A unified hoop conjecture for black holes and horizonless compact stars
aa r X i v : . [ g r- q c ] J a n A unified hoop conjecture for black holes and horizonless compact stars
Yan Peng , ∗ School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China and Center for Gravitation and Cosmology, College of Physical Science and Technology,Yangzhou University, Yangzhou 225009, China
Abstract
We propose a unified version of hoop conjecture valid for various black holes and horizonlesscompact stars. This conjecture is expressed by the mass to circumference ratio 4 πM in /C
1, whereC is the circumference of the smallest ring that can engulf the object in all azimuthal directions and M in is the mass within the engulfing sphere. PACS numbers: 11.25.Tq, 04.70.Bw, 74.20.-z ∗ [email protected] I. INTRODUCTION
The famous hoop conjecture introduced almost five decades ago asserts that the existence of black hole hori-zons is characterized by the mass and circumference relation 4 π M /C > M is usually interpreted asthe asymptotically measured total ADM mass of black holes [3]-[29].For horizonless compact stars, the curved spacetimes should be characterized by the relation 4 π M /C < M is still interpreted as the total ADM mass, this relation 4 π M /C < M in the mass to circumference ratio [1]. And it has been proved that therelation 4 π M /C < M is interpreted as the mass contained within theengulfing sphere (not as the total ADM mass) [32, 33].It is natural to assume that a unified hoop conjecture may be 4 π M /C > π M /C < M is the total ADM mass or the mass contained in the engulfing sphere[34]. In the background of horizonless stars, the calculations in [32] imply that the term M should be themass contained in the sphere and cannot be the total ADM mass. However, in the black hole spacetimes, Hodfound that the hoop conjecture holds if M is the ADM mass and cannot be the mass contained in the sphere[34]. So it seems that there is no unified hoop conjecture valid for both black holes and compact stars.In this paper, we propose another version of unified hoop conjecture, which holds for Schwarzschild blackholes, neutral Kerr black holes, RN black holes, Kerr-Newman black holes and horizonless charged compactstars. It may be a general property for both black holes and horizonless compact stars. II. TEST THE UNIFIED HOOP CONJECTURE
In the background of horizonless charged compact stars, it has recently been proved that the relation4 π M /C < M is interpreted as the total ADM mass and the relation holds if M = M in ,where M in is the mass contained within the engulfing sphere [30–33]. In addition, the numerical data in [34]implies that 4 πM in /C πM in /C , (1)where C is the circumference of the smallest ring that can engulf the object in all azimuthal directions and M in is the mass within the engulfing sphere. The relation (1) is natural according to the idea that the ring ofthe engulfing sphere is related to the mass within the sphere (not the mass in the total spacetime).In the following, we give examples supporting the bound (1) following studies in [30–34]. A Kerr-Newmanblack hole spacetime is described by the curved line element [34] ds = − ∆ − a sin θρ dt + ρ ∆ dr − asin θ (2 Mr − Q ) ρ dtdφ + ρ dθ + ( r + a ) − a ∆ sin θρ sin θdφ , (2)where M is the ADM mass, Q is the electric charge, J = M a is the angular momentum, ∆ = r − M r + Q + a and ρ = r + a cos θ . Black hole horizons are expressed as r ± = M ± ( M − Q + a ) / , (3)which are determined by the zeros of the metric function ∆( r ) = 0.Setting dt = dr = dθ = 0 , θ = π , we get the relation ds = r + a r + dφ. (4)Also taking ∆ φ = 2 π , we obtain the equatorial circumference C = 2 π r + a r + . (5)For Schwarzschild black holes, there is Q = 0 and a = 0. The mass to circumference ratio is4 πM in C = 4 πMC = 4 πM πr + = 4 πM π (2 M ) = 1 . (6)For RN black holes with Q = 0 and a = 0, the ratio satisfies4 πM in C = 4 π ( M − Q r + )2 πr + = 4 M r + − Q r = 4 M ( M + p M − Q ) − Q M + p M − Q ) = 1 . (7)The neutral Kerr black holes correspond to Q = 0 and a = 0. It yields the relation4 