Comment on "Extended Born-Infeld theory and the bouncing magnetic universe"
aa r X i v : . [ g r- q c ] O c t Comment on “Extended Born-Infeld theory and the bouncing magnetic universe”
Ricardo Garc´ıa-Salcedo,
1, a
Tame Gonzalez,
2, b and Israel Quiros
3, c Centro de Investigacion en Ciencia Aplicada y Tecnologia Avanzada - Legaria del IPN, M´exico D.F., CP 06050, M´exico. Departamento de Ingenier´ıa Civil, Divisi´on de Ingenier´ıa,Universidad de Guanajuato, Guanajuato, CP 36000, M´exico. Departamento de Matem´aticas, Centro Universitario de Ciencias Ex´actas e Ingenier´ıas,Universidad de Guadalajara, Guadalajara, CP 44430, Jalisco, M´exico. (Dated: September 24, 2018)In a recent paper [Phys. Rev. D , 023528 (2012)] the authors proposed a generalized Born-Infeldelectrodynamics coupled to general relativity which produces a nonsingular bouncing universe. Fora magnetic universe the resulting cosmic evolution inevitably interpolates between asymptotic deSitter states. Here we shall show that (i) the above theory does not have the standard weak fieldMaxwell limit, (ii) a sudden curvature singularity – not better than the big bang – arises, (iii) thespeed of sound squared is a negative quantity signaling instability against small perturbations of thebackground energy density, and that (iv) the conclusion about the inevitability of the asymptoticvacuum regime in a magnetic universe is wrong. PACS numbers: 98.80.Cq
I. INTRODUCTION
Studying the equations of the non-linear electrody-namics (NLED) is an attractive subject of research ingeneral relativity thanks to the fact that such quantumphenomena as vacuum polarization can be implementedin a classical model through their impact on the prop-erties of the background space-time. Even if the NLEDmodels coupled to general relativistic cosmology describehypothetical systems reminiscent of the fields in the realworld, these models comprise interesting dynamical be-havior that is worthy of independent investigation.The prototype of a NLED theory is provided by theoriginal Born-Infeld Lagrangian [1]: L = − γ (cid:16)p F/ γ − (cid:17) , (1)where F stands for the electromagnetic (EM) invariant F := F µν F µν , and γ is a free constant parameter. Themotivation of the authors was to have regular field con-figurations without singularities. The gravitational fieldwas not included in their analysis. If one introduces grav-itational effects through the theory of general relativity,a drawback of Born-Infeld proposal unfolds: there is noplace for a regular cosmological scenario with the com-bined effects of gravity and NLED.In the reference [2], motivated by the original Born-Infeld’s idea of having regular field configurations with a Electronic address: [email protected] b Electronic address: [email protected] c Electronic address: [email protected] the EM field bounded – this time in a magnetic uni-verse – the authors focused in a modification of the La-grangian (1) by the inclusion of a term quadratic in thefield F within the square root. Besides, in [2] the La-grangian term ∝ γ in (1) was removed. According tothe authors, the resulting cosmological model is charac-terized by a bounce at some non vanishing value of thescale factor and, the corresponding cosmological evolu-tion inevitably interpolates between asymptotic vacuum(de Sitter) states.In this comment we shall show that the latter conclu-sion is wrong and – what is more relevant – that theirtheory does not have the standard weak field (linear)Maxwell limit. Besides, a sudden curvature singularityis inevitable in the resulting cosmological model which,together with the fact that the speed of sound squaredcan be a negative quantity – signaling instability againstsmall perturbations of the magnetic background – leadsto concluding that the theory should be ruled out. II. NONLINEAR ELECTRODYNAMICSCOUPLED TO GENERAL RELATIVITY
