On the Hoop conjecture in Einstein gravity coupled to nonlinear electrodynamics
K.K. Nandi, R.N. Izmailov, G.M. Garipova, R.R. Volotskova, A.A. Potapov
aa r X i v : . [ g r- q c ] J u l On the Hoop conjecture in Einstein gravity coupled to nonlinear electrodynamics
K.K. Nandi ∗ Zel’dovich International Center for Astrophysics, Bashkir State Pedagogical University,3A, October Revolution Street, Ufa 450008, RB, RussiaDepartment of Physics & Astronomy, Bashkir State University,47A, Lenin Street, Sterlitamak 453103, RB, Russia andHigh Energy and Cosmic Ray Research Center, University of North Bengal, Siliguri 734 013, WB, India
R.N. Izmailov † and G.M. Garipova ‡ Zel’dovich International Center for Astrophysics, Bashkir State Pedagogical University,3A, October Revolution Street, Ufa 450008, RB, Russia
R.R. Volotskova § Salavat Industrial College, 27 Matrosova Boulevard, Salavat 453259, RB, Russia
A.A. Potapov ¶ Department of Physics & Astronomy, Bashkir State University,47A, Lenin Street, Sterlitamak 453103, RB, Russia
The famous hoop conjecture by Thorne has been claimed to be violated in curved spacetimes cou-pled to linear electrodynamics. Hod [10] has recently refuted this claim by clarifying the status andvalidity of the conjecture appropriately interpreting the gravitational mass parameter M . However,it turns out that partial violations of the conjecture might seemingly occur also in the well knownregular curved spacetimes of gravity coupled to nonlinear electrodynamic s. Using the interpretationof M in a generic form accommodating nonlinear electrodynamic coupling, we illustrate a novelextension that the hoop conjecture is not violated even in such curved spacetimes. We introduce aHod function summarizing the hoop conjecture and find that it surprisingly encapsulates the tran-sition regimes between ”horizon and no horizon” across the critical values determined essentially bythe concerned curved geometries. I. INTRODUCTION
In 1972, Kip S Thorne [1, 2] introduced a mathematically elegant and influential conjecture, called the hoopconjecture, that is widely believed to reflect a fundamental aspect of classical general relativty. The conjecture assertsthat a self-gravitating matter configuration of mass M will form an engulfing horizon if its circumferential radius R = C/ π is equal to (or less than) the corresponding Schwarzschild radius 2 M (in units G = 1 , c = 1). That is, thehoop conjecture states that [1] a black hole horizon exists if C ≤ πM ( R ) . (1)This relation has been supported by several studies (see, e.g., [3–7]). Nevertheless, there has been some intriguingclaims in the literature, based on a na¨ıve application of the above relation, that the famous hoop conjecture canbe violated in charged curved spacetimes coupled to linear eletrodynamics [8, 9]. Hod [10] has recently refuted thisclaim by clarifying the status and validity of the conjecture suggesting that the mass parameter on the r.h.s of (1) beappropriately interpreted as the gravitational mass M ( R ) contained within the engulfing hoop of radius R and notas the total (asymptotically measured) mass M ∞ of the entire spacetime.In this paper, we shall be concerned with three well known curved spacetimes of Einstein gravity coupled to nonlinear electrodynamics that are exact, everywhere regular including at the origin, asymptotically flat with ADMmass M ∞ and charge Q . It turns out that, with M = M ∞ , the hoop conjecture can be partially violated in those ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: fmfi[email protected] ¶ Electronic address: [email protected] spacetimes as well. To show that this need not be the case, we shall consider two classes of Ay´on-Beato and Garc´ıa(AG) spacetimes [11, 12] and the Bardeen spacetime [13] and examine the validity of the hoop conjecture in the lightof Hod’s interpretation [10] taken in its generic form accommodating nonlinear electrical energy.