πM in C = 4 πM π r + a r + = 2 M r + r + a = 2 M ( M + √ M − a )( M + √ M − a ) + a = 1 . (8)In the background of Kerr-Neuman black holes, both charge and rotating parameters are nonzero as Q = 0and a = 0. The equatorial circumference is C = 2 π r + a r + and the mass contained within the black holehorizon is M in = M − Q r + [1 + r + a ar + arctg ( ar + )] [35]. In this case of Kerr-Neuman black holes, numerical datain [34] implies an upper bound on the ratio 4 πM in /C . (9)The horizonless charged compact star satisfies the inequality [32, 33]4 πM in /C < . (10)According to relations (6-10), we propose the unified hoop conjecture (1), which may be a general propertyfor both black holes and horizonless compact stars. III. CONCLUSIONS
We proposed a unified hoop conjecture, which is valid for various black holes and horizonless compact stars.Our conjecture is that the mass and circumference relation 4 πM in /C M in is the gravitating masswithin the engulfing sphere. Our statement is in accordance with the idea that the ring of the engulfing sphereshould be related to the mass within the sphere (not the mass in the total spacetime). Acknowledgments
This work was supported by the Shandong Provincial Natural Science Foundation of China under GrantNo. ZR2018QA008. This work was also supported by a grant from Qufu Normal University of China underGrant No. xkjjc201906. [1] K.S. Thorne, in Magic Without Magic: John Archibald Wheeler, ed. by J. Klauder (Freeman, San Francisco,1972).[2] C.W.Misner,K.S.Thorne,J.A.Wheeler,Gravitation(Freeman,San Francisco, 1973).[3] I.H. Redmount, Phys. Rev. D 27, 699 (1983).[4] A.M. Abrahams, K.R. Heiderich, S.L. Shapiro, S.A. Teukolsky, Phys. Rev. D 46, 2452 (1992).[5] S. Hod, Phys. Lett. B 751, 241 (2015).[6] P. Bizon, E. Malec, and N. ´o Murchadha, Trapped surfaces in spherical stars, Phys. Rev. Lett. 61, 1147 (1988).[7] P. Bizon, E. Malec, and N. ´o Murchadha, Class. Quantum Grav. 6, 961 (1989).[8] D. Eardley, Gravitational collapse of vacuum gravitational field configurations, J. Math. Phys. 36,3004(1995).[9] J. Guven and N. ´o Murchadha,Sufficient conditions for apparent horizons in spherically symmetric initial data,Phys. Rev. D 56,7658(1997).[10] J. Guven and N. ´o Murchadha,Necessary conditions for apparent horizons and singularities in spherically symmetricinitial data, Phys. Rev. D 56, 7666(1997).[11] E. Malec, Event horizons and apparent horizons in spherically symmetric geometries, Phys. Rev. D 49, 6475(1994).[12] E. Malec and ´o Murchadha,The Jang equation, apparent horizons, and the Penrose inequality, Class. QuantumGrav. 21,5777(2004). [13] T. Zannias, Phys. Rev. D 45, 2998 (1992).[14] T. Zannias, Phys. Rev. D 47, 1448 (1993).[15] E. Malec,Isoperimetric inequalities in the physics of black holes, Acta Phys. Pol. B 22, 829 (1991).[16] M. Khuri,The Hoop Conjecture in Spherically Symmetric Spacetimes, Phys. Rev. D 80, 124025 (2009).[17] H. Bray and M. Khuri, Asian J. Math. 15, 557 (2011).[18] R. Schoen and S.-T. Yau, Commun. Math. Phys. 90, 575(1983).[19] Takeshi Chiba, Takashi Nakamura, Ken-ichi Nakao, Misao Sasaki, Hoop conjecture for apparent horizon formation,Class. Quant. Grav. 11(1994)431-441.[20] Takeshi Chiba, Apparent horizon formation and hoop concept in nonaxisymmetric space, Phys. Rev. D60(1999)044003.[21] Ken-ichi Nakao, Kouji Nakamura, Takashi Mishima, Hoop conjecture and cosmic censorship in the brane world,Phys. Lett. B 564(2003)143-148.[22] G.W. Gibbons,Birkhoff’s invariant and Thorne’s Hoop Conjecture, arXiv:0903.1580[gr-qc].[23] M. Cvetic, G.W. Gibbons, C.N. Pope,More about Birkhoff’s Invariant and Thorne’s Hoop Conjecture for Horizons,Class. Quant. Grav. 28(2011)195001.[24] John D. Barrow, G. W. Gibbons, Maximum Tension: with and without a cosmological constant, Mon. Not. Roy.Astron. Soc. 446(2014)3874-3877.[25] John D. Barrow, G.W. Gibbons, A maximum magnetic moment to angular momentum conjecture, Phys. Rev. D95(2017)064040.[26] Edward Malec, Naqing Xie, Brown-York mass and the hoop conjecture in nonspherical massive systems, Phys.Rev. D 91(2015)no.8,081501.[27] Fabio Anz` aa