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The correspondence between a scalar field and an effective perfect fluid ∗ Valerio Faraoni Physics Department and STAR Research Cluster, Bishop’s University, 2600 College Street, Sherbrooke, Québec, Canada J1M 1Z7 It is widely acknowledged that, for formal purposes, a minimally coupled scalar field is equivalent to an effective perfect fluid with equation of state determined by the scalar potential. This correspondence is not complete because the Lagrangian densities L1 =P and L2 =−ρ, which are equivalentfor a perfect fluid,are not equivalentfor a minimally coupled scalar field. The exchange L1←→L2 amountsto exchanging a “canonical” scalar field with a phantom scalar field. PACSnumbers: 04.40.-b,04.20.Fy,04.20.-q 2 INTRODUCTION using the other side ([4, 5], see also [6]). 1 The non-equivalence between these two Lagrangian 0 densitiesforaperfectfluidcouplingexplicitlytotheRicci 2 Scalar fields have been present in the literature on curvature of spacetime (which has been suggested as a n gravity and cosmology for many decades and are par- possible alternative to dark matter in galaxies [7]) has a J ticularly important in the contexts of inflation in the been recently discussed in [8], and the equivalence be- early universe and of quintessence in the late universe. tween a k-essence scalar field and a barotropic perfect 6 It is widely recognized that a minimally coupled scalar fluid was discussed in [5]. ] fieldinGeneralRelativitycanberepresentedasaperfect We adopt the notations of Ref. [9]; in particular, the c fluid (see, e.g., Ref. [1] for a detailed discussion). From signature of the spacetime metric is +++. q − - theconceptualpointofview,itisclearthatascalarfield r g and a perfect fluid are very different physical systems; a MIMICKING A PERFECT FLUID WITH A [ fluid is obtained by averaging microscopic quantities as- MINIMALLY COUPLED SCALAR FIELD sociatedwithits constituent particles,andthe fluidlaws 1 can be obtained via a kinetic theory using microscopic v 8 models of the fluid particles and of their interactions. A Let us mimick a perfect fluid using a minimally cou- 4 scalar field, instead, is more fundamental and does not pled scalar field φ in a curved spacetime self-interacting 4 derive from an average; it is not made of point-like par- through a potential V(φ). We assume standard General 1 ticles and does not need an average to be defined—it is Relativity and the issue is to determine whether both . 1 already a continuum. As a consequence, the scalar field Lagrangian densities 1 = P and 2 = ρ correctly de- L L − 0 stress-energytensor(givenbyeq.(2)below)involvesfirst scribe the scalar field effective fluid. It is well known 2 orderderivativesofthescalarfield,whiletheperfectfluid that these Lagrangiandensities are equivalent for a per- 1 : stress-energy tensor Tab =(P +ρ)uaub+Pgab does not fect fluid [2]. v involve derivatives explicitly but can be described using A minimally coupled scalar field φ obeys the Klein- Xi only the energy density ρ, pressure P, and four-velocity Gordon equation ar fineoltdauna.idHenetnificec,attihoen,flubiudtdoenslcyriaptcioonnvoefnaiensctaclaorrrfiesepldonis- (cid:3)φ− ddVφ =0 (1) dence for formalpurposes. However,while regardingthe whichisobtained(when cφdoesnotvanishidentically) scalarfield/perfectfluiddualityasaformalone,onestill ∇ from