No-hair conjecture and black-hole quadrupole moments

Determine whether a black hole can possess a quadrupole moment, an issue addressed by the no-hair conjecture invoked in restricting the Kerr-like metric to horizonless deformed compact objects.

Background

The paper studies perturbations of a Kerr-like metric containing a mass-quadrupole parameter q, intended to model rotating compact objects that are not perfectly spherical. Near the Kerr radial coordinate r_+, the perturbation equation contains terms that prevent the authors from reducing it to the Kerr equation or separating it in the usual way.

To avoid interpreting the r → r_+ limit as a black-hole horizon limit, the authors invoke the no-hair conjecture, according to which black holes cannot possess a quadrupole moment. Under that assumption, the Kerr-like objects considered in the paper have surfaces outside r_+, and the relevant boundary conditions must instead be imposed at the material surface. The quoted passage therefore explicitly identifies the conjectural status of the assertion that excludes quadrupolar black holes.