The non-integrability of the Zipoy-Voorhees metric

TL;DR

This paper demonstrates through numerical analysis that the Zipoy-Voorhees spacetime, previously thought to be integrable, is actually non-integrable, impacting gravitational wave modeling for testing black hole nature.

Contribution

It provides the first numerical evidence that the Zipoy-Voorhees spacetime is non-integrable, challenging prior assumptions of its integrability.

Findings

Numerical examples show non-integrability of ZV spacetime.

Contradicts previous claims of ZV being an integrable system.

Implications for gravitational wave templates and black hole tests.

Abstract

The low frequency gravitational wave detectors like eLISA/NGO will give us the opportunity to test whether the supermassive compact objects lying at the centers of galaxies are indeed Kerr black holes. A way to do such a test is to compare the gravitational wave signals with templates of perturbed black hole spacetimes, the so-called bumpy black hole spacetimes. The Zipoy-Voorhees (ZV) spacetime (known also as the $\gamma$ spacetime) can be included in the bumpy black hole family, because it can be considered as a perturbation of the Schwarzschild spacetime background. Several authors have suggested that the ZV metric corresponds to an integrable system. Contrary to this integrability conjecture, in the present article it is shown by numerical examples that in general ZV belongs to the family of non-integrable systems.