The Einstein-Hilbert action of gravity coupled toNLED is given by S = Z d x √− g [ R + L m + L ( F, G )] , (2)where R is the curvature scalar, L m – the Lagrangianof the background matter, and L ( F, G ) is the gauge-invariant electromagnetic (EM) Lagrangian, which is afunction of the EM invariants F := F µν F µν and G := ǫ αβµν F αβ F µν (see, for instance, Ref. [3]). As usual,the EM tensor is defined as F µν := A ν,µ − A µ,ν (here thecomma denotes partial derivative in respect to the space-time coordinates, while the semicolon denotes covariantderivative instead). Standard (linear) Maxwell electro-dynamics is given by the Lagrangian L ( F ) = − F/ g µν , to obtain: G µν = T m µν + T em µν ,where T m µν = ( ρ m + p m ) u µ u ν − p m g µν , and T em µν = g µν [ L ( F ) − GL G ] − F µα F αν L F , with ρ m = ρ m ( t ), p m = p m ( t ) – the energy density and barotropic pressureof the background fluid, respectively, while L F ≡ dL/dF , L F F ≡ d L/dF , etc. Variation with respect to the com-ponents of the EM potential A µ yields to the EM fieldequations ( F µν L F + ǫ αβµν F αβ L G / ; µ = 0.In this comment we shall consider a homogeneousand isotropic Friedmann-Robertson-Walker (FRW) back-ground metric with flat spatial sections: ds = − dt + a ( t ) δ ij dx i dx j ( i, j = 1 , , a ( t ) is the cosmo-logical scale factor. In order to meet the requirements ofhomogeneous and isotropic cosmology, the energy densityand the pressure of the NLED field should be evaluatedby averaging over volume. To do this, we define the vol-umetric spatial average of a quantity X at the time t by(for details see, for instance, [4, 5]): X ≡ lim V → V V Z d x √− g X, where V = R d x √− g and V is a sufficiently large time-dependent three-volume. Following the above averagingprocedure, for the electromagnetic field to act as a sourcefor the FRW model we need to impose that (the Latinindexes run over three-space); E i = 0 , B i = 0 , E i B j = 0 ,E i E j = − E g ij , B i B j = − B g ij . Additionally it has to be assumed that the EM fields,being random fields, have coherent lengths that aremuch shorter than the cosmological horizon scales. Un-der these assumptions the energy-momentum tensor ofthe EM field can be written in the form [3]: T em µν =( ρ em + p em ) u µ u ν − p em g µν , where ρ em = − L + GL G − L F E , and p em = L − GL G − B − E ) L F /
3, with E and B being the averaged electric and magnetic fields,respectively. III. EXTENDED BORN-INFELD THEORY
Following [2], in order to simplify the analysis, here weshall consider a flat FRW universe filled with a ”magnetic fluid”, i. e., the electric component E will be assumedvanishing or, in other words, only the average of the mag-netic part B is different from zero. This choice leads tothe so called magnetic universe which is characterized bythe following barotropic parameters: ρ b = − L, p b = L − L F F, F = 2 B . (3)Here we shall focus in the study of a cosmologicalmodel proposed in [2], which is based upon the followingmodified EM Born-Infeld Lagrangian density [6]: L = − γ W / , W := 1 + F γ − α F , (4)where γ and α are free parameters. As seen from Eq.(3), in a magnetic universe the energy density associatedwith the EM field ρ b = − L = γ s F γ − α F . (5)From this expression it follows that ρ b vanishes at F = F ± = 1 ± p α γ α γ , (6)i. e., the field F is bounded both from above and frombelow. The value of the field F = F − is unphysicalsince F − < F = 2 B ], only non negative values0 ≤ F ≤ F ≡ F +0 , are to be considered (see FIG. 1). The energy density ofthe magnetic field is a maximum at F = F ∗ = 14 α γ ⇒ ρ b = ρ max b = p α γ α . (7)A distinctive feature of the model of Ref. [2] is that,even at vanishing magnetic field F = 0, the energy den-sity ρ b = γ is non null. So that, one wonders, where This situation turns out to be relevant in cosmology as long asthe averaged electric field E is screened by the charged primordialplasma, while the magnetic field lines are frozen [7]. Notice that in Ref. [2] the values F ± are incorrectly associatedwith extrema of the EM field. FIG. 1: Plot of the energy density of the magnetic field ρ b vs F for arbitrarily chosen α = 0 . γ = 1 (left-hand andcenter figures), and of ρ b vs scale factor a (right-hand figure). The solid curve is for the theory of Ref. [2], which is givenby equations (4), and (5), while the dashed curve is for the model (8). For the theory of [2] only that part of the curve { ρ b = ρ b ( F ) : 0 ≤ F ≤ F = 6 . } is physically meaningful. At F = 2 . ρ b is a maximum in both cases[ ρ max b = 1 .
27 for (5) and 0 .
