Removing black-hole singularities with nonlinear electrodynamics
aa r X i v : . [ g r- q c ] M a r Removing black hole singularities withnonlinear electrodynamics Christian Corda and Herman J. Mosquera Cuesta
September 3, 2018 Associazione Galileo Galilei, Via Pier Cironi 16 - 59100 Prato, Italy; Instituto de Cosmologia, Relatividade e Astrof`ısica (ICRA-BR), CentroBrasilero de Pesquisas Fisicas, Rua Dr. Xavier Sigaud 150, CEP 22290 -180Urca Rio de Janeiro - RJ Brazil
E-mail addresses: [email protected]; [email protected] Abstract
We propose a way to remove black hole singularities by using a par-ticular nonlinear electrodynamics Lagrangian that has been recently usedin various astrophysics and cosmological frameworks. In particular, weadapt the cosmological analysis discussed in a previous work to the blackhole physics. Such analysis will be improved by applying the Oppen-heimer–Volkoff equation to the black hole case. At the end, fixed the ra-dius of the star, the final density depends only on the introduced quintessen-tial density term ρ γ and on the mass. Keywords : Black holes, removed singularity.
It is well known that the concept of black hole has been considered very fascinat-ing by scientists even before the introduction of Einstein’s general relativity (seeRef. [1] for a historical review). However, an unsolved problem concerning suchobjects is the presence of a spacetime singularity in their core. Such a problemwas present starting by the first historical papers concerning black holes [2]-[4]and was generalized in the famous paper by Penrose [5]. It is a common opinionthat this problem could be solved when a correct quantum gravity theory willbe, finally, constructed (see Ref. [6] for recent developments).In this letter, we propose a way to remove black hole singularities at a clas-sical level. The idea is to use a particular nonlinear electrodynamics Lagrangianto address this issue. Such Lagrangian has been recently used in various analy-ses in astrophysics, like surface of neutron stars [7] and pulsars [8] and also oncosmological contexts [9]. In particular, we will adapt the cosmological analysisin Ref. [9] to the black hole case. The analysis will be improved by applying the1ppenheimer–Volkoff equation [15] to the black hole case. In this way, we showthat the density singularity of a black hole is also removed. At the end, fixedthe radius of the star, the final density depends only on ρ γ , a a quintessentialdensity term which will be introduced in the following, and also on the starmass.At this point it is very important to provide to the reader some elements thatwere part of our motivations for using NonLinear Electrodynamics (NLED) inthe search for physical methods to eliminate the black hole singularity in generalrelativity [19]. Those elements provide a good guide to understand the formalanalysis that we are going to present below.As it has been carefully explained in Refs. [7] and [8], the effects arisingfrom a NLED become quite important in super-strongly magnetized compactobjects, such as pulsars, and particular neutron stars. Some examples includethe so-called magnetars and strange quark magnetars. In particular, NLEDmodifies in a fundamental basis the concept of Gravitational Redshift as com-pared to the well-established method introduced by standard general relativity[7]. The analysis in Ref. [7] proved that unlike general relativity, where theGravitational Redshift is independent of any background magnetic field, whena NLED is incorporated into the photon dynamics, an effective GravitationalRedshift appears, which happens to depend decidedly on the magnetic field per-vading the pulsar. An analogous result has also been obtained in Ref. [8] formagnetars and strange quark magnetars. The resulting Gravitational Redshifttends to infinity as the magnetic field grows larger [7, 8], as opposed to thepredictions of standard general relativity.What it is important for our goal in this letter is that the GravitationalRedshift of neutron stars is connected to the mass–radius relation of the object,see Refs. [7] and [8]. Thus, NLED effects turn out to be important as regard tothe