January16,2010 15:34 WSPC-ProceedingsTrimSize:9.75inx6.5in Manuscript ON SELF-GRAVITATING ELEMENTARY SOLUTIONS OF NON-LINEAR ELECTRODYNAMICS J.DIAZ-ALONSO,D.RUBIERA-GARCIA LUTH, Observatoire de Paris, CNRS, Université Paris Diderot. 5 Place Jules Janssen, 92190 0 Meudon, France and 1 Departamento de Fisica, Universidad de Oviedo. Avda. Calvo Sotelo 18, E-33007 Oviedo, Spain 0 2 The electrostatic, spherically symmetric solutions of the general class of non-linear n abeliangaugemodels,minimallycoupledtogravity,areclassifiedanddiscussedinterms a oftheADMmassandtheelectromagnetic energyoftheassociatedflat-spacesolutions. J 6 Generalizations of the Reissner-Nordstro¨m (RN) solution of the Einstein- 1 Maxwellfieldequationsthroughnon-linearelectrodynamics(NED)coupledtograv- itationhavebeenstudiedforseveraldecades.ThisisthecaseoftheBorn-Infeld(BI) ] h lagrangian,1 originally introduced to obtain a classical theory with finite-energy, t - electrostatic,sphericallysymmetric(ESS)solutions.Inpresenceofthegravitational p fieldsuchsolutionswereanalyzedinseveralpapers.2Similaranalysishavebeenalso e h performed for ESS solutions of some other models.3 [ However, while these studies are concerned with particular cases of NEDs, the 1 main structure of the gravitating ESS solutions of general physically admissible v NED models can be largelycharacterizedby the vacuum andboundary behaviours 6 oftheirlagrangiandensityfunctions,nomattertheirexplicitformseverywhere.Let 3 8 2 R . I = d4x√ g ϕ(X,Y) , (1) 1 Z − (cid:18)16πG − (cid:19) 0 be the action of our theory, where ϕ(X,Y) is an arbitrary function of the two field 0 1 invariants (X = 1F Fµν, Y = 1F F∗µν), F = ∂ A ∂ A being the v: field strength tens−or2anµνd Fµ∗ν = εµνα−β2Fαµβνits dual.µWνe conµstrνai−n thνeseµmodels by i several“admissibility”conditions:i) ϕmust be single-branchedandC1-class onits X domain of definition of the X Y plane, which must include the vacuum. ii) For r − a parity invariance, the condition ϕ(X,Y) = ϕ(X, Y) must hold. iii) The positive − definite character of the energy functional for any field configuration requires4 ρ . . ≥ √X2+Y2+X ϕ +Yϕ ϕ(X,Y) 0,whereϕ =∂ϕ/∂X andϕ =∂ϕ/∂Y. X Y X Y − ≥ (cid:0) As a consequ(cid:1)ence of the source symmetry T0 = T1, and the static spherically 0 1 symmetricmetricmaybewrittenasds2 =g(r)dt2+g−1(r)dr2 r2(dθ2+sin2θdϑ2). Thefieldequations (ϕ Fµν+ϕ F∗µν)=0forthesemode−ls lead,forESSfields µ X Y ∇ (E~(r)=E(r)~r,H~ =0),toafirst-integralr2E(r)ϕ =q,whichformsacompatible r X system with the Einstein equations G =8πT . Its solution can be obtained as5 µν µν 2m(r) g(r)=1 , (2) − r where m(r) = M ε (r,q) contains the ADM mass M and ε (r,q) = ex ex ∞ − 4π R2T0(R,q)dR, the (flat-space) energy of the ESS field outside of the sphere r 0 R 1 January16,2010 15:34 WSPC-ProceedingsTrimSize:9.75inx6.5in Manuscript 2 of radius r. For admissible models ε (r,q) can be shown to be a monotonically ex decreasing and concave function of r, for fixed values of the electric charge q.5 Let us now consider the class of admissible NED models supporting flat-space finite-energyESSsolutions.4 Forthetotalenergyoftheelectromagneticfieldinflat ∞ space, ε(q) = 4π r2T0(r,q)dr = q3/2ε(q = 1), to be finite, we must have fields 0 0 vanishingatinfiniRtyasE(r) β/rp,withp>1(12:caseB3, ∼ andp=2:caseB2;whichcorrespondtoaslowerthan,fasterthan,andCoulombian behaviours,respectively)whileatr =0therearetwopossiblebehaviours:A1,where E(r) βrp, 1 < p < 0 and A2, with E(r) a brσ , β, a and σ(> 0) being ∼ − ∼ − universalconstants for a givenmodel, while b is a function ofq. For these fields the integral defining ε (r,q) is convergentfor any r. Let us stress that any admissible ex NED model supporting flat-space finite-energy ESS solutions belongs to one of the B-cases at r and to one of the A-cases as r 0. Using only these data the → ∞ → associatedgravitatingESSsolutionscanbefullyclassified.Suchaclassificationcan be done by looking for the horizons (g(r )=0) present in each configuration. The h general mass-horizon radius relation, obtained from Eq.