Abstract
The hoop conjecture, introduced by Thorne almost five decades ago, asserts that black holes are characterized by the mass-to-circumference relation 4pi {mathcal {M}}/{mathcal {C}}ge 1, whereas horizonless compact objects are characterized by the opposite inequality 4pi {mathcal {M}}/{mathcal {C}}<1 (here {mathcal {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}} 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}} 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 4pi {mathcal {M}}/{mathcal {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.
Highlights
IntroductionThe interesting physical results presented in [3,4,5,6,7] indicate that if there is any hope to formulate the hoop conjecture (1) in a unified way, which is valid for both black-hole spacetimes and spatially regular (horizonless) self-gravitating compact objects, the mass term M should not be interpreted as the total ADM mass of the spacetime
Critical circumference Ccritical = 4π M can be placed around the self-gravitating matter distribution and rotated in 360◦
It has recently been proved [6] that if the mass M is interpreted as the gravitating mass Min ≡ M(R) contained within an engulfing sphere of radius R [8], horizonless self-gravitating charged compact objects are characterized by the sub-critical dimensionless relation H < 1 and respect the hoop conjecture (1), see the physically interesting related works [9,10,11,12,13]
Summary
The interesting physical results presented in [3,4,5,6,7] indicate that if there is any hope to formulate the hoop conjecture (1) in a unified way, which is valid for both black-hole spacetimes and spatially regular (horizonless) self-gravitating compact objects, the mass term M should not be interpreted as the total ADM mass of the spacetime. We shall explicitly show below that if the mass M in the hoop relation (1) is interpreted as the quasilocal Einstein–Landau–Lifshitz–Papapetrou and Weinberg mass Min contained within the black-hole horizon, charged and spinning Kerr–Newman black holes are characterized by the sub-critical mass-to-circumference ratio 4π Min/C < 1 and violate the hoop conjecture (1)
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