mass–radius relation, and one can also reasonably expect important effectsin the case of black holes, where the mass–radius ratio is even more importantthan for a neutron star. Then, from a physical point of view, the formal anal-ysis presented in the letter displays a correct procedure to estimate the crucialphysical properties stemming from NLED effects in the presence of super strongmagnetic fields. In fact, the formal discussion developed here shows that thequintessential density term permits one to construct a model of star supportedagainst self-gravity entirely by radiation pressure.In this sense, the approach here is similar to the one introduced in the recentpapers of Refs. [20] and [28], where it has been shown that it is possible to haveradiation pressure supported stars at arbitrary value of high mass. The resultis that trapped surfaces as defined in general relativity are not formed duringa gravitational collapse, and hence the singularity theorem on black holes, asproposed in Ref. [5], cannot be applied.It is well known that the conditions concerning the early era of the universe,when very high values of curvature, temperature and density were present [1],and where matter should be identified with a primordial plasma [1], [9]-[11],are similar to the conditions concerning black holes physics. This is exactly themotivation because the singularity theorem on black holes has been generalized2o the Universe [1, 10, 11]. In the literature there are various cases where aparticular analysis on black holes has been applied to the Universe [1, 10, 11]and vice versa [1, 10, 11, 14].We will re-discuss the simple case of a homogeneous and isotropic spherethat was analysed in the historical paper of Oppenheimer and Snyder [12] and,in a different approach, by Beckerdoff and Misner [13]. See also Ref. [1] fora review of the issue. In such a case, the well-known Robertson–Walker line-element can be used, which represents comoving hyper-spherical coordinates forthe interior of the star [1], and in terms of conformal time η it reads ds = a ( η )( − dη + dχ + sin χ ( dθ + sin θdϕ ) , (1)where a ( η ) is the scale factor of a conformal space-time. Using sin χ we arechoosing the case of positive curvature, which is the only one of interest becauseit corresponds to a gas sphere whose dynamics begins at rest with a finite radius[1]. In Refs. [1, 12] and [13] the simplest model of a “star of dust” has beendiscussed, i.e. the case of zero pressure. In this case the stress-energy tensor is T = ρu ⊗ u, (2)where ρ is the density of the star and u the four-vector velocity of the matter.Thus, working with 8 πG = 1, c = 1 and ~ = 1 (natural units), following Ref.[9], the Einstein field equations give only one meaningful relation( dadη ) + a = ρ a , (3)which admits the familiar cycloidal solution a = a η ) , (4)where a is a constant, that is singular for η = π. Now, let us realize our modified model. We re-introduce a more generalstress-energy tensor, i.e. the one regarding a relativistic perfect fluid [1, 9]: T = ρu ⊗ u − pg, (5)where p is the pressure and g is the metric. In this case, by restoring theordinary time with the substitution [1, 9] dt = adη, (6)the Einstein field equations give the relation( ˙ a ) = ρ a − , (7)where dot represents time-derivative.3e will use the non-linear electrodynamics Lagrangian studied in Ref. [9].It describes the Heisenberg-Euler NLED L m ≡ − F + c F + c G , (8)where G = η αβµν F αβ F µν , F ≡ F µν F µν is the electromagnetic scalar and c and c are constants [9].Thus, we also use the equation of state p = 13 ρ − ρ γ , (9)where (see Eq. (25) of Ref. [9]) ρ γ ≡ c B . (10) B is the strength of the magnetic field associated to F . In Ref. [9] the equationof state Eq. (9) has been obtained by performing an averaging on electric andmagnetic fields. The presence of the quintessential density term ρ γ will permitto violate the Null-Energy-Condition of the Penrose’s Theorem [5].The equation of state (9) is no longer given by the Maxwellian value. Thus,by inserting Eq. (9) and Eq. (10) into Eq. (7) one obtains [9]˙ a = B a (cid:18) − c B a (cid:19) − , (11)which can be solved as obtained by [9] t = ˆ a ( t ) a (0) dz ( B z − c B z − − . (12)This expression is not singular for values of c >
