Further evidence for the non-existence of a unified hoop conjecture

The hoop conjecture, introduced by Thorne almost five decades ago, asserts that black holes are characterized by the mass-to-circumference relation $$4\pi {\mathcal {M}}/{\mathcal {C}}\ge 1$$ 4 π M / C ≥ 1 , whereas horizonless compact objects are characterized by the opposite inequality $$4\pi {\mathcal {M}}/{\mathcal {C}}<1$$ 4 π M / C < 1 (here $${\mathcal {C}}$$ C is the circumference of the smallest ring that can engulf the self-gravitating compact object in all azimuthal directions). It has recently been proved that a necessary condition for the validity of this conjecture in horizonless spacetimes of spatially regular charged compact objects is that the mass $${\mathcal {M}}$$ M be interpreted as the mass contained within the engulfing sphere (and not as the asymptotically measured total ADM mass). In the present paper we raise the following physically intriguing question: is it possible to formulate a unified version of the hoop conjecture which is valid for both black holes and horizonless compact objects? In order to address this important question, we analyze the behavior of the mass-to-circumference ratio of Kerr–Newman black holes. We explicitly prove that if the mass $${\mathcal {M}}$$ M in the hoop relation is interpreted as the quasilocal Einstein–Landau–Lifshitz–Papapetrou and Weinberg mass contained within the black-hole horizon, then these charged and spinning black holes are characterized by the sub-critical mass-to-circumference ratio $$4\pi {\mathcal {M}}/{\mathcal {C}}<1$$ 4 π M / C < 1 . Our results provide evidence for the non-existence of a unified version of the hoop conjecture which is valid for both black-hole spacetimes and spatially regular horizonless compact objects.