Further evidence for the non-existence of a unified hoop conjecture
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Regular Article - Theoretical Physics
Further evidence for the non-existence of a unified hoop conjecture Shahar Hod1,2,a 1 2
The Ruppin Academic Center, 40250 Emek Hefer, Israel The Hadassah Academic College, 91010 Jerusalem, Israel
Received: 24 July 2020 / Accepted: 14 October 2020 © The Author(s) 2020
Abstract The hoop conjecture, introduced by Thorne almost five decades ago, asserts that black holes are characterized by the mass-to-circumference relation 4π M/C ≥ 1, whereas horizonless compact objects are characterized by the opposite inequality 4π M/C < 1 (here 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 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 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 4π 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.
1 Introduction The influential hoop conjecture has been suggested by Thorne [1] as a simple necessary and sufficient condition for the formation of black holes in dynamical gravitational collapse scenarios. In particular, the hoop criterion asserts that a self-gravitating matter configuration of mass M would collapse to form a black hole if and only if a circular hoop of a a e-mail:
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critical circumference Ccritical = 4π M can be placed around the self-gravitating matter distribution and rotated in 360◦ to form an engulfing sphere. The hoop conjecture therefore implies the simple relation [1] H≡
4π M ≥ 1 ⇐⇒ Black-hole horizon exists. C
(1)
In his original formulation of the hoop conjecture, Thorne [1] has not provided an explicit definition for the mass term M in his dimensionless mass-to-circumference ratio H ≡ 4π M/C [2]. Interestingly, as explicitly demonstrated in [3–5], the hoop conjecture (1) can be violated in curved spacetimes of horizonless charged compact objects if M is interpreted as the total ADM mass of the spacetime. In particular, as nicely shown in [5], spherically symmetric horizonless thin shells of radius R,
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