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Extended Klein-Gordon Action, Gravity and Non-Relativistic Fluid

We consider a scalar field action for which the Lagrangian density is a power of the massless Klein-Gordon Lagrangian. The coupling of gravity to this matter action is considered. In this case, we show the existence of nontrivial scalar field configurations with vanishing energy-momentum tensor on any static, spherically symmetric vacuum solutions of the Einstein equations. These configurations in spite of being coupled to gravity do not affect the curvature of spacetime. The properties of this particular matter action are also analyzed. For a particular value of the exponent, the extended Klein-Gordon action is shown to exhibit a conformal invariance without requiring the introduction of a nonminimal coupling. We also establish a correspondence between this action and a non-relativistic isentropic fluid in one fewer dimension. This fluid can be identified with the (generalized) Chaplygin gas for a particular value of the power. It is also shown that the non-relativistic fluid admits, apart from the Galileo symmetry, an additional symmetry whose action is a rescaling of the time.


I. INTRODUCTION
The fundamental tenet of General Relativity is the manifestation of the curvature of spacetime produced by the presence of matter. This phenomena is encoded through the Einstein equations that relate the Einstein tensor (with or without a cosmological constant) to the energy-momentum tensor of the matter, Since the energy-momentum tensor depends explicitly on the metric, both sides of the equations must be solved simultaneously.
In the first part of this paper, we address the following question: for a fixed geometry solving the vacuum Einstein equations, is it possible to find a matter source coupled to this spacetime without affecting it. Concretely, this problem consists of examining a particular solution of the Einstein equations (1) for which both sides of the equations vanish, i. e. G µν + Λg µν = 0 = κ T µν . ( In three dimensions, such gravitationally undetectable solutions have been obtained in the context of scalar fields nonminimally coupled to gravity with a negative cosmological constant [1]. Recently, the same problem has been considered in higher dimensions but with a flat geometry [2]. In our case, we show that for any static, spherically symmetric vacuum solutions of the Einstein equations (eventually with a cosmological constant), a particular matter source action can be coupled to this geometry yielding to non-trivial solutions of the equations (2). This matter source is described in (D, 1) dimensions by an * Electronic address: hassaine-at-inst-mat.utalca.cl extended Klein-Gordon action, where θ is the dynamical field and α is a real parameter whose range will be fixed later. For α = 1, this action reduces to the well-known massless Klein-Gordon action.
The derivation of solutions of the equations (2) is shown to be equivalent of finding a null vector field k µ (for the vacuum metric g µν ) which derives from the scalar field θ. For a static, spherically symmetric spacetime, the derivation of vector fields satisfying these conditions is always possible. In this case, the potential θ can be expressed as a general smooth function that depends only on a null coordinate.
The existence of these gravitationally undetectable solutions is essentially due to the particular form of the matter action (3). For this reason, we investigate some properties of this matter action in the second part of this paper. In particular, the symmetries of the extended Klein-Gordon action (3) are analyzed. It is shown that for the particular value α = (D + 1)/2, this action exhibits a conformal invariance for which the conformal weight of the scalar field θ is zero. In contrast with the massless Klein-Gordon action, this conformal symmetry is achieved without requiring the introduction of a nonminimal coupling. Finally, we establish a correspondence between the relativistic action (3) and a nonrelativistic isentropic fluid defined in one fewer dimension. The clue of this correspondence lies in the fact that the non-relativistic space-time can be viewed as the quotient of a higher-dimensional Lorentz manifold by the integral curves of a covariantly, lightlike vector field [3]. For α = 1/2, this non-relativistic fluid is identified with the Chaplygin gas while for 0 < α < 1/2, the model corresponds to the generalized Chaplygin gas. Note that the Chaplygin cosmology provides an interesting setup to explain that the expansion of the universe is accelerating [4], [5] and [6]. The Chaplygin gas has also raised a growing attention because of its connection with the Nambu-Goto action [7], its rich symmetrical structure [8] and its supersymmetric extension [9]. For a general review on this topics see [10] and works on supersymmetric fluid models are reported in [11]. The features of the Chaplygin gas can also be understood in a Kaluza-Klein type framework [12] as well as in its relativistic formulation [13]. The symmetries of the non-relativistic fluid are also investigated and it is shown the existence, apart from the Galileo symmetry, of an additional symmetry whose action is a time rescaling.
The paper is organized as follows. We first derive solutions of the equations (2) for the energy-momentum tensor associated to the extended Klein-Gordon action (3). In the remaining sections, we discuss some general features of the matter action. In particular, we explain the origin of the conformal invariance of the extended Klein-Gordon action for the particular value α = (D + 1)/2. The last part of the paper is devoted to the correspondence between the relativistic matter action and a nonrelativistic isentropic and polytropic fluid in one fewer dimension. The symmetries of this non-relativistic fluid are also analyzed.

