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Note on Scalars, Perfect Fluids, Constrained Field Theories, and All That

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Note on Scalars, Perfect Fluids, Constrained Field Theories, and All That Physics Letters B 727 (2013) 27–30 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Note on scalars, perfect fluids, constrained field theories, and all that Alberto Diez-Tejedor Departamento de Física, División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, León 37150, Mexico article info abstract Article history: The relation of a scalar field with a perfect fluid has generated some debate along the last few years. Received 6 May 2013 In this Letter we argue that shift-invariant scalar fields can describe accurately the potential flow of an Received in revised form 14 October 2013 isentropic perfect fluid, but, in general, the identification is possible only for a finite period of time. Accepted 14 October 2013 After that period in the evolution the dynamics of the scalar field and the perfect fluid branch off. Available online 18 October 2013 The Lagrangian density for the velocity-potential can be read directly from the expression relating the Editor: M. Trodden pressure with the Taub charge and the entropy per particle in the fluid, whereas the other quantities of interest can be obtained from the thermodynamic relations. © 2013 Elsevier B.V. All rights reserved. 1. Introduction For the particular case of a potential flow, i.e. a fluid with no vorticity, Ωμν =∇μVν −∇ν Vμ = 0, we can always write Vμ = Recently many papers have addressed the question: can we ∂μΦ,withΦ a velocity-potential and Vμ = vuμ the Taub cur- identify a scalar field with the potential flow of a perfect fluid? For rent [4].Herev = h/s is a measure for the enthalpy per unit of a representative sample of these works please see Refs. [1].Here entropy, and h = ρ + p is the enthalpy density. For “standard” flu- we will concentrate on classical, relativistic field theories, but of ids (see for instance Ref. [5]) the Taub charge is positive-definite, course one could extend our results and conclusions to the Newto- v > 0, with v = 0 only in vacuum, i.e. s = ni = 0foralli. nian regime by taking the appropriate limit. For a brief discussion Leaving fluids aside, a local, minimally coupled to gravity, on the quantum aspects of these models see Section 5 at the end Lorentz invariant field theory with no more than two derivatives of this Letter. acting on a real field φ is described by the action A perfect fluid is one with no dissipation effects [2].Fora √ perfect fluid in general relativity [3] the energy–momentum ten- S = d4x −gM4L φ/M, X/M4 . (1) sor and the entropy flux can be written in the form Tμν = (ρ + p)uμuν + pgμν , and Sμ = suμ, respectively, with ρ, p and For our purposes φ is a Lorentz scalar measured in units of energy, μ s the energy density, pressure, and entropy density measured by and X =−∂μφ∂ φ/2 is the kinetic term. We are taking units with an observer at rest with respect to the fluid. The velocity uμ is c = h¯ = 1, and M is an energy scale. A scalar field described by an a four-vector pointing to the future, u0 > 0, and normalized to action of the form (1) is usually dubbed k-essence [6]. In this lan- μ uμu =−1, with the spacetime metric gμν taking the mostly- guage the Lagrangian density of a canonical scalar field takes the plus-signature. Spacetime indexes are raised and lowered using the form L = X − M4 V (φ/M),withV a potential term. In order to in- ν spacetime metric, e.g. uμ = gμν u . Both quantities, the energy– clude the dynamics of the gravitational interaction we only have momentum tensor and the flux of entropy are covariantly con- to add a Einstein–Hilbert term to the expression above; however, μν μ served for a perfect fluid, ∇μT =∇μ S = 0. for the purposes of this Letter and with no loss of generality, we ∇ μ = will restrict our attention to fixed background spacetime configu- Additionally we can have other conserved currents, μNi 0, such as those associated to the particle or baryon numbers. Here rations. the letter i is a label for these currents. For a perfect fluid the Invariance under local Lorentz transformations defines energy μ μ and momentum. The energy–momentum tensor associated to the conserved currents are all parallel, and we can write N = ni u , i scalar field can be obtained varying the action in Eq. (1) with re- with ni a charge density. As usual, in order to close the system spect to the spacetime metric, we need a relation of the form ρ = ρ(s,ni); ultimately it should be provided by the micro-physics. The other quantities of interest