The correspondence between a scalar field and an effective perfect fluid

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by  Valerio Faraoni

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, Qu´ebec, Canada J1M 1Z7
It is widely acknowledged that, for formal purposes, a minimally coupled scalar field is equiva-
lent 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 Poincar´e 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.B¨ohmer,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 Poincar´e 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).