II. STEALTH SOLUTIONS
We consider the action of (D + 1)−dimensional gravity coupled to a scalar field θ with dynamics described by the extended Klein-Gordon action (3), where R is the scalar curvature and κ is the gravitational constant. The field equations obtained by varying the metric and the scalar field read respectively where the energy-momentum tensor is given by We look for a particular solution of the Einstein equations (5a) for which both sides of the equations vanish (2). We first show that the equation T µν = 0 implies some restrictions on the scalar field θ and on the real parameter α. Indeed, independently of the background metric, the vanishing of the energy-momentum tensor is possible only if the scalar field satisfies ∇ σ θ∇ σ θ = 0, otherwise the metric would be proportional to and, hence its determinant would vanish. Consequently, the condition T µν = 0 can be consistently achieved only for a scalar field satisfying ∇ σ θ∇ σ θ = 0 and for α > 1 because of the form of the energy-momentum tensor (6). It is interesting to note that under these conditions, the extended Klein-Gordon equation (5b) is automatically satisfied.
In sum, the system described by the particular Einstein equations (2) together with the extended wave equation (5b) is equivalent of finding a vacuum metric and a null vector field that derives from θ, We now show that a static and spherically symmetric spacetime geometry always admits such vector field (7). Indeed, let us consider such geometry which in addition is supposed to solve the vacuum Einstein equations. The line element is given by where dΣ 2 D−1 is the line element of the (D − 1)− dimensional sphere. It is simple to see that the vector field defined by is null and derived from the potential Hence, any smooth function of this potential (10) is a solution on the vacuum spacetime metric of the constraint This result is not surprising and can be explained as follows. The null coordinates u and v associated to the spacetime (8) can be defined as and, in these coordinates, the metric (8) takes the following form Consequently, any smooth function F of one of the null coordinate, i. e. F (u) or F (v), will provide a nontrivial solution to the equation (12). In the original coordinates (8), this solution becomes precisely (11). Thus, we have shown that the curvature of vacuum spacetime (8) may not react to the presence of matter source described by the action (3) provided the scalar field is given by the expression (11). The existence of such configurations is a direct consequence of the particular form of the matter action. In what follows, we analyze some properties of this action.

III. PROPERTIES OF THE EXTENDED KLEIN-GORDON ACTION
We first show that the matter action (3) can be derived from the action of a massive complex scalar field with Lagrangian given by where V denotes a potential that depends only on the module of the scalar field. Decomposing the scalar field as where m represents the mass of the complex scalar field, the Lagrangian density (15) becomes As shown below, this expression reduces to the extended Klein-Gordon Lagrangian (3) provided the first term in (17) is neglected or eliminated and also for a particular election of the potential. One possibility to eliminate the first term is to consider an effective potential that depends also on the derivatives of the scalar field, In this case the starting Lagrangian (15) with this potential reduces to One can also reach to this expression (18) by neglecting the first term of (17) by considering the scale of the inhomogeneities as corresponding to the spacetime variations of f on scales greater than m −1 , i. e. ∂ µ f << mf . This has been considered in the context of cosmology to show that the generalized Chaplygin gas can be described by a generalized Born-Infeld Lagrangian [5]. The field equations resulting from the variation of the Lagrangian (18) read where we have set the mass to unity for simplicity. We consider a self-interacting potential V given by where λ is a constant. In this case, the action (18) becomes and the function f can be eliminated in favor of the phase θ through the equation of motion (19a), Finally, the substitution of this last expression into the Lagrangian (21) yields the extended Klein-Gordon action (3).

