Massive stealth scalar fields from field redefinition method

We propose an uni-parametric deformation method of action principles of scalar fields coupled to gravity which generates new models with massive stealth field configurations, i.e. with vanishing energy-momentum tensor. The method applies to a wide class of models and we provide three examples. In particular we observe that in the case of the standard massive scalar action principle, the respective deformed action contains the stealth configurations and it preserves the massive ones of the undeformed model. We also observe that, in this latter example, the effect of the energy-momentum tensor of the massive (non-stealth) field can be amplified or damped by the deformation parameter, alternatively the mass of the stealth field.


Introduction
It is generally believed that matter curves the space, a consequence of the interpretation of the equations of gravity-matter systems, which tells that the energy-momentum tensor of matter fields feedback the curvature of the geometry equations. However, it seems mathematically possible the existence of non-trivial matter-field configurations with vanishing energy-momentum tensor, such that the first statement will not be always truth. Indeed, there are examples of systems where this happens [1][2][3][4][5][6][7][8][9]. In these references the authors impose separately the vanishing Einstein tensor equation, the vanishing energy-momentum tensor equation (obtained from some scalar theory), and the matter-field equations of motion. If these three sets of equations are satisfied the scalar field would exist but it will not be detected by the background geometry, since it will not deform it. This is what a "stealth field" means. It is worth to mention that stealth solutions appeared also in the references [10][11][12][13], and that an analogous result, for a Dirac fermion field was found in [14]. Though this behavior of matter may seem strange, the mathematical possibility on the existence of stealth fields encourage their study.
The purpose of this paper is to present a method to construct models with massive stealth fields. This is, the method takes a given (original ) action-principle and produces a related (deformed ) one which contains massive stealth configurations. Just one restriction on the original action principle is needed, that the trivial vacuum (with vanishing VEV) must exist. We shall provide three non-trivial examples of application of our method. Advancing one interesting result, we shall observe that stealth field mass produces a re-scaling of the the energy-momentum tensor of a (non-stealth) massive configuration. Hence the stealth field, though undetectable by the background space-time geometry, can amplify (or reduce) the gravitational effect of regular matter fields, which may be interesting from a cosmological point of view. This paper is organized as follow. In section 2 we introduce the notation and define what a stealth field is. In section 3 we define the deformation of the action principle and obtain the respective equations of motion, as deformations of the original equations of motion. We prove that the deformed theories contain massive fields with mass inversely proportional to the deformation parameter. This is done on full generality, without reference to any particular model. In section 4 we construct some examples and characterize their solutions, and in section 4 we present some conclusions.

Stealth scalar field definition
In generic space-time background metric g µν , µ " 0, ..., D´1, with diagonal components signature p´,`,`, ...q, consider the gravity-matter system, Srg, φs " S G rgs`S M rg, φs, where S G rgs is the gravitational action principle for a given theory of gravity (e.g. general relativity or modified models) and S M rg, φs is the action principle for a scalar matter field φ coupled to the g µν .
The variation with respect to the (inverse) metric tensor g µν yields, δSrg, φs δg µν " ?´g pH µν rgs´Ξ µν rg, φsq, which represents a differential operator acting on the field φ. In what follows, for functionals F rf s of a function f pxq valued in the point x, we shall declare the dependence on this point as F rf spxq, whenever is necessary, as in e.g. δ{δg µν pxq, δ{δφpxq, H µν rgspxq, Ξ µν rgspxq.
The equations of motion of the theory (2.1) are given by, H µν rgs´Ξ µν rg, φs " 0, Υrg, φs " 0, (2.5) respectively for the variation of the metric and the matter field, and for a non-degenerated metric tensor ?´g ‰ 0.
By definition, a stealth scalar field is a non trivial field which satisfies the equations of motion and its energy-momentum tensor vanishes, respectively, Υrg, φs " 0, Ξ µν rg, φs " 0 . (2.6) As consequence of (2.5), the generalized Einstein tensor must also vanish, so that in presence of a stealth φ the metric tensor must satisfy identical equations of motion than in the vacuum φ " 0. The stealth scalar field does not feedback the metric background.