(2) reads r h M(r ) =ε (r ,q), (3) h ex h − 2 and thus the horizonradii (if any) of the different gravitating solutions are defined by the cut points of the curves ε (r,q) (fixed q) with the beam of straight lines ex M r/2, given by different values of the ADM mass M. In this sense, a tangency − ′ cut point (g (r) = 0) leads to an extreme black hole (EBH), defined by the |rh condition8πr2T0(r,q)=1,whichgivestheEBHmassM (q)throughEq.(3).For 0 extr completenessletusalsoconsiderthecaseofNEDsleadingto(flat-space)divergent- energy solutions, which can be tackled following the same method as in the finite- energycases.Thisprocedureleadstothefollowinggravitatingstructures(seeFig.1): y ¶HqL IIIa IIIc IIIb II I 0 r Fig. 1. Behaviours of εex(r,q) for the ESS solutions of admissible NED models: (I) Divergent- energycase; (II) Finite-energy(A-1);(III) Finite-energy(A-2)(IIIa: 16πqa<1,IIIb: 16πqa>1, IIIc: 16πqa=1).ThedashedstraightlinescorrespondtodifferentvaluesofM inM−rh/2. January16,2010 15:34 WSPC-ProceedingsTrimSize:9.75inx6.5in Manuscript 3 (Flat-space) divergent-energysolutions: In this case there are three possible structures (see Fig.1, curve I): i) M = M (q): EBH, ii) M < M (q): Naked extr extr singularity (NS), iii) M > M (q): Two-horizon (Cauchy and event) BH. Thus extr the behaviour of any admissible NED model of this class is similar to that of the RN solution of the Einstein-Maxwell equations (ϕ(X,Y)=αX, α a constant). Finite-energy solutions: Case A-1: At the center ε (r,q) ε 16πqβ rp+1 ex ∼ − (2−p)(1+p) which implies a negative divergent slope there (see Fig.1, curve II). There are five classes of solutions: i) M = M (q): EBH, ii) M < M (q): NS, iii) M (q) < extr extr extr M <ε(q):Two-horizonBH,iv)M =ε(q):Single-horizonBHtowhichthesequence − ofsolutionsofcaseiii)converges(forr=0)whenM ε(q) 0 ,v)M >ε(q):BH 6 − → withasinglehorizon.AnexampleofthisfamilyisprovidedbytheEuler-Heisenberg lagrangian,where ϕ(X,Y)=X/2+ξ(X2+(7/4)Y2), ξ >0.5 ′ Finite-energy solutions: Case A-2: At the center ε (0,q) = 8πqa R 1/2, ex − − thus we have three different behaviours for ε (r,q) (see Fig.1, curves III): A2a ex ↔ curve IIIa) If 16πqa < 1 there are three cases: i) M < ε(q): NS, ii) M > ε(q): Single-horizon BH, iii) M = ε(q): NS with g(0) = 1 16πqa > 0. A2b curve − ↔ IIIb) If 16πqa>1 we have five cases: i) M =M (q): EBH, ii) M ε(q): Single-horizon BH, v) extr M = ε(q) : Single-horizon BH with g(0) = 1 16πqa < 1. A2c curve IIIc) − ↔ If 16πqa = 1 the charge is fixed and this case is similar to the A2a, expecting for M =ε(q)where g(0)=1 16πqa=0 andwehavean“extremeblackpoint”.5 The − BI model ϕ(X,Y)=2β2(1 1 β−2X (√2β)−4Y2) belongs to this family. −q − − ManyotherpropertiesofthegravitatingESSsolutionsofadmissibleNEDmod- elscanbeestablishedthroughthisprocedure.Inparticular,forthethermodynamic analysisone must takeinto accountthat both the zerothand firstlawsof BHther- modynamics hold for NEDs.6 The Hawking temperature of the ESS solutions is givenby T = k ;k = 1g′(r) =( 1 4πr T0(r ,q)) andits analysisleads,aside 2π 2 |rh 2rh − h 0 h from“Schwarzschild-like”and“RN-like”behaviours,toothercaseswithveryspecial features.Asanexample,for16πqa=1(A2ccase)andM =ε(q),wefindvanishing- T(σ > 1), finite-T(σ = 1) or divergent-T(σ < 1) extreme black points(r = 0). hextr The analysisofthese andother propertiesofthe solutionsis currentlyinprogress.7 References 1. M. Born and L. Infeld, Proc. R. Soc. London A144, 425 (1934). 2. A.Garcia,H.SalazarandJ.F.Plebanski,Nuovo. Cim.84,65(1984);M.Demianski, Found.ofPhys.16,187(1986);G.W.GibbonsandD.A.Rasheed,Nucl.Phys.B454, 185 (1995); N.Breton, Phys. Rev. D67, 124004 (2003). 3. H. P. de Oliveira, Class. Quant. Grav. 11, 1469 (1994); H. H. Soleng, Phys. Rev. D52, 6178 (1995); E. Ayón-Beato and A. Garc´ıa, Phys. Rev. Lett. 80, 5056 (1998); M. Hassaine and C. Martinez, Class. Quant. Grav. 25, 195023 (2008). 4. J. Diaz-Alonso and D.Rubiera-Garcia, Ann. Phys. 324, 827 (2009). 5. J. Diaz-Alonso and D.Rubiera-Garcia, arXiv:0908.3303[hep-th]. 6. D.A. Rasheed, arXiv:hep-th/9702087. 7. J. Diaz-Alonso and D.Rubiera-Garcia, to be published.

On self-gravitating elementary solutions of non-linear electrodynamics

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