the covariant conservation equation bT = 0 for needs to be careful because this correspondence is not ∇ ab the scalar field energy-momentum tensor perfect even for purely formal purposes. In this short note we discuss a property of perfect fluids which is not T [φ]= φ φ 1g cφ φ Vg . (2) ab a b ab c ab satisfied by the effective fluid associated with a scalar ∇ ∇ − 2 ∇ ∇ − field, namely the possibility of describing a perfect fluid ThetensorT [φ]assumestheformofaneffectiveperfect ab with the two equivalent Lagrangian densities L1 = P fluid stress-energy tensor Tab = (P +ρ)uaub +Pgab if and 2 = ρ. These two Lagrangian densities, which cφ is a timelike vector field [1]. The fluid four-velocity L − ∇ are equivalent for a perfect fluid [2, 3], are not equiva- is lent for a scalar field “fluid”, as shown in the following φ a section. This property leads to a caveat against regard- ua = ∇ (3) cφ φ ing the fluid/scalar field duality as a perfect one, which |∇ ∇c | is relevant in the light of the recent interest in this cor- assuming cφ cφ = 0,pwith uaua = sign( cφ cφ). ∇ ∇ 6 ∇ ∇ respondence motivated by the possibility of computing The energy density and pressure relative to a comov- inflationary perturbations on one side of the duality by ing observer with four-velocity ua are given by ρ = 2 T [φ]uaub and P = T [φ]hab/3, respectively, where yields ab ab h g +u u is the Riemannian 3-metric in the 3- ab ab a b ≡ dV space orthogonal to uc (this 3+1 decomposition makes d4x√ g ( cδφ) φ+ δφ =0. (14) c − − ∇ ∇ dφ sense when cφ is timelike). One easily obtains Z (cid:20) (cid:21) ∇ Usingtheidentity( cδφ) φ= c(δφ φ) δφ(cid:3)φand 1 ∇ ∇c ∇ ∇c − ρ = cφ φ V sign( cφ φ) , (4) discarding the boundary term, one obtains c c 2∇ ∇ − ∇ ∇ (cid:18) (cid:19) dV 1 1 (cid:3)φ+ =0. (15) P = 1+ sign( cφ φ) cφ φ dφ c c 3 − 2 ∇ ∇ ∇ ∇ (cid:26)(cid:20) (cid:21) Eq. (15) is not the Klein-Gordon equation because of [4+sign( cφ φ)]V(φ) . (5) the incorrect sign of the potential derivative term. The c − ∇ ∇ } difference between eq. (15) and the Klein-Gordon equa- If we restrict ourselves to the case in which cφ is time- tion (1) disappears, of course, if V = constant. In this ∇ like, cφ cφ<0, we have case the potential term gabV(φ) in the stress-energy ∇ ∇ − tensorT [φ]canbeattributedentirelytoacosmological ab ρ = 1 cφ φ+V(φ), (6) constant, i.e., to gravity instead of matter. If this con- c −2∇ ∇ stant vanishes, V 0, then eqs. (6) and (7) yield ρ=P ≡ and eq. (15) coincides with the Klein-Gordon equation. 1 P = cφ φ V(φ), (7) However, there is still something wrong: the scalar field c −2∇ ∇ − sourcing gravity and appearing in the total action and R S =S +S = d4x√ g +S[φ] (16) total gravity matter − 2κ (P +ρ)uaub+Pgab (8) Z (where κ 8πG) will give the incorrect field equations ≡ 1 = φ φ g cφ φ Vg (9) 1 a b ab c ab ∇ ∇ − 2 ∇ ∇ − R g R= κT [φ] (17) ab ab ab − 2 − T [φ] (10) instead of the Einstein equations ab ≡ 1 in addition to ucuc = −1. The last equation shows Rab− 2gabR=κTab[φ]. (18) not only that a minimally coupled scalar field can be given a perfect fluid description, but also that any per- When V 0, the correct Lagrangian density would be ≡ fectfluidwithbarotropicequationofstateP =P(ρ)can = P = ρ instead of = ρ. (Then = ρ 2 3 2 L L − L ≡ −L be mimicked by a scalar field with appropriate poten- can only describe a perfect fluid with stiff equation of tial