Introducing the inverse hoop conjecture for black holes

It is conjectured that stationary black holes are characterized by the inverse hoop relation ${\cal A}\leq {\cal C}^2/\pi$, where ${\cal A}$ and ${\cal C}$ are respectively the black-hole surface area and the circumference length of the smallest ring that can engulf the black-hole horizon in every direction. We explicitly prove that generic Kerr-Newman-(anti)-de Sitter black holes conform to this conjectured area-circumference relation.


I. INTRODUCTION
The isoperimetric inequality [1] in a two-dimensional Euclidean space states that the area A of a connected domain is bounded from above by the simple relation where C is the circumference length of the two-dimensional domain. The equality in (1) may be attained by an engulfing circular ring.
On the other hand, the area A of a deformed (or wrinkled) two-dimensional patch which is embedded in a three-dimensional space can violate the area-circumference relation (1) [2]. Likewise, the surface area of a (3 + 1)-dimensional black hole may in principle grow unboundedly with respect to its (squared) circumference length.
Intriguingly, however, it is well known that black holes in three spatial dimensions behave in many respects as two-dimensional objects. In particular, a black hole is characterized by a thermodynamic entropy [3,4] which is proportional to its two-dimensional surface area (and not to its effective volume). One can therefore expect that, in analogy with the twodimensional relation (1), the surface area of a black hole may be bounded from above by a quadratic function of its circumference length.
The main goal of the present compact paper is to raise the inverse hoop conjecture, according to which the surface areas of all stationary (3 + 1)-dimensional black holes are bounded from above by the simple functional relation where C s is the circumference length of the smallest ring that can engulf the black-hole horizon in all azimuthal directions [5][6][7][8][9].
where the metric functions ∆ r , ∆ θ , ρ, and I are given by the functional expressions [10,11] and Asymptotically flat Kerr-Newman black holes are characterized by the simple relation Λ = 0, whereas non-asymptotically flat Kerr-Newman-de Sitter and Kerr-Newman-anti-de Sitter black-hole spacetimes are characterized respectively by the relations Λ > 0 and Λ < 0.
The horizon radii of the black-hole spacetime (3) are determined by the roots of the radial metric function ∆ r (r) [10,11,14]. In particular, where r + is the radius of the black-hole event horizon.
From Eqs. (3) and (8) one finds the compact expressions C eq = 2π r 2 + + a 2 r + I and A = 4π for the equatorial circumference and the horizon surface area of the Kerr-Newman-(anti)-de Sitter black hole.
Interestingly, from Eqs. (9) and (10) one finds the compact dimensionless ratio for generic Kerr-Newman-(anti)-de Sitter black holes [15]. The conjectured inverse hoop relation asserts that stationary (3 + 1)-dimensional black holes are characterized by the simple relation Taking cognizance of Eqs. (7) and (11), one finds that asymptotically flat Kerr-Newman black holes (with Λ = 0 and therefore I = 1) and Kerr-Newman-anti-de Sitter black holes (with Λ < 0 and therefore I < 1) conform to the inverse hoop relation (12). It is easy to show that Kerr-Newman-de Sitter black holes (with Λ > 0) are characterized by the relation Λr 2 + ≤ 1 [16] and therefore also respect the inverse hoop relation (12).

III. SUMMARY AND DISCUSSION
The famous Thorne hoop conjecture [5] Taking cognizance of the fact that the irreducible mass of a black hole is related to its horizon surface area A by the simple relation one realizes that the inverse hoop relation (13) is a statement about the geometric properties of the black-hole horizon, bounding its surface area in terms of the squared circumference of the smallest ring that can engulf the horizon in every direction: If true, the conjectured inverse hoop relation (15) implies that the black-hole surface area cannot be unboundedly wrinkled [17].
Finally, it is worth noting that there is an important numerical evidence [18] for the validity of the inverse hoop conjecture (15) in non-stationary (dynamical) black-hole spacetimes. In particular, in a very interesting work, East [18] has studied numerically the full non-linear gravitational collapse of self-gravitating spheroidal matter configurations. Remarkably, it has been explicitly demonstrated in [18] that, in accord with the weak cosmic censorship conjecture [19], the final state of the collapse is a black hole. Interestingly, the initially distorted dynamically formed horizons obtained in [18] are characterized by damped oscillations between being prolate and oblate (see Figure 1. of [18]).
Intriguingly, and most importantly for our analysis, the numerical data presented in [18] (see, in particular, Figure 1. of [18]) reveals the fact that, within the bounds of the numerical accuracy [20], the dynamically formed black holes presented in [18] are characterized by the where C eq and C p are respectively the time-dependent (oscillating) equatorial and polar circumferences of the non-stationary black-hole horizons. Thus, the dynamically formed black holes presented in [18] seem to respect the conjectured inverse hoop relation (15).