T = L ∂ φ∂ φ − Lg . (2) can be read from the thermodynamic relations, such as the tem- μν μ ν μν = = perature, T (∂ρ/∂s)ni , the chemical potential, μi (∂ρ/∂ni )s,n j , From now on and in order to simplify the notation we will omit and so on. All these variables are related by the Euler equation, the scale M. Here the prime denotes the derivative with respect to ρ + p − Ts− μini = 0. the kinetic term. Using the relations 0370-2693/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2013.10.030 28 A. Diez-Tejedor / Physics Letters B 727 (2013) 27–30 ∂μφ uμ = √ , p = L, ρ = 2XL − L, (3) with c1 and c2 two arbitrary constants. We can use the family of 2X two-parametric solutions in Eq. (6) to construct the kinetic scalar as a function of the scale factor, we can identify the energy–momentum tensor of a scalar field with that of a perfect fluid, as long as the kinetic scalar is posi- 2 2 1 c2 c1 tive definite, X > 0. In addition, if Eq. (1) is invariant under shift- X = − . (7) 2 a6 a2 transformations, φ → φ + const., that is, if the Lagrangian density does not depend explicitly on the scalar field, L = L(X),wecan In order to identify the scalar field with the velocity-potential of also identify the Noether current a perfect fluid we need to satisfy X > 0. From Eq. (7) it is evi- dent that this condition is verified at early times, when a < acrit = √ Jμ = L ∂μφ (4) |c2/c1| (the particular value of this quantity depends on the initial conditions for the scalar field). However, for a acrit,the with the flux of entropy in a perfect fluid, inequality X > 0 does no longer holds. √ The moral is simple: not all the particular solutions to a shift- = L s 2X . (5) invariant scalar field satisfy the condition necessary to mimic a √ perfect fluid, X > 0. But even if they do at a certain initial time, From the identities in Eqs. (3) and (5) we can read v = 2X.The t , the evolution of the system can change this behavior. Imag- energy–momentum tensor (2) and the Noether current (4) are both 0 ine for instance a universe filled with a scalar field, like in the covariantly conserved. Fields with no potential term are known as inflationary model, and assume that this field is invariant under purely-kinetics, and have been considered for their possible role to shift-transformations. If there were some small perturbations to the dark matter and/or dark energy problems [7]. the homogeneous and isotropic background, we could not guaran- Associated to Eq. (1) there are no other conserved charges apart tee a perfect-fluid-solution, even if the identification of the scalar from the energy, momentum, and Noether charge. Then, if there field with a perfect fluid were possible in the early universe. But, were any other thermodynamic charges in the fluid, they should how is it possible that something that looks like a perfect fluid, be distributed on a trivial configuration, i.e. the fluid should be and evolves like a perfect fluid, reaches a state that does not look isentropic, s¯ = s/n = const.; this guarantees a barotropic relation i i like a perfect fluid? of the form p = p(ρ). If on the contrary extra thermodynamic charges do not exist, e.g. a gas of photons, we can simply iden- 3. The constraints tify the Taub charge with the temperature in the fluid, v = T ;see Euler equation. In order to improve our understanding of the previous section, According to the previous lines, as long as the kinetic term we find it appropriate to start from the action principle describing is positive definite, it seems possible to identify a shift-invariant a perfect fluid in general relativity. Following Schutz formalism [8] scalar field φ with the velocity-potential of an isentropic, per- (see also Refs. [9], and Ref. [10] for a Newtonian description), the fect, rotation-less fluid Φ. But, what happens if the kinetic scalar Lagrangian density of an isentropic, rotation-less perfect fluid with changes sign? Naturally the identifications in Eqs. (3) and (5),and equation of state ρ = ρ(s,ni ) is given by in particular that for the vector uμ, break down. If the dynamical evolution of the scalar field prevented a sign inversion, we could √ = 4 − − − μ + μ + 2 forget this concern. Something similar happens, for instance, when S d x g ρ(s,ni ) S ∂μΦ λ Sμ S s . (8) a perfect fluid is isentropic, or rotation-less, at a given instant of time, i.e. on a given Cauchy hypersurface: the dynamics main- Eq. (8) is a functional of the spacetime metric, gμν ,theentropy μ tains constant entropy per particle and no-vorticity along the fluid flux, S , the velocity-potential, Φ, the entropy density, s, and a ¯ = evolution.
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