V(φ). Roughlyspeaking,prescribingthe equationof state P =ρ.) state P = P(ρ) corresponds to assigning a suitable po- The conclusion is that, for a perfect fluid, = P is 1 L tential,butthecorrespondencebetweenequationofstate theLagrangiandensityreproducingthecorrectequations and scalar field potential is not one-to-one [10]. of motion, while = ρ (or =ρ) provides incorrect 2 3 L − L The Klein-Gordon Lagrangian density for the scalar equations of motion. field is the well known [9] For completeness, we conclude this section by going back to eqs. (4) and (5) and considering the case of a = 1 cφ φ V(φ), (11) spacelike cφ, although this choice obviously does not KG c ∇ L −2∇ ∇ − correspond to any physical fluid. If cφ φ > 0 it is c ∇ ∇ u = φ/√ cφ φ and, using eqs. (3)-(7), whichcoincideswith 1 =P andthe variationoftheac- a ∇a ∇ ∇c L tion SKG = d4x√ gP with respect to φ reproduces ucu = 1, (19) − c the Klein-Gordon equation (1), as is also well known R [9]. Let us try to adopt instead the other candidate La- 1 grangian density ρ = cφ cφ V(φ), (20) 2∇ ∇ − 1 L2 =−ρ= 2∇cφ∇cφ−V(φ); (12) P = 1 cφ cφ 5V(φ). (21) −6∇ ∇ − 3 then, the variational principle Taking again =P yields the incorrect field equation 1 L dV δS δ d4x√ g =0 (13) (cid:3)φ 5 =0, (22) 2 ≡ − L2 − dφ Z 3 while taking = = ρ yields again one obtains the incorrect energy-momentum tensor 2 L ±L ∓ dV (cid:3)φ+ =0. (23) Sab = ∂aφ∂bφ+ 1ηab∂cφ∂ φ Vηab , (28) dφ (2) − 2 c − Hence, the Lagrangians , all give incorrect field L1 ±L2 which does not reproduce Tab[φ] and the Klein-Gordon equations for a scalar field with spacelike gradient. equation unless V 0. Finally, we consider a null cφ. In this case, the pur- ≡ ∇ To reiterate the argument, one can consider an- pose of considering a scalar field representationof a per- other situation in which the Noether symmetry ap- fectfluidwouldbethemodellingoftheonlyperfectfluid proach is applicable: that of a spatially homo- withnullfour-velocitythatmakessensephysically,i.e.,a geneous and isotropic Friedmann-Lemaˆıtre-Robertson- nulldustwithassociatedstress-energytensorT =u u ab a b withu uc =0,describingadistributionofcoherentmass- Walker universe with line element c less φ-waves [11, 12]. In this case, it must be V = 0 and and ,whicharebothproportionalto cφ φ, ds2 = dt2+a2(t) dx2+dy2+dz2 (29) 1 2 c − L L ∇ ∇ vanish identically and the usual Lagrangiandensity (11) (cid:0) (cid:1) cannot describe the null dust. In fact, a minimally cou- incomovingcoordinates(for simplicity, we considerhere pled scalar field cannot work as a model of conformally only the spatially flat metric). The minimally coupled invariantfluidsuchasanulldust. Tomodelsuchafluid, Klein-Gordonfieldinthisspacetimedependsonlyonthe the physics of the scalar field would need to be confor- comoving time, φ = φ(t), and its energy density and mally invariant, which can only be achieved by coupling pressure are conformallythescalartotheRiccicurvatureofspacetime R via the introduction of the term Rφ2/12 in the ac- ρ(t) = 1φ˙2+V(φ), (30) tion S [13–15]. Moreover,the sca−lar φ must be either 2 KG free (V 0), or have a quartic self-coupling V = λφ4 1 [14]. In≡general, mimicking a null fluid or an imperfect P(t) = φ˙2 V(φ). (31) 2 − fluidwithascalarfieldrequiresthatthelatterbecoupled non-minimally to the curvature [1, 16]. By adopting the Lagrangiandensity THE NOETHER APPROACH 1 a,φ,φ˙ =P , (32) L (cid:16) (cid:17) In flat spacetime there is an independent line of ap- a suitable point-like Lagrangianis proach to the Lagrangian description of a perfect fluid. TheNoethertheoremappliedtothetranslationalKilling φ˙2 L = √ g = a3 =a3 V =a3P . (33) fields of the Poincaré group for a field φ described by 1 L1 − L1 2 − ! the Lagrangian density [φ,∂ φ,η ] (where η is the a ab ab L Minkowski metric) yields the (independent) canonical The Euler-Lagrangeequation energy-momentum tensor [9] d ∂L ∂L Sab = ∂L ∂bφ ηab . (24) dt ∂φ˙1 − ∂φ1 =0 (34) ∂(∂aφ) − L (cid:18) (cid:19) Thistensorisconservedandcoincideswiththecanonical then yields the correct Klein-Gordon equation T [φ] of eq. (2) up to a constant [9]. By adopting ab dV 1 φ¨+3Hφ˙ + =0. (35) ηab∂ φ∂ φ V(φ) (25) dφ 1 a b L ≡−2 − one obtains By contrast, the second point-like Lagrangian 1 S(a1b) =−∂aφ∂bφ+ 2ηab∂cφ∂cφ+Vηab =−Tab[φ]. L = √ g =a3 = a3 φ˙2 +V = a3ρ (36) 2 2 2 (26) L − L − 2 ! − As is well known, the conservation equation bT [φ] = ab ∇ 0 reproduces the Klein-Gordon equation. By contrast, yields the incorrect equation of motion using dV 1 φ¨+3Hφ˙ =0. (37) 2 ηab∂aφ∂bφ+V(φ) (27) − dφ L ≡−2 4 CONCLUSIONS The duality between a minimally coupled scalar field ∗ [email protected] andaperfectfluidiswidelyacknowledged,butitisnota [1] M.S. Madsen, Class. Quantum Grav. 5, 627 (1988);see complete one. Lagrangiandensities which are equivalent also M. Madsen, Astrophys. Sp. Sci. 113, 205 (1985). [2] R.L. Seliger and G.B. Whitham, Proc. R. Soc. (Lon- for a perfect fluid cease to be equivalent for an effective don) A305, 1 (1968); B. Schutz 1970 Phys. Rev. D, 2 fluid constructed out of a scalar field. Specifically, the 2762(1970);J.D.Brown,Class.QuantumGrav.10,1579 change =P to = ρ, which does not have conse- L1 L2 − (1993). quences for the equations of motion of a real fluid, does [3] S.W. Hawking and G.F.R. Ellis, The Large Scale Struc- change the Klein-Gordonequation of motion of a scalar. ture of Space-Time (Cambridge University Press, Cam- Could this change point to the possibility that a scalar bridge, 1973). field exist in nature which satisfies eq. (15) instead of [4] L.Boubekeur,P.Creminelli, J. Norena,andF.Vernizzi, J. Cosmol. Astropart. Phys. 0808, 028 (2008). eq. (1)? In the cosmological literature, such a field (sat- [5] A. Diez-Tejedor and A. Feinstein, Int. J. Mod. Phys. D isfying eq. (37) instead of (35)) is known as a phantom 14, 1561 (2005); F. Arroja and M. Sasaki, Phys. Rev. D field [17–19]. This hypothetical scalar field would cause 81, 107301 (2010). superaccelerationoftheuniverse(i.e.,Hubbleparameter [6] S. Unnikrishnan and L. Sriramkumar, Phys. Rev. D 81, H a˙/aincreasingwithtime, H˙ >0)andleadstoaBig 103511 (2010). ≡ Rip singularity at a finite time in the future [18, 19]. [7] O.Bertolami,C.G.Böhmer,T.Harko,andF.S.N.Lobo, Phys. Rev. D 75, 104016 (2007); O. Bertolami and J. Phantom fields have been the subject of much at- Pa´ramos, Phys. Rev. D 77, 084018 (2008); O. Berto- tention in cosmology, due to repeated reports from as- lami, F.S.N. Lobo, and J. Pa´ramos, Phys. Rev. D 78, tronomers that the effective equation of state parameter 064036 (2008); O. Bertolami, J. Pa´ramos, T. Harko, w P/ρ of the cosmic fluid lies in a range which does and F.S.N. Lobo, in The Problems of Modern Cosmol- ≡ not exclude, or evenfavours, w < 1: this is a signature ogy, P.M. Lavrov ed. (Tomsk State