A. Conformal symmetry
In (D + 1) dimensions, the extended Klein-Gordon action (3) possesses the conformal invariance for a value of the parameter α given by Indeed, it is simple to see that the trace of the energymomentum tensor (6) vanishes for this value, The implementation of this symmetry on the scalar field θ is with a zero conformal weight, Various comments can be made with respect to this conformal symmetry. Firstly, in contrast with the standard Klein-Gordon action this conformal symmetry is achieved without the introduction of the nonminimal coupling. Moreover, it is interesting to note that the scalar field action (3) remains unchanged against the transformations (23), i. e.
Finally, the origin of this conformal invariance can be explained as follows. It is well-known that in (D + 1) dimension with D > 1, the following generalized Klein-Gordon Lagrangian density is conformally invariant. Here R represents the scalar curvature of the metric and the last term in (24) is the unique potential that preserves the conformal invariance. For convenience, we rewrite this action in terms of the module and the phase, Φ = f e iθ , and the implementation of the conformal symmetry on the dynamical fields is given by A simple but tedious computation shows that the term proportional to Rf 2 is precisely introduced to cancel the variation of the term ∂ µ f ∂ µ f under the conformal transformation (26). Consequently, if one drops this kinetic term (as we did in the previous derivation), the following Lagrangian also exhibits the conformal invariance. This expression corresponds precisely to the Lagrangian (21) for the conformal value of the parameter α given by (22).

B. Non-relativistic fluid
We now establish a correspondence between the relativistic action (3) defined in (D, 1) dimensions and a non-relativistic fluid in (D − 1, 1) dimensions through a Kaluza-Klein type framework. The clue of this correspondence lies in the fact that the quotient of a Lorentz (D, 1)−manifold by the integral curves of a covariantly constant and lightlike vector field ξ is a (D − 1, 1)−manifold which carries the geometric structure of non-relativistic space-time [3].
We first illustrate this framework with a simple example. On the Lorentz manifold we consider a coordinate system given by (t, x, z) where (t, x) are the coordinates on the non-relativistic space-time and z represents the additional coordinate. Let Φ be a complex scalar field satisfying the wave equation in (D, 1) dimensions together with an equivariance condition This system of equations (28) has been shown to be strictly equivalent to the free Schrödinger equation in (D − 1, 1) dimensions on a general Newton-Cartan spacetime [3]. In our case, we consider the action defined in (21) which has been shown to be equivalent to the extended Klein-Gordon action (3). The field equations associated to (21) read where for convenience we have substituted f 2 by ρ. In order to fix the set up of the Kaluza-Klein type framework, we consider the Minkowski metric written in the light-cone coordinates for which the lightlike vector field ξ can be chosen to be ξ µ ∂ µ = ∂ z . Since ρ is the module and θ the phase of the complex scalar field (16), the analogue of the equivariance condition (28) is given by It is easy to see that on the flat background (30) and for fields ρ and θ satisfying the conditions (31), the relativistic field equations (29) project onto the following (D − 1, 1)−dimensional non-relativistic equations These equations turn out to be the non-relativistic equations of an isentropic, irrotational and polytropic fluid. Indeed, identifying Θ as the potential velocity, i. e. v = ∇Θ and ρ as the density, the first equation (32a) is a continuity relation while the gradient of the second equation (32b) yields an Euler equation of an isentropic fluid with pressure given by For a potential strength λ > 0 and for α = 1/2 (resp. for 0 < α < 1/2), the relation (33) represents the state equation of the Chaplygin gas (resp. the generalized Chaplygin gas). Finally, we remark that the field equations (32) can be derived from the following action principle In what follows, we study the symmetries of the nonrelativistic fluid model whose dynamics is described by the equations (32).

