Introducing the inverse hoop conjecture for black holes
aa r X i v : . [ g r- q c ] J a n Introducing the inverse hoop conjecture for black holes
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: January 15, 2021)
Abstract
It is conjectured that stationary black holes are characterized by the inverse hoop relation
A ≤ C /π , where A and C are respectively the black-hole surface area and the circumferencelength of the smallest ring that can engulf the black-hole horizon in every direction. We explicitlyprove that generic Kerr-Newman-(anti)-de Sitter black holes conform to this conjectured area-circumference relation. . 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 A ≤ C / π , (1)where C is the circumference length of the two-dimensional domain. The equality in (1) maybe attained by an engulfing circular ring.On the other hand, the area A of a deformed (or wrinkled) two-dimensional patch whichis 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 growunboundedly with respect to its (squared) circumference length.Intriguingly, however, it is well known that black holes in three spatial dimensions behavein many respects as two-dimensional objects. In particular, a black hole is characterized bya 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 two-dimensional relation (1), the surface area of a black hole may be bounded from above by aquadratic 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 arebounded from above by the simple functional relation A ≤ C /π , (2)where C s is the circumference length of the smallest ring that can engulf the black-holehorizon in all azimuthal directions [5–9]. II. THE INVERSE HOOP CONJECTURE IN CHARGED AND SPINNINGKERR-NEWMAN-(ANTI)-DE SITTER BLACK-HOLE SPACETIMES
A Kerr-Newman-(anti)-de Sitter black-hole spacetime of mass M , angular momentum J ≡ M a , electric charge Q , and cosmological constant Λ is characterized by the curved lineelement [10–13] ds = − ∆ r ρ (cid:16) dtI − a sin θ dφI (cid:17) + ∆ θ sin θρ h adtI − ( r + a ) dφI i + ρ (cid:16) dr ∆ r + dθ ∆ θ (cid:17) , (3)2here the metric functions ∆ r , ∆ θ , ρ , and I are given by the functional expressions [10, 11]∆ r ≡ r − M r + Q + a −
13 Λ r ( r + a ) , (4)∆ θ ≡ a cos θ , (5) ρ ≡ r + a cos θ , (6)and I ≡ a . (7)Asymptotically flat Kerr-Newman black holes are characterized by the simple relationΛ = 0, whereas non-asymptotically flat Kerr-Newman-de Sitter and Kerr-Newman-anti-deSitter black-hole spacetimes are characterized respectively by the relations Λ > < r ( r ) [10, 11, 14]. In particular,∆ r ( r + ) = 0 , (8)where r + is the radius of the black-hole event horizon.From Eqs. (3) and (8) one finds the compact expressions C eq = 2 π r + a r + I (9)and A = 4 π r + a I (10)for the equatorial circumference and the horizon surface area of the Kerr-Newman-(anti)-deSitter black hole.Interestingly, from Eqs. (9) and (10) one finds the compact dimensionless ratio H ( M, Q, a, Λ) ≡ π AC = Ir r + a (11)for generic Kerr-Newman-(anti)-de Sitter black holes [15]. The conjectured inverse hooprelation asserts that stationary (3 + 1)-dimensional black holes are characterized by thesimple relation H ≤ . (12)Taking cognizance of Eqs. (7) and (11), one finds that asymptotically flat Kerr-Newmanblack holes (with Λ = 0 and therefore I = 1) and Kerr-Newman-anti-de Sitter black holes3with Λ < I <
1) conform to the inverse hoop relation (12). It is easy toshow that Kerr-Newman-de Sitter black holes (with Λ >
0) are characterized by the relationΛ r ≤ III. SUMMARY AND DISCUSSION
The famous Thorne hoop conjecture [5] asserts that black-hole spacetimes of suitablydefined mass M are characterized by the relation M ≥ C / π . Since there are many differentdefinitions of mass (energy) in curved spacetimes, it is natural to ask: what is the exactphysical meaning of the mass (energy) term M in the hoop relation? To the best of ourknowledge, in his original work Thorne [5] has not provided a specific definition for the massterm M in the intriguing hoop conjecture.In the present compact paper we have explicitly demonstrated that if the mass term M is interpreted as the irreducible mass M irr of the black hole, then generic Kerr-Newman-(anti)-de Sitter black-hole spacetimes conform to the inverse hoop relation M irr ≤ C s / π . (13)Taking cognizance of the fact that the irreducible mass of a black hole is related to itshorizon surface area A by the simple relation M irr ≡ p A / π , (14)one realizes that the inverse hoop relation (13) is a statement about the geometric propertiesof the