Investigations on hoop conjecture for horizonless spherical charged stars
aa r X i v : . [ g r- q c ] F e b Investigations on hoop conjecture for horizonless spherical charged 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
For horizonless spherical stars with uniform charge density, the hoop conjecture was tested basedon the interior solution. In this work, we are interested in more general horizonless spherical chargedstars. We test hoop conjecture using the exterior solution since all types of interior solutions corre-spond to the same exterior Reissner-Nordsr¨ o m solution. Our analysis shows that the hoop conjectureis violated for very compact stars if we express the conjecture with the total ADM mass. And thehoop conjecture holds if we express the conjecture using the mass in the 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 holehorizons is characterized by the mass and circumference relation C π M M is usually interpreted asthe asymptotically measured total ADM mass [3]-[31].For horizonless curved spacetimes, the hoop conjecture should be characterized by the opposite inequality C π M > M is interpreted as the ADM mass, the relation C π M > M is the mass contained in the engulfing sphere [32–35]. In contrast, black holes canviolate the hoop relation if the mass term is interpreted as the mass in the sphere [36]. Considering thedifferent appearances of hoop conjecture in black holes and horizonless stars with uniform charge density, it isinteresting to test hoop conjecture in the background of more general horizonless compact stars. In particular,it is meaningful to examine the case of horizonless stars compact nearly to form horizons.We extend the discussion in [32–35] by considering the exterior solution since all types of interior solutionscorrespond to the same exterior Reissner-Nordsr¨ o m solution. For general horizonless spherical charged stars,our analysis shows that the hoop relation is violated if the mass term is interpreted as the ADM mass andthe hoop relation holds if we use the gravitating mass within the sphere. II. STUDIES OF THE MASS IN HOOP CONJECTURE
We are interested in general horizonless spherical charged stars. And the spacetime reads [32–35] ds = − e ν dt + e λ dr + r ( dθ + sin θdφ ) . (1)The metric functions ν and λ only depend on the radial coordinate r. The sphere surface radius is located at r . In the exterior region r > r , the background is the Reissner-Nordsr¨ o m solution e ν = e − λ = 1 − Mr + Q r , (2)where M is the ADM mass of the spacetime and Q is the star charge. In this work, we pay attention tothe case of M > Q . The would be horizon position is at r h = M + p M − Q . Since we are interested inhorizonless stars, there is the relation r > r h . We can simply set the surface radius as r = (1 + ε ) r h , where ε is a small positive parameter.The circumference C is given by C = 2 πr = 2 π (1 + ε )( M + p M − Q ) . (3)For horizonless curved spacetimes, the hoop conjecture should be characterized by [1, 2] C π M > . (4)If we interpret the mass M as the ADM mass M. The mass to circumference ratio is C π M = 2 π (1 + ε )( M + p M − Q )4 πM = (1 + ε )( M + p M − Q )2 M . (5)For parameters satisfying ε M − √ M − Q M + √ M − Q , the relation (5) yields that C π M . (6)So for very compact stars with ε M − √ M − Q M + √ M − Q , the hoop conjecture is violated if M is the ADM mass.If we interpret M as the mass in the engulfing sphere, the mass is M = M − Q r . (7)The hoop conjecture is expressed by the mass to circumference ratio C π M = 2 πr π ( M − Q r ) = (1 + ε ) r h M (1 + ε ) r h − Q = r h + 2 εr h + ε r h M r h − Q + 2 M εr h > r h + 2 εr h M r h − Q + 2 M εr h . (8)Since r h is the horizon satisfying 1 − Mr h + Q r h = 0, there is the relation r h = 2 M r h − Q . (9)Considering r h = M + p M − Q > M , we get2 εr h > εM r h . (10)The relations (8), (9) and (10) imply that C π M > . (11)It means the hoop conjecture holds for horizonless stars if we use the mass within the engulfing sphere. Sincewe consider the exterior solution, our conclusion holds in the exterior region of various horizonless stars. Herewe analytically show that for exterior solutions of compact stars, Thorne hoop conjecture may generally holdif the mass term is interpreted as mass contained within the engulfing sphere. III. CONCLUSIONS
The famous hoop conjecture is expressed by the mass to circumference ratio. For horizonless sphericalstars with uniform charge density, the hoop conjecture was tested based on the interior solution in [32–35].We investigated hoop conjecture in the exterior region of more general spherical horizonless compact stars.We tested hoop conjecture using the exterior solution since all types of interior solutions correspond to thesame exterior Reissner-Nordsr¨ o m solution. Our analysis showed that the hoop conjecture cannot hold for verycompact stars if the mass is interpreted as the ADM mass in the total spacetime. And the hoop conjectureholds if we interpret the mass as the gravitating mass contained within the engulfing sphere. Acknowledgments
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