Non-relativistic symmetries
We analyze the dynamical symmetries of the nonrelativistic isentropic fluid described by the equations (32). This model being non-relativistic possesses the appropriate symmetry, namely the Galileo symmetry. The action of this symmetry on the coordinates is given by where R ∈ SO(D − 1), ǫ, δ and β are the parameters associated to the rotations, the time translations, the space translations and the Galileo boosts respectively. The implementation of the Galileo symmetry on the dynamical fields reads ρ →ρ(t, x) = ρ(T, X), The application of the Noether theorem yields to the following constants of motion which corresponds to the energy ǫ, the momentum δ, the rotations R and the Galileo boosts β respectively. The equations are also invariant under a shift of the velocity potential by constant,Θ → Θ + constant, and the associated conserved charge is given by The corresponding Lie algebra generated by H, P , M ij , G and N is the Galileo algebra with a central extension given by (36) and corresponds to the Galileo 2−cocycle.
• For a generic value of the parameter α, there exists an additional symmetry which does not belong to the Galileo group. This symmetry acts on the coordinates by rescaling only the time while its action on the dynamical fields is given by ρ →ρ(t, x) = e 2(α−1)ω ρ(T, X), or infinitesimally The associated conserved quantity reads This expression is clearly not defined for α = −1/2 in spite of the fact that this value is not singular at the level of the transformations (37) and (38). In fact, a careful application of the Noether procedure shows that for α = −1/2 the associated conserved quantity does not involve the energy density and instead is given by In the case of the Chaplygin gas, α = 1/2, the last piece of the expression (40) vanishes and the Noether charge corresponds to the one derived in [8]. Interestingly, for α = 1/2, the two conserved charges (40) where H represents the density energy (35a). For α = 1/2, this variation (42) reduces to a surface term and, hence a direct application of the Noether theorem yields to the conserved charge (40). For α = 1/2, the reason for which the transformation (37) still acts as a symmetry is due to the fact that the original action can also be written as a surface term. Indeed, using the equations of motion (32), the action (34) can be expressed on-shell as As a consequence, for α = 1/2, the variation ∆S of the action (42) under the time dilatation transformation (37) is also a surface term. This proves that these transformations act as a symmetry for the action and also the reason for which the conserved quantities (40) and (41) involve the space coordinate x. For a generic value of the parameter α, the group structure of the symmetries is given by the semidirect sum of the Galileo group with central extension G with the generator associated to this extra symmetry (40), There exist two particular values of the parameter α for which the symmetry group can be larger than the one discussed before (44).
• For the value α = 1/2, which corresponds to the Chaplygin gas, it has been shown that, apart from the time dilation (37), there exists an extra symmetry whose action is field dependent, i. e. and where |J| is the Jacobian of the transformation, J = det (∂X µ /∂x σ ), [8]. The Noether conserved quantity is given by and the group structure generated by the quantities (35), (40) and (47) is the Poincaré group in one higher dimension, namely in (D, 1) dimensions. The existence of this symmetry is due to the fact that for α = 1/2, the action expressed in terms of Θ can be written as a square root, and this later can be seen to descend from a Nambu-Goto action in one higher dimension in the light cone parametrization [7]. This explains the arising of the Poincaré symmetry only for α = 1/2 since for the other values of α, the action is not a square root and hence it is not longer parametrization invariant.
• In (D − 1, 1) dimensions and for a value of α given by (D + 1)/2, the polytropic fluid exhibits a Schrödinger symmetry for which the group structure is the semidirect sum of the static Galileo group with SL(2, IR), (see [14], [15] and for an extension to discontinuous flows see [16]). The arising of this symmetry is a consequence of the conformal symmetry of the relativistic model in one higher dimension.

IV. DISCUSSION
Here, we have considered the Einstein equations with a matter source scalar field. The dynamics of the scalar field is described by an extended Klein-Gordon action that depends on a real parameter α. We have restricted ourselves to a particular class of solutions for which both sides of the Einstein equations vanish. In this case, we have shown that for any static, spherically symmetric vacuum solutions of the Einstein equations, a nontrivial scalar field with zero energy-momentum on-shell can be derived. This means that this matter source in spite of being coupled to gravity does not affect the curvature of the spacetime. An interesting work will consist to see whether scalar field with a non-zero energy momentum tensor can be coupled to black hole geometry. The existence of these undetectable solutions is essentially due to the form of the matter action. For this reason, some properties of this action have been analyzed. In particular, we have shown that the extended Klein-Gordon action possesses a conformal invariance for a particular value of the exponent α. This invariance does not require the introduction of the nonminimal coupling as it is the case for the standard Klein-Gordon action. We have also established a correspondence between this extended Klein-Gordon dynamics and a non-relativistic isentropic fluid in one fewer dimension. This gas can be identified with the (generalized) Chaplygin gas for specific values of the parameter α. The non-relativistic model corresponds to an irrotational, isentropic and polytropic fluid. This fluid admits, apart from the Galileo symmetry, an additional symmetry. The action of this extra symmetry on the coordinate consists of a rescaling of the time only. An interesting work will consist of finding solutions of this non-relativistic fluid and to make use of this extra symmetry to generate non trivial solutions.