black-hole horizon, bounding its surface area in terms of the squared circumferenceof the smallest ring that can engulf the horizon in every direction: A ≤ C /π . (15)If true, the conjectured inverse hoop relation (15) implies that the black-hole surface areacannot be unboundedly wrinkled [17].Finally, it is worth noting that there is an important numerical evidence [18] for thevalidity of the inverse hoop conjecture (15) in non-stationary (dynamical) black-hole space-times. In particular, in a very interesting work, East [18] has studied numerically the fullnon-linear gravitational collapse of self-gravitating spheroidal matter configurations. Re-markably, it has been explicitly demonstrated in [18] that, in accord with the weak cosmic4ensorship conjecture [19], the final state of the collapse is a black hole. Interestingly, theinitially distorted dynamically formed horizons obtained in [18] are characterized by dampedoscillations 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 numericalaccuracy [20], the dynamically formed black holes presented in [18] are characterized by therelation max {C eq ( t ) , C p ( t ) } π M irr ≥ , (16)where C eq and C p are respectively the time-dependent (oscillating) equatorial and polarcircumferences of the non-stationary black-hole horizons. Thus, the dynamically formedblack holes presented in [18] seem to respect the conjectured inverse hoop relation (15). ACKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I would like to thankProfessor W. E. East for sharing with me his interesting numerical data. I would also liketo thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulatingdiscussions. 5
1] See R. Osserman, Bulletin of the American Mathematical Society, , 6 (1978) and referencestherein.[2] For a two-dimensional patch embedded in a three-dimensional space, the parameter C maybe defined as the circumference length of the smallest ring that can engulf that patch in alldirections.[3] J. D. Bekenstein, Phys. Rev. D , 2333 (1973).[4] S. W. Hawking, Commun. Math. Phys. , 199 (1975).[5] K. S. Thorne, in Magic without Magic: John Archibald Wheeler , edited by J. Klauder (Free-man, San Francisco, 1972).[6] The mathematically compact and physically influential Thorne hoop conjecture [5] asserts thatblack-hole spacetimes are characterized by the relation
M ≥ C / π . It is worth noting thatthe exact physical meaning of the mass (energy) term M in the hoop relation has not beenspecified in the pioneering work of Thorne [5]. In the present compact paper we shall explicitlydemonstrate that if the mass term M is interpreted as the irreducible mass M irr ≡ p A / π of the black hole, then generic Kerr-Newman-(anti)-de Sitter black-hole spacetimes conformto the inverse hoop relation M irr ≤ C / π .[7] See [8, 9] and references therein for recent studies of the Thorne hoop conjecture [5].[8] S. Hod, The Euro. Phys. Jour. C , 1013 (2018) [arXiv:1903.09786].[9] Y. Peng, The Euro. Phys. Jour. C , 943 (2019).[10] B. Carter in Les Astres Occlus , edited by B. DeWitt, C. M. DeWitt, (Gordon and Breach,New York, 1973).[11] H. Suzuki, E. Takasugi and H. Umetsu, Prog. Theor. Phys. , 491 (1998).[12] We use natural units in which G = c = ~ = 1.[13] Here we use the familiar Boyer-Lindquist spacetime coordinates ( t, r, θ, φ ) [10, 11].[14] Note that generic Kerr-Newman and Kerr-Newman-anti-de Sitter black-hole spacetimes arecharacterized by two (Cauchy and event) horizons, whereas generic Kerr-Newman-de Sitterblack-hole spacetimes are characterized by three (Cauchy, event, and cosmological) horizons.[15] Note that spinless ( a = 0) black holes saturate the inverse hoop conjecture (2) [see Eqs. (7)and (11)].
16] Note that an extremal Schwarzschild-de Sitter black hole whose event horizon coincides withthe cosmological horizon is characterized by the simple dimensionless relation Λ r = 1.[17] It is worth emphasizing that the conjectured inverse hoop relation (15) is expected to be validonly for black holes. In particular, it is straightforward to imagine non-black hole objects thatviolate the area-circumference relation (15). For example, a moon-like object whose surface iscovered with craters can violate the relation (15). Likewise, a non-black hole Coronavirus-likeobject, whose surface is covered with spikes, can violate the area-circumference relation (15).[18] W. E. East, Phys. Rev. Lett. , 231103 (2019).[19] R. Penrose, Riv. Nuovo Cim. , 252 (1969).[20] W. E. East, Private communication., 252 (1969).[20] W. E. East, Private communication.