II. HOD’S INTERPRETATION
The energy outside a charged ball of radius R is E elec ( r > R ) = Z ∞ R T πr dr = Q R , (2)where T ( r > R ) = Q πr is the electric energy density in linear electrodynamics. Thus, for a charged ball of radius R , electric charge Q , the gravitational mass contained within ( r ≤ R ) of the ball is given by M ( r ≤ R ) = M ∞ − Q R , (3)and so, according to the interpretation by Hod [10], this mass should be used in the hoop conjecture instead of theasymptotic mass M ∞ , then: C ( R )4 πM ( r ≤ R ) ≤ ⇒ black hole horizon. (4)This relation was used to show that the curved spacetime coupled to linear electrodynamic in [8, 9] actually obeysthe hoop conjecture.Since we are concerned in this paper with spacetimes coupled to nonlinear electrodynamics, we shall use the genericformula for gravitational mass integrating the corresponding T : M ( r ≤ R ) = M ∞ − E elec = M ∞ − Z ∞ R T πr dr. (5)A peculiarity common to the three curved spacetimes considered below is that the asymptotic ADM mass M ∞ is independent of the charge parameter Q , supporting the original idea of Born and Infeld [14] to use nonlinearelectrodynamics for proving the electromagnetic nature of mass. Therefore, Hod’s interpretation embodied in (5)entails that the gravitational mass M is plainly divided between two electromagnetic masses, one asymptotic M ∞ andthe other E elec . Despite the curved spacetime coupled to nonlinear electrodynamics, such a straightforward divisionof masses surprisingly works well as far as the conjecture is concerned, as we will see shortly. We shall use (5) tostudy the validity of hoop conjecture (4). III. CURVED SPACETIMES COUPLED TO NONLINEAR ELECTRODYNAMICS
The three spacetimes under consideration follow from the gravitational action S with source of nonlinear electro-dynamics S = Z √− gd x (cid:20) π R − π L ( F ) (cid:21) , (6)where R is the Ricci scalar and L is a function of F = F µν F µν . We omit further details and come directly to therelevant solutions. (a) Ay´on-Beato and Garc´ıa class 1 spacetime (AG1) The asymptotically flat metric is given by [11] dτ = − A ( r ) dt + 1 A ( r ) dr + r ( dθ + sin θdψ ) , (7) A ( r ) = 1 − M r ( r + Q ) / + Q r ( r + Q ) , (8)with the associated asymptotically vanishing electric field E given by E = Qr (cid:20) r − Q ( r + Q ) + 152 M ( r + Q ) / (cid:21) , (9)where the M is the asymptotic ADM mass, hereinafter to be understood as M ∞ , and Q is related to the electriccharge. AG define two dimensionless parameters s, x as s = Q M ∞ , x = rQ , (10)and by numerically solving two simultaneous equations A ( x c , s c ) = 0 and ∂ x A ( x c , s c ) = 0 , (11)they find two critical values, s c and x c , given by s c = 0 . , x c = 1 . , (12)Keeping x c = 1 .
58 fixed, the transition between ”no horizon to black hole horizon” regime is marked by the criticalparameter s c as follows [11]: s > s c ⇒ no horizon (13) s < s c ⇒ black hole horizon (14) s = s c ⇒ two coincident horizons. (15)Let us look at Eq.(5). The electric energy density T can be obtained from the Einstein equations G αβ = 8 πT αβ ,which yield [15] − T ( r > R ) = − π G g = Q (cid:16) r − Q + 6 M ∞ p r + Q (cid:17) π ( r + Q ) . (16)Integrating, we find the energy outside a ball of radius R to be E elec ( r > R ) = Z ∞ R T πr dr = Q R R + Q ) + 2 M ∞ h Q p R + Q + R (cid:16)p R + Q − R (cid:17)i R + Q ) / . (17)Therefore, Eq.(4) becomes M ( r ≤ R ) R = M ∞ − E elec R = 2 b − a (cid:0) √ a − b (cid:1) a ) / , (18)where we have used the dimensionless parameters a, b defined by a = QR , b = M ∞ R ⇒ s = a b . (19)Taking into account Eq.(5), the hoop conjecture (4) yields what one may call the Hod function H ( b, s ) for brevity: C ( R )4 πM ( r ≤ R ) = (1 + 4 b s ) / b − b s (cid:0) √ b s − b (cid:1) ≡ H ( b, s ) , (20)the subscript 1 refers to solution AG1. Eq.(19) says that there are three variables connected by one equation s = a b ,so we can choose two independent variations in b and s . Since the transitions in (13-15) are described only in termsof s c , so we need to vary s through s c , and b through b c given by2 b c = 2 M ∞ R (cid:12)(cid:12)(cid:12)(cid:12) c = QR (cid:12)(cid:12)(cid:12)(cid:12) c × M ∞ Q (cid:12)(cid:12)(cid:12)(cid:12) c (21)= 1 x c × s c = 11 . × .