Pedagogical Univer- − ofaphantomscalarfieldwhichcausessuperacceleration. sity, 2009) [arXiv:0811.2876]; V. Faraoni, Phys. Rev. D 76, 127501 (2007); O.Bertolami and J. Pa´ramos, Class. There is consensus that phantom scalar fields are unsta- Quantum Grav. 25, 245017 (2008); T.P. Sotiriou, Phys. ble, classically and, even more, quantum mechanically Lett. B 664, 225 (2008); O. Bertolami and M.C. Se- [18, 20], therefore a phantom field is extremely unlikely. queira, Phys. Rev. D 79, 104010 (2009); O. Bertolami However, a phantom can appear in low-energy effective and J. Pa´ramos, arXiv:0906.4757; O. Bertolami and M. actions which are eventually modified by higher order Carvalho Sequeira, arXiv:0910.3876; T.P. Sotiriou and corrections. In the cosmological literature, a duality be- V.Faraoni,Class.Quantum Grav.25,205002(2008);D. tween a canonical scalar field and a phantom one is is PuetzfeldandYuriN.Obukhov,Phys.Rev.D78,121501 (2008). obtained by changing the sign of the kinetic energy den- [8] V. Faraoni, Phys. Rev. D 80, 124040 (2009). sitytermineqs.(30)and(31). Ourdiscussionputsanew [9] R.M. Wald, General Relativity (Chicago University twistonthisduality,inthataphantomscalarcanbeob- Press, Chicago, 1984). tained froma canonicalone by the exchange 1 2. [10] V. Faraoni, Am. J. Phys. 69, 372 (2001). L ←→L Itispresentlyunknownwhetherthisexchangehasamore [11] H.Stephani,D.Kramer,M.MacCallum,C.Hoenselaers, fundamental meaning. and E. Hertl, Exact Solutions of Einstein’s Field Equa- tions, 2nd edition (Cambridge University Press, Cam- We have mentioned that a null fluid could be con- bridge, 2003). structed using a scalar field with potential V(φ) = λφ4 [12] J.BicakandK.V.Kuchar,Phys.Rev.D56,4878(1997). conformally coupled to the curvature and with lighlike [13] N.A. Chernikov and E.A. Tagirov, Ann. Inst. H. gradient. Ingeneral,anon-minimallycoupledscalarfield Poincaré A 9, 109 (1968). corresponds to an imperfect fluid whose stress-energy [14] C.G. Callan Jr., S. Coleman and R. Jackiw, Ann. Phys. tensor contains terms which can be interpreted as heat (NY) 59, 42 (1970). [15] F.G. Friedlander, The Wave Equation on a Curved currents and anisotropic stresses [1]. Since a Lagrangian Spacetime (Cambridge University Press, Cambridge, description of imperfect fluids has not yet been devel- 1975). oped, we cannot comment on this aspect of the non- [16] V. Faraoni, Int. J. Theor. Phys. 40, 2259 (2001). minimally coupled scalarfield/imperfect fluid duality. It [17] R.R. Caldwell, Phys. Lett. B 545, 23 (2002). is plausible, however, that the obstruction to a perfect [18] R.R. Caldwell, M. Kamionkowski, and N.N. Weinberg, duality found for perfect fluids will carry over to (ef- Phys. Rev. Lett. 91, 071301 (2003). fective) imperfect fluids constructed with non-minimally [19] S.M. Carroll, M. Hoffman, and M. Trodden, Phys. Rev. D 68, 023509 (2003). coupled scalar fields, if they are found to admit a La- [20] S. Nojiri and S.D. Odintsov, Phys. Lett. B 562, 147 grangian description. (2003);V.Faraoni,Phys.Rev.D69,123520(2004);J.M. The author thanks a referee for constructive remarks Cline,S.Jeon,andG.D.Moore,Phys.Rev.D70,043543 and the Natural Sciences and Engineering Research (2004); S.D.H. Hsu, A. Jenkins, and M.B. Wise, Phys. Council of Canada for financial support. Lett. B 597, 270 (2004); R.V. Buniy and S.D.H. Hsu, 5 Phys. Lett. B 632, 543 (2006).