θ-deformation of scalar field theories
Consider now the following field redefinition: where θ is a real-valued parameter and is the Laplace-Beltrami operator acting upon φ.
We define the matter field θ-deformed action principle, consisting on the replacement of φ by φ θ . The deformation of the gravity plus matter field is defined similarly, S θ rg, φs :" Srg, φ θ rg, φss " S G rgs`S θ M rg, φs . (3.10) Note that the pure gravity sector S G rgs is not affected by the deformation, and that the deformed action will be of higher order in derivatives with respect to the original action.
Let us assume that the minimal degree of the matter field action as a functional of φ is ą 1, such that the respective equations of motion admit the vacuum solution φ " 0. Then the value of the action, S M rg, 0s " 0, must be an extremal. This implies that S M rg, φ θ " 0s " 0 also vanishes, which occurs for a massive scalar field φ m of mass m " θ´1, since vanishes. Since φ θ rg, φs is of degree 1 and homogeneous in φ, the degree of S θ M rg, φs " S M rg, φ θ s in φ must be the same than the degree of the original action principle S M rg, φs. It must be truth then that S θ M rg, 0s " 0 in the vacuum φ " 0. Morally speaking, this means that the massive stealth field φ " φ m (3.12) and the vacuum φ " 0 are at the same foot, since the deformed action principle vanishes for both of them. We should expect therefore that both, the vacuum φ " 0 and the massive mode φ " φ m , are solutions of the equations of motion provided by the deformed action principle (cf. (2.5)). Extending these arguments to the energy-momentum tensor (3.15), we may ask whether it vanishes for φ " φ m or not. We shall verify this is indeed truth. Let us prove this in full generality.
The equations of motion of deformed theory are given by: where H µν rgs was defined in (2.3) and Note that in (3.14) the generalized Einstein tensor H µν rgs remains undeformed, since the field redefinition (3.8) does not affect the metric tensor.

Variation of the action with respect to φ
The variation with respect to φ should be carried out taking into account that the action depends on φ implicitly by means of φ θ rg, φs, so that we should consider the chain rule for functional derivation (see e.g. appendix A in [15]). Indeed, given two functionals F rf s and Grf s, and its composition F rGrf ss, the generalized chain rule reads, Applying this in the computation of (3.16), in the point y, we obtain the variation of the deformed action, where we have used the definition (3.9) in r.h.s. The latter expression is equivalent to, and it is understood that and therefore equivalent to the original operator (2.4) valued in the field redefinition φ θ rg, φs.

Variation of the action with respect to g µν
Let L M rg, φs the Lagrangian density, such that, In correspondence with the substitution (3.9), the θ-deformed action principle is given by Considering again the functional chain rule (3.17), and that the deformed action functional (3.25) depends on g µν explicitly and also implicitly, by means of the functional φ θ " φ θ rg, φs, we obtain that the variation of the deformed action principle (3.9) is equivalent to, Here, the first term on the r.h.s. corresponds to the variation with respect to the explicit dependence of the action, while the second encodes its implicit dependency. Alternatively, (3.26) can be written also as, δS M rg, φ θ rg, φss δg µν pyq "´?´g Ξ θ µν rg, φspyq`ż d D zΥ θ rg, φspzq δφ θ pzq δg µν pyq , (3.27) where we have defined Ξ θ µν rg, φs :" Ξ µν rg, φ θ s , and considered (2.3), and (3.20)-(3.21). From (3.27) and (3.15), the energy-momentum tensor provided by the deformed matter action is given by, To complete the calculation, we need to replace the functional in (3.28), which after integration by parts yields,
Again, since Υrg, 0s " 0, the second and third term on the r.h.s. vanish, while the first term Ξ µν rg, 0s " 0, as we argued that the vacuum solution φ " 0 of the original theory must exist and it has, naturally, a vanishing energy-momentum tensor. Hence, the equations of motion (3.32) are reduced to H µν rgs " 0 , which correspond to the equation of motion of the original gravity theory S G rgs. Hence, φ m is stealth, since in its presence, the gravity background will satisfy identical equations than in the vacuum. Note that if other (regular) matter fields are incorporated in the original action, we would obtain similar results after the deformation. This means that the respective deformed theory, when valued in the stealth scalar solution, will produce an effective theory equivalent to the original one in the trivial vacuum φ " 0, i.e. with standard matter sourcing the space-time curvature.

Examples
In this section we shall present three examples which illustrate the results of our deformation method.

