On self-gravitating elementary solutions of non-linear electrodynamics

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by  J. Diaz-Alonso

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, Universit´e 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
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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(1<p<2:caseB1,p>2: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.
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(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 <M : NS,
extr extr
iii) M (q) < M < ε(q): Two-horizon BH, iv) 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. Ay´on-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.