317 = 1 . . (22)Note that horizon properties of AG1 are described with fixed x c = 1 .
58, hence b (= sx c ) depends only on s . Now,take a value in the ”no-horizon” range, say, s = 0 .
4. To protect the hoop conjecture in this case with M = M ∞ , onewould need to show that C πM ∞ >
1. On the other hand, using (10) and (19), we find C πM ∞ = 12 b = sx c = 0 . < , (23)which violates the conjecture. However, the other half of the story, viz., the horizon range s ≤ .
317 is consistentwith the hoop conjecture C πM ∞ = sx c ≤
1. So, overall, the use of M = M ∞ partially violates the hoop conjecturebut this violation is only apparent. Using (4,5) we can restore the validity of the conjecture in the entire range of s .This argument can be applied to the remaining two solutions as well, hence will not be reproduced further. (b) Ay´on-Beato and Garc´ıa class 2 spacetime (AG2) The asymptotically flat metric is given by [12] dτ = − A ( r ) dt + 1 A ( r ) dr + r ( dθ + sin θdψ ) , (24) A ( r ) = 1 − Mr (cid:20) − tanh (cid:18) Q M r (cid:19)(cid:21) , (25)with the associated asymptotically vanishing electric field E given by E = Q M r (cid:20) − tanh (cid:18) Q M r (cid:19)(cid:21) × (cid:20) M r − Q tanh (cid:18) Q M r (cid:19)(cid:21) . (26)Like before, the electric energy density is [15] − T ( r > R ) = Q πr sec h (cid:18) Q M ∞ r (cid:19) . (27)Integrating, we find the energy outside a ball of radius R to be E elec ( r > R ) = Z ∞ R T πr dr = M ∞ tanh (cid:18) Q M ∞ r (cid:19) . (28)For AG2, the two dimensionless parameters are s = Q M ∞ , x = 2 M ∞ rQ (29)Defining as before a = QR , b = M ∞ R ⇒ a = 2 bs The Hod function follows as H ( b, s ) = 12 b [1 − tanh (2 bs )] , (30)the subscript 2 refers to solution AG2. The critical values are s c = 0 . , x c = 1 .
56 (31) ⇒ b c = 2 M ∞ R (cid:12)(cid:12)(cid:12)(cid:12) c = 11 . × . = 2 . (c) Bardeen spacetime he asymptotically flat metric is given by [13] dτ = − A ( r ) dt + 1 A ( r ) dr + r ( dθ + sin θdψ ) , (33) A ( r ) = 1 − M r ( r + Q ) / , (34)with the associated electromagnetic field tensor F µν given by F µν = 2 δ θ [ µ δ ψν ] Q sin θ. (35)The charge parameter Q was originally interpreted as describing the electric charge, but later identified by Ay´on-Beato and Garc´ıa [10] as representing a magnetic monopole coupled to nonlinear electrodynamics. Like before, theelectric energy density is [15] − T ( r > R ) = 3 M ∞ Q π ( r + Q ) / . (36)Integrating, we find the energy outside a ball of radius R to be E elec ( r > R ) = Z ∞ R T πr dr = M ∞ (cid:16) Q p Q + R + R (cid:16)p Q + R − R (cid:17)(cid:17) ( Q + R ) / . (37)As in AG1, the two dimensionless parameters are s = Q M ∞ , x = rQ (38)with their critical values s c = 2 √ , x c = √ , b c = √ √ . . (39)Proceeding exactly as in (a), the Hod function follows as H ( b, s ) = (cid:0) b s (cid:1) / b , (40)the subscript 3 referring to the Bardeen solution [13].The 3D plots of H , H and H , where s and b are varied through their critical values, are combined and shown inFig.1. We have checked that the plots are somewhat insensitive to the variation of b . However, the combined 3D plotis not very transparent for reading out the transition points stated in (13-15). For clarity, we show in Fig.2 the plotsof H , H and H , which are just a section of Fig.1 at some average value of b around b c , say b = 1. Fig.2 excellentlyshows the transitions points between ”no horizon and horizon” regimes corresponding to each solution in (a)-(c) . FIG. 1: The functions H i ( b, s ), i = 1 , , b ∈ [0 . , .