27 for (8)]. does the energy density of the EM field come from? Theanswer is that the present theory does not have the stan-dard (linear) Maxwell electrodynamics limit L = − F/ γ → ∞ ( α = 0). Actually, at this limit one gets fromEq. (4), L ≈ − γ − F/
4. But, as discussed in Sec. VII of[2], this feature is at the heart of the present model. Thequestion then is, what is the meaning of an EM theorywhich at weak field does not have the Maxwell limit?Let us to modify the Born-Infeld Lagrangian (1) justby adding the term quadratic in F within the square root,but without removing the Lagrangian term ∝ γ (see Eq.(31) of Ref. [2] with W given by Eq. (4)): L = − γ (cid:16) W / − (cid:17) ,ρ b = γ s F γ − α F − ! . (8)Notice that this ρ b vanishes at vanishing field value F = 0( B = 0) as it should be to recover the correct weak fieldlinear behavior. Contrary to what is stated in [2], inthis case, the energy density of the magnetic field is nonnegative definite [ ρ b ≥
0] in the finite interval (see theFIG. 1) 0 ≤ F ≤ α γ , where the upper bound on F is obtained, precisely, by re-quiring the energy density ρ b to be non negative. Hence,the theory (8) meets the requirement which inspired the model of Ref. [2]: boundedness of the field F = 2 B andof the associated energy density.The energy density ρ b in (8) is a maximum at the samefield value F ∗ as ρ b in Eq. (5): F ∗ = 14 α γ ⇒ ρ max b = p α γ − αγ α . (9)Nevertheless, at the maximum, the energy density of themagnetic field is smaller in this case than in the theoryof [2] (compare with Eq. (7)).From this simple analysis it follows that, the main ar-gument stated in Ref. [2] against the model (8) relatedwith the non positivity of ρ b in (8), is not a valid one andthat the conclusion about the inevitability of the asymp-totic vacuum regime in a magnetic universe is wrong. IV. COSMOLOGICAL EVOLUTION
Assuming plain magnetic universe – no backgroundmatter fluid other than the EM field – the cosmologicalequations can be written in the following form:3 H = ρ b , H = − ( ρ b + p b ) = 43 L F F, ˙ ρ b + 3 H ( ρ b + p b ) = 0 , (10)where H = ˙ a/a is the Hubble parameter, and L F = − − α γ F √ W , (11)for both models (4) and (8). The continuity equation in(10) can also be written as˙ F + 4 HF = 0 ⇒ ˙ B = − HB. (12)These equations can be easily integrated B ( a ) = B a − , F ( a ) = 2 B a − , (13)where B is an integration constant.Worth noting that, since − H = ρ b + p b = − L F F, (14)whenever L F F = 0, the magnetic fluid behaves as a cos-mological constant and drives the de Sitter evolution ofthe universe:˙ H = 0 ⇒ F = 0 || F = F ∗ = 14 α γ . (15)Hence, at both field values [ F = 0 and F = F ∗ ] H = H .For the model of [2] at F = 0, since according to (5) ρ b (0) = γ , one has H = ± γ/ √
3. Meanwhile, for themodel (8), at F = 0, since ρ b = 0, then H = 0. Inconsequence, while for the former theory at vanishingfield value one has de Sitter cosmological evolution, forthe latter model one has a static universe instead. Inboth cases [theory (4) and theory (8)], at F = F ∗ wherethe energy density is a maximum ρ max b , the universe is ina stage of de Sitter expansion a ( t ) ∝ e H t , H = p ρ max b / . V. THE BOUNCE
At the bounce, since the scale factor is a minimumwhile H changes sign (contraction turns into expansion),then H = 0. In general, the sufficient conditions for abounce are [8]:˙ a = 0 , ¨ a ≥ ⇒ H = 0 , ˙ H ≥ , (16)where the above quantities are evaluated at the bounce. For the theory of reference [2] – equations (4), (5) –the condition H = 0 can not be reconciled with a non-vanishing minimum of the energy density ρ b in respectto t – the cosmic time, as it has been incorrectly statedin [2] (see section VI.D of the mentioned reference and,in particular, the FIG. 4 where the time evolution of theenergy density is shown, and a non-vanishing minimumof ρ b is associated with the bounce). Actually, at a min-imum of ρ b in respect to t , ˙ ρ b = 0, which means that H ( ρ b + p b ) = 0 . This condition is fulfilled even if ρ b + p b = 0, since at thebounce H = 0. However, according to the Friedmannequation in (10), H = 0 ⇒ ρ b = 0. Therefor a non-vanishing ρ b = 0 at the bounce is not compatible withthe Friedmann constraint. This means that in the theoryof Ref. [2] the bounce, if any, can be attained only atvanishing ρ b , i. e., at the upper bound of the F -field F = F , where W vanishes (see equations (4), (6)). Notice, inpassing, that at this point ˙ H ∝ L F F ∝ W − / – see Eq.