0. In fact, by using ellipticfunctions of the first and of second kinds, one gets a parabolic trend near aminimum value in t = t f for a ( t ), where t f is the time which corresponds to theconformal time η = π. In concrete terms, by calling l, m, n the solutions of the equation 8 c B − B x + 3 x = 0 , it is [9] t = [ − ( m − l ) A (arcsin q z − lm − l , q l − ml − n )+ n ( m − l ) − B (arcsin q z − lm − l , q l − ml − n )] | z = a ( t ) z = a (0) , (13)where A ( x, y ) ≡ ˆ sin x dz [(1 − z ) − (1 − y z ) − ] (14)is the elliptic function of the first kind and B ( x, y ) ≡ ˆ sin x dz [((1 − z ) − ) − ((1 − y z ) − ) ] (15)4s the elliptic function of the second kind.Then, by defining the minimum value of the scale factor like a f ≡ a ( t = t f ) , (16)and recalling that the Schwarzschild radial coordinate, in the case of theblack hole geometry (1), is [1] r = a sin χ , (17)where χ is radius of the surface in the coordinates (1), one gets r f = a f sin χ > M if B has an high strength, where M is the black holemass and 2 M the gravitational radius in natural units [1]. Thus, we find thatthe mass of the star generates a curved space-time without event horizons.Now, let us consider the famous Oppenheimer - Volkoff Equation [15] ( r theradial coordinate) dpdr = − ( M + 4 πr p )( p + ρ ) r ( r − M ) . (18)Using Eq. (9) one gets ρ f = ρ − r f ˆ r [3 M + 4 πr ( ρ − ρ γ )](4 ρ − ρ γ ) r ( r − M ) dr, (19)where r f ≡ r ( t f ), r ≡ r (0), ρ f ≡ ρ ( t f ), ρ ≡ ρ (0).By defining ρ f ≡ ρ f + ρ f + ρ f + ρ f + ρ f , ρ f ≡ ρ − ´ r f r ρMr ( r − M ) dr,ρ f ≡ ρ + ´ r f r ρ γ Mr ( r − M ) dr, ρ f ≡ ρ − ´ r f r πr ρ r ( r − M ) dr,ρ f ≡ ρ + ´ r f r πr ρρ γ r ( r − M ) dr, ρ f ≡ ρ − ´ r f r πr ρ γ r ( r − M ) dr, (20)the integral (19) can be computed by separating variables.One then gets ρ f = ρ r f r r − Mr f − M + ρ + ρ γ ln r r f r f − Mr − M ++ { ρ − π [ r ( r − r f ) + ( r − r f ) + 4 M ln r f − Mr − M ] } − ++ ρ exp 60 π [ r ( r − r f ) + ( r − r f ) + 4 M ln r f − Mr − M ]++ ρ + 36 πρ γ [ r ( r − r f ) + ( r − r f ) + 4 M ln r f − Mr − M ] . (21)5 reasonable assumption is ρ → r → r , i.e. at the beginning ofthe collapse the initial density is considered negligible with respect to the finaldensity. In this case it is ρ = ρ = ρ f = ρ = ρ ≃ , and the finaldensity reduces to ρ f = 92 ρ γ ln r r f r f − Mr − M + 36 πρ γ [ r ( r − r f ) + 12 ( r − r f ) + 4 M ln r f − Mr − M ] . (22)Thus, we have shown that the density singularity has also been removedand that, fixed the radius of the star, the final density depends only by theintroduced quintessential density term ρ γ and by the mass. • Notice that in this last updated version we correct two typos which werepresent in Eqs. (21) and (22) in the version of this letter which has beenpublished in Mod. Phys. Lett. A 25, 2423-2429 (2010). In the presentversion, both of Eqs. (21) and (22) are dimensionally and analyticallycorrect.In summary, adapting to the black hole case a cosmological analysis in Ref.[9], we have shown that the introduction of the Heisenberg-Euler NLED (8) inthe black hole physics can, in principle, remove the black hole singularity forvalues of c > ρ γ and bythe mass of the black hole.We are going to further improve the analysis by using the Born-Infeld La-grangian in the context of black holes physics in a future work [16].Finally, we take the chance to stress that this result follows in the same di-rection of the papers on this issue pioneered by Mitra [17, 21, 22] and Robertson,Leiter and Schild [18], [23]-[27], who have shown that the formation of trappedsurfaces during the gravitational collapse can be avoided if the collapsing stardevelops a powerful magnetic field of strength of > G , or equivalently, anintrinsic magnetic moment on the order of 10 G − cm , in the case of galacticblack hole candidates.For a sake of completeness [29], we recall previous works attempting toremove black holes singularities using nonlinear electrodynamics by Ayon-Beatoand Garcia in [30]-[34]. Acknowledgements
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