Deformation of a square-potential
The simplest action principle one can imagine is the "mass term" where M is the coupling constant proportional to the "mass". The equation of motion of this action is, Υrg, φs :" δS M rg, φs δφ "´M 2 ?´g φ " 0 . Let us construct the deformed action (3.10), Following the usual procedure to compute the equation of motion for φ yields, r Υrg, φs :" δS θ M rg, φs δφ "´M 2 ?´g`φ´2 θ 2 lφ`θ 4 l 2 φ˘" 0 , (4.37) which turns out to be equivalent to, r Υrg, φs " p1´θ 2 lq`´M 2 ?´g p1´θ 2 lqφ˘" p1´θ 2 lqΥ θ rg, φs " 0 , Now let us demonstrate that the gravitational energy-momentum tensor vanishes. A direct calculation of variational derivative of the matter deformed action (4.36) with respect to the inverse metric tensor yields, which evidently vanishes for φ " φ m . Hence the action (4.36) describes a massive stealth field.
We can conclude therefore that when we consider the gravity-matter action principle, S G rgsS M rg, φs, for any reasonable S G rgs, and with the matter action given by (4.34), the only solution for φ is the trivial one φ " 0 while the space of solutions for the gravity sector remain invariant. Now, when we θ-deform the matter action, as given in (4.36), the original space of solutions (φ " 0) is extended by the stealth configurations φ " φ m (up to double degeneracy), while the gravitational space of solutions remains invariant.

Deformation of the massless field action
Consider now the action of a massless scalar field, According to the deformation recipe, the deformed action is given now by, Now let us obtain the gravitational energy-momentum tensor directly from the action (4.42), we find that the solutions are given by the trivial one, φ " 0, the massless one, lφ " 0, and the doubledegeneracy massive stealth φ " φ m . Let us evaluate the massless solutions in the energy-momentum tensor (4.44). We obtain, i.e., the deformed energy-momentum takes the same value than the undeformed energy-momentum tensor. Therefore, the space of solutions of the gravitational sector remains invariant with respect to the original theory (4.40). This means that, in terms of gravity effects, after deformation the stealth field yields the same results than the trivial matter vacuum, while for the massless configurations the geometry is sourced by the same energy-momentum tensor than in the original theory.

Deformation of the massive field action
In the case of the scalar field φ, with mass M the matter action principle is given by, which corresponds to the sum of the massless field action (4.40) with the square potential action (4.34), and whose equation of motion for φ is proportional to the Klein-Gordon equation, Υrg, φs " ?´g`l´M 2˘φ " 0 . (4.48) Following the θ-deformation recipe of the action (4.47), which produces a linear combination of (4.36) and (4.42), we obtain the equation of motion, r Υrg, φs " ?´g´1´θ 2 l¯2`l´M 2˘φ " 0 , (4.49) which has the general form (3.23), as expected. As for the energy momentum-tensor, the θ-deformed action yields again a linear combination of (4.39) and (4.44), i.e. respectively where φ θ :" p1´θ 2 lqφ as it has been already defined in (3.8). so that their energy-momentum tensor (4.50) is given by, which means that the energy-momentum tensor of the original gravity-matter system is rescaled by factor λ 2 in the deformed theory.
Hence, the space of solutions for the scalar field in the undeformed theory consist of the trivial solution φ " 0 and the massive solution φ M . In the deformed theory this space is extended by the massive stealth fields φ m . The energy-momentum tensor (4.53) vanishes for the trivial solution and for the massive stealth fields but for the solutions φ " φ M the deformed theory yields a rescaled energymomentum tensor. Note that this can be interpreted also as a rescaling of the Newton coupling constant by a factor, G N Ñ λ 2 G N , in the standard nomenclature. Hence, the mass of the stealth field (equivalently the deformation parameter) can be used to smooth or amplify the effects of the massive field of mass M on the gravitational background.

Overview and remarks
In this paper we show that for a wide class of scalar field action principles in curved space we can construct a deformation which admits massive stealth configurations. The theory here presented consists of a method for the construction of action principles which insures the existence of massive stealth fields, independently of the solutions of the gravity field equations. In the proof, we just need to assume that the Klein-Gordon equation admits non-trivial solutions.
As for new developments, it would be interesting to extend our construction to gauge theories. For example, one possible way is to couple the scalar fields to (non-)abelian gauge fields, in a curved background, and use the correspondent generalization of the field redefinition to obtain stealth solutions with non-trivial gauge charges. Also, in a similar spirit but in absence of scalar fields, we can consider redefinitions of gauge fields to produce deformation of gauge theories with non-trivial propagating degrees of freedom, in spite of which they will not feedback the gravitational background. Indeed, non-linear electrodynamics can produce stealth configurations [8]