3] and s ∈ [0 . , . H i ( b, s ), i = 1 , , s at some fixed average value of b around b c , say b = 1. For H , transition point in the metric AG1 is s = s c = 0 . H , transition point in the metric AG2is s = s c = 0 .
53 and for H , the transition point in the Bardeen metric is s = s c = √ = 0 . s ≤ s c , corresponding to existence of ”horizon (14,15)”, we find H i ≤ s > s c ,corresponding to existence of ”no horizon (13)”, we find H i >
1, all nicely showing that the hoop conjecture is not violated.
IV. CONCLUSIONS
Hod [7] has recently shown that Thorne’s hoop conjecture is not violated, despite claims to the contrary, in thecharged curved spacetimes if the concerned gravitational mass M is appropriately interpreted (we called it Hod’sinterpretation). Present paper is an extension of this result to curved spacetimes coupled to nonlinear electrodynamics ,where a na¨ıve application of the conjecture using M ∞ would also appear to lead to its partial violation, as arguedaround (23). This need not be the case.The novel result we obtained is that, despite electrodynamic nonlinearity, the hoop conjecture is not violated in thespacetimes as exemplified in the text , provided Hod’s interpretation, embodied in Eq.(5), is taken with generic formof E elec accommodating the nonlinearity. The Hod function H ( b, s ) we introduced in (20), (30) and (40) summarizesthe conjecture. Fig.1 plots H i ( b, s ) in 3D for intervals of ( b, s ) that contain their respective critical values (it may beverified that the 3D plots are actually almost insensitive to b ). For clarity, we fix an average value around the criticalvalues of b , say b = 1, without any loss of rigor. It can be immediately seen that the curves in Fig.2 surprisinglyencapsulate the transition between the ”no horizon and horizon” regimes across the critical values s c determined purely by the geometry. [1] K.S. Thorne, in Magic without Magic: John Archibald Wheeler , edited by J. Klauder (Freeman, San Francisco, 1972).[2] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [3] I.H. Redmount, Phys. Rev. D , 699 (1983).[4] A.M. Abrahams, K.R. Heiderich, S.L. Shapiro and S.A. Teukolsky, Phys. Rev. D , 2452 (1992).[5] S. Hod, Phys. Lett. B , 241 (2015).[6] Y. Peng, Eur. Phys. J. C , 943 (2018).[7] S. Hod, Eur. Phys. J. Plus , 106 (2019)[8] J.P. de Le´on, Gen. Relativ. and Grav. , 289 (1987).[9] W.B. Bonnor, Phys. Lett. A , 424 (1983).[10] S. Hod, Eur. Phys. J. C , 1013 (2018).[11] E. Ay´on-Beato and A. Garc´ıa, Phys. Rev. Lett., , 5056 (1998).[12] E. Ay´on-Beato and A. Garc´ıa, Phys. Lett. B , 25 (1999).[13] E. Ay´on-Beato and A. Garc´ıa, Phys. Lett. B , 149 (2000).[14] M. Born and L. Infeld, Proc. Roy. Soc. Lond., A , 410 (1934); A , 425 (1934).[15] V.S. Manko and E. Ruiz, Phys. Lett. B760