(11) – blows up, signaling a curvature singularity. Sinceat F ⇒ ρ b = 0, from ρ b + p b ∝ − L F F – see equation (10)– it follows that it is the parametric pressure p b whichblows up at F = F . This, in addition to the fact that at F the scale factor is a finite quantity, means that at thisfield value a sudden curvature singularity arises. Hencein the theory (4) exposed in [2] the bounce, if it exists atall, is to be associated with a curvature singularity, whichis not better than the initial cosmological singularity instandard general relativity.For the theory (8) the condition H = 0 for a bounce oc-curs at vanishing field F = 0 [ ˙ H = 0] where, consistentlywith the standard linear Maxwell limit, the associatedenergy density ρ b vanishes. Regrettably, since F ∝ a − ,a vanishing F is associated with an infinitely large valueof the scale factor, so that the bounce does not actu-ally occur. In this case the value F in Eq. (6) is neverattained. Recall that the upper bound on F is12 α γ < F = 1 + p α γ α γ , which means that ˙ H (also H ) is always a finite quantity,i. e., the resulting cosmological evolution is regular – nocurvature singularity at all – unlike for the theory of [2]. We have to be careful since a given quantity, say ρ b = ρ b ( F ( t )),can have extrema in respect to the variable F , which might notcoincide with extrema of the same quantity in respect to theimplicit variable t . FIG. 2: Plot of the speed of sound (squared) c s vs F for arbitrarily chosen α = 0 . γ = 1. The left-hand and centerfigures depict c s = c s ( F ) for the models (4) and (8) – these coincide in this case – while the right-hand figure shows c s ( F ) forthe mentioned models (solid curve) and for the model (17) (dashed curve). It is seen that for the models (4) and (8) the speedof sound is a negative quantity in the field interval 0 ≤ F < F ∗ = 2 .
5. Behind the point F = F ∗ – where there is a verticalasymptote and the speed of sound is undefined – c s for these models is a positive quantity. For the theory of Ref. [2] c s blowsup also at F = F = 6 .
53 which is associated with a sudden curvature singularity. Meanwhile, for the model (17) the speed ofsound squared is always positive (dashed curve) and blows up at F = 2 γ = 2. VI. DISCUSSION AND CONCLUSION
In the reference [2] a generalized Born-Infeld EM the-ory was proposed which, for a magnetic universe – ac-cording to the authors – inevitably leads to a cosmo-logical history which interpolates between asymptotic deSitter states. The main motivation to modify the Born-Infeld Lagrangian (1) in [2] was to have a regular mag-netic universe driven by a bounded field F = 2 B . Un-fortunately, the resulting theory (4) does not have thestandard weak field (linear) Maxwell limit. To worsenthings, as shown in the former section, a sudden curva-ture singularity – not better than the big bang singularityof general relativity – arises at the upper bound of thefield F = F , which is the only field value which couldbe associated with a bounce (if any) in this theory.Perhaps a simpler modification of the Born-Infeld the-ory can be given by the following Lagrangian L = γ s − F γ − ! . (17)This theory has the correct linear Maxwell limit at weakfield (formal limit γ → ∞ ): L = − F/ Notice the subtle sign differences with the original Born-InfeldLagrangian (1). netic universe of interest in this comment, allows for anupper bound on the field F = 2 γ [ B = γ ], which was themotivation of [2]. In the present case integration of Eq.(12) yields to expressing the cosmic time as a monotonicfunction of the field F = 2 B : t ( F ) − t = p / γ arc tanh s − p − F/ γ + s − p − F/ γ , (18)where t is an integration constant and, to get F = F ( t )one has to invert (18). The starting point of the cosmicevolution is associated with the upper bound of the field F = 2 γ [ B = γ ], while at the future infinity t → + ∞ thefield vanishes F →
0. Regrettably, since for the model(17) p b = − ρ b + F p − F/ γ , the start of the cosmological expansion is associated witha sudden singularity: at F = 2 γ , H = γ/ √ p b → ∞ blows up (as we shall see below at thisfield value the speed of sound squared also blows up).Another quantity of cosmological importance is thespeed of sound squared which, for the cases of interestin this comment, can be written as c s := dp b dρ b = dp b /dFdρ b /dF = 13 + 4 L F F L F F. (19)If consider small perturbations of the background en-ergy density ρ b ( t, x ) = ρ b ( t ) + δρ b ( t, x ), the conservationof energy-momentum T µν ; ν = 0, leads to the wave equa-tion [9]: δ ¨ ρ = c s ∇ δρ , which solution for positive c s > δρ b = δρ b0 exp( − iωt + i k · x ) while, for negative c s < δρ b = δρ b0 exp( ωt + i k · x ). In the latter case theenergy density perturbation uncontrollably grows result-ing in an instability of the cosmological model. For themodels (4) and (8) the speed of sound is the same c s = 13 γ (cid:20) γ − α γ (1 − α γ F ) W (cid:21) , where W is defined in (4), while for the model (17) c s = 13 γ (cid:18) γ + 11 − F/ γ (cid:19) . A plot of c s vs F is shown in the FIG. 2 for the theo-ries (4), (8) – solid curves – and (17) – dashed curve – forarbitrarily chosen α = 0 . γ = 1. It is seen that fortheories (4) and (8) the speed of sound squared is nega-tive in the field interval 0 ≤ F ≤ F ∗ and that, at F = F ∗ where the magnetic background energy density is a max-imum, c s is undefined (there is a vertical asymptote).This signals a fundamental instability against small per-turbations of the background energy density in a mag-netic universe described by both theories in this field in-terval. For the theory of [2] the speed of sound is alsoundefined at F = F where, as shown in the former sec-tion, a sudden curvature singularity arises. For the most simple theory (17) the speed of sound squared c s is al-ways a non negative quantity and blows up at F = 2 γ ,where also a sudden curvature singularity occurs.Our main conclusion is that none of the modificationsof the Born-Infeld theory where 1 + F/ γ is replacedby 1 + F/ γ − α F – theories given by (4) and (8) –can be an adequate cosmological model since, thanks tothe fact that the speed of sound squared can be negative,the resulting theory is unstable against small perturba-tions of the background energy density. The most simplesuch modification of the Born-Infeld theory which (i) hasthe correct weak field Maxwell limit, (ii) for a magneticuniverse the field F is bounded, and (iii) is free of cur-vature singularities, is the one given by the Lagrangian(8), which was incorrectly ruled out in [2] but which,nevertheless, is to be ruled out due to the mentioned in-stability. In this theory the fate of the cosmic evolutionis a static universe and, besides, there is no place for thebounce. The alternative theory (17) is free of the in-stability against small perturbations of the backgroundenergy density although a sudden curvature singularityis also unavoidable in this theory.Summarizing: the main conclusion of [2] about theinevitability of the asymptotic vacuum regime in a mag-netic universe is wrong. Besides, the theory of Ref. [2];(i) does not have the Maxwell limit, (ii) in a cosmologicalsetting a sudden curvature singularity is inevitable in it,and (iii) it is unstable against small perturbations of thebackground magnetic energy density, so that it should beruled out.The authors thank SNI of Mexico for support. Thework of R G-S was partly supported by SIP20131811,COFAA-IPN, and EDI-IPN grants. I Q thanks ”Pro-grama PRO-SNI, Universidad de Guadalajara” for sup-port under grant No 146912. [1] M. Born, Nature (1933) 282; Proc. R. Soc. A (1934) 410; M. Born, L. Infeld, Proc. R. Soc. A (1934)425.[2] M. Novello, J. M. Salim, A. N. Araujo, Phys. Rev. D (2012) 023528.[3] V. A. De Lorenci, R. Klippert, M. Novello, J. S. Salim,Phys. Rev. D (2002) 063501.[4] M. Gasperini, G. Marozzi, G. Veneziano, JCAP (2010) 009 [arXiv:0912.3244]; M. Gasperini, G. Marozzi,G. Veneziano, JCAP (2009) 011 [arXiv:0901.1303]. [5] R. Tolman, P. Ehrenfest, Phys. Rev. (1930) 1791.[6] M. Novello, M. Makler, L. S. Werneck, C. A. Romero,Phys. Rev. D (2005) 043515 [astro-ph/0501643].[7] D. Lemoine, M. Lemoine, Phys. Rev. D (1995) 1955.[8] M. Novello, S. E. Perez Bergliaffa, Phys. Rept. (2008)127 [arXiv:0802.1634].[9] P. J. E. Peebles, B. Ratra, Rev. Mod. Phys.75