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 $4\pi {\cal M}/{\cal C}\geq1$, whereas horizonless compact objects are characterized by the opposite inequality $4\pi {\cal M}/{\cal C}<1$ (here ${\cal C}$ 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 ${\cal M}$ 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 ${\cal M}$ 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 $4\pi {\cal M}/{\cal C}<1$. 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.


I. 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 critical circumference C critical = 4πM 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 ≡ 4πM C ≥ 1 ⇐⇒ Black-hole horizon exists .
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 ≡ 4πM/C [2]. Interestingly, as explicitly demonstrated in [3][4][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, electric charge Q, and total ADM mass M can be characterized by the dimensionless relation M/R → 1 − in the Q/M → 1 − limit. Thus, these horizonless charged shells are characterized by the super-critical dimensionless relation 4πM/C → 2 − and can therefore violate the hoop conjecture (1) if M is interpreted as the total ADM mass M of the spacetime.
Intriguingly, it has recently been proved [6] (see also [7]) that if the mass M is interpreted as the gravitating mass M in ≡ M(R) contained within an engulfing sphere of radius R [8] (and not as the total ADM mass of the spacetime), then horizonless self-gravitating charged compact objects are characterized by the sub-critical dimensionless relation H < 1 and therefore respect the hoop conjecture (1), see also the physically interesting related works [9][10][11][12][13].
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, then the mass term M should not be interpreted as the total ADM mass of the spacetime. In particular, if a valid and unified formulation of the conjecture (1) do exists, then the results presented in [3][4][5][6][7] indicate that the mass M should be interpreted as the mass M in contained within the boundaries of the compact object (a black hole or a self-gravitating horizonless object).
Motivated by the intriguing results of [3][4][5][6][7], in the present compact paper we raise the following physically interesting question: Is it possible to formulate the hoop conjecture (1) in a unified way which is valid for both black-hole spacetimes and spatially regular horizonless compact objects?
In order to address this intriguing question, we shall analyze the functional behavior of the dimensionless mass-to-circumference ratio H(Q, a) ≡ 4πM/C of charged and spinning where ∆ ≡ r 2 − 2Mr + Q 2 + a 2 ; ρ 2 ≡ r 2 + a 2 cos 2 θ .
The radii of the (outer and inner) black-hole horizons are determined by the zeros of the metric function ∆(r).
Substituting dt = dr = dθ = 0, θ = π/2, and ∆φ = 2π into the curved line element (2), one finds the simple functional expression for the equatorial circumference of an engulfing ring which is located at a fixed radial coor- just outside (ǫ ≪ 1) the outer horizon (4) of the Kerr-Newman black-hole [19]. Taking cognizance of Eq. (4), one can express the black-hole equatorial circumference (5) in the dimensionless form [20] C eq (Q,ā) = 4π · 1 −Q are respectively the dimensionless circumference, the dimensionless electric charge, and the dimensionless angular momentum of the Kerr-Newman black hole.
Before proceeding, it is worth pointing out that the hoop relation (1) is respected by all Kerr-Newman black holes if the mass term M is interpreted as the total ADM mass M of the spacetime. In particular, taking cognizance of Eqs. (6) and (7), one finds the simple dimensionless relation for the charged and spinning Kerr-Newman black holes.
However, as emphasized above, it has been demonstrated explicitly in [3][4][5] for the quasilocal Einstein-Landau-Lifshitz-Papapetrou and Weinberg mass M in contained within the horizon of a Kerr-Newman black hole. Taking cognizance of Eqs. (4) and (9), one can express the engulfed gravitational mass M in of the charged and spinning Kerr-Newman black hole (2) explicitly in terms of its dimensionless physical parametersQ andā: In Table I we present the dimensionless mass-to-circumference ratio [see Eqs. (6) and (10)] of charged and spinning Kerr-Newman black holes for various values of the black-hole dimensionless physical parametersQ andā. Intriguingly, the data presented in Table I reveals the fact that all charged and spinning (Q = 0,ā = 0) Kerr-Newman black holes are characterized by the sub-critical mass-to-circumference relation [23] H(Q,ā) < 1 .
In particular, from the data presented in Table I one (6) and (10)]. Interestingly, the dimensionless ratio H(Q,ā) is found to be a monotonically decreasing function of the black-hole physical parametersā andQ. In particular, one finds that charged and spinning Kerr-Newman black holes are characterized by the sub-critical mass-to-circumference ratio H(Q = 0,ā = 0) ≡ 4πM/C eq < 1.
function of the black-hole angular momentum parameterā. In addition, one finds that, for a given value of the black-hole angular momentumā, the mass-to-circumference ratio H is a monotonically decreasing function of the black-hole electric chargeQ. These facts indicate that the dimensionless mass-to-circumference function H(Q,ā) is minimized by an extremal black hole. SubstitutingQ 2 +ā 2 = 1 for extremal Kerr-Newman black holes [24] into Eqs. (6) and (10), one obtains the compact functional expression [see Eq. (12)] for the mass-to-circumference ratio of extremal Kerr-Newman black holes. From (14) one finds minQ ,ā {H} = 0.9468 for (Q,ā) = (0.7776, 0.6287) . (15)

III. SUMMARY AND DISCUSSION
The Thorne hoop conjecture [1] serves as a boundary between black-hole configurations and horizonless compact objects. In particular, this physically influential conjecture asserts that black holes should be characterized by the mass-to-circumference inequality 4πM/C ≥ 1, whereas horizonless compact objects should be characterized by the opposite inequality 4πM/C < 1.
The physical meaning of the mass term M in the hoop conjecture (1) has not been specified by Thorne [1]. However, it is known that the hoop relation (1) can be violated by self-gravitating horizonless charged objects if the mass M is interpreted as the total ADM mass of the spacetime [3][4][5]. On the other hand, it has recently been proved [6,7] that spherically symmetric horizonless charged objects respect the hoop relation (1) if the mass M is interpreted as the gravitating mass M in ≡ M(R) contained within an engulfing sphere of radius R (and not as the total mass of the spacetime).
The physical results presented in [3][4][5][6][7] imply that a unified version of the hoop relation we have explicitly proved that charged-spinning (Q = 0,ā = 0) Kerr-Newman black holes are characterized by the sub-critical dimensionless mass-to-circumference ratio (see the data presented in Table I) [25][26][27] 4πM in C eq < 1 for charged and spinning Kerr-Newman black holes .
(2) It has been shown that, for Kerr-Newman black holes, the dimensionless mass-tocircumference ratio H(Q,ā) ≡ 4πM in /C eq is a monotonically decreasing function of the [24] Note that extremal Kerr-Newman black holes are characterized by the simple relation r − = r + [see Eq. (4)].
[25] It is worth noting that the Komar [26] and Møller [27] prescriptions for calculating energy densities in general relativity yield the expression M in ≡ M (r = r + ) = M − Q 2 2r + 1 + r 2 + +a 2 ar + · arctan a r + for the mass M in contained within the horizon of a Kerr-Newman black hole. This expression for M in is smaller than the corresponding expression (9) for the mass contained within the black-hole horizon according to the Einstein, the Landau-Lifshitz, the Papapetrou, and the Weinberg prescriptions for calculating energy densities in general relativity [21]. Thus, according to the Komar and Møller prescriptions for calculating the mass contained within the black-hole horizon, Kerr-Newman black-hole spacetimes are characterized by smaller massto-circumference ratios than the ones presented in Table I Table I which, as emphasized above, are based on the Einstein, Landau-Lifshitz, Papapetrou, and Weinberg prescriptions for calculating energy densities in general relativity [21].