for space-time geometries with cylindrical isometries. This is done by investigating some well-known exact solutions of Einstein's field equations with whole-cylindrical symmetry, for which the physical interpretation is very clear, but for which the standard Weyl interpretation would give contradictory results. For planar or spherical geometries, however, the standard interpretation works for both, static and dynamical space-times. It is argued that one reason for the failure in the cylindrical case is that for waves spreading in two spatial dimensions there is no local criterion to distinguish incoming and outgoing waves already at the linear level. It turns out that Thorne's local energy notion, subject to certain qualifications, provides an efficient diagnostic tool to extract the proper physical interpretation of the space-time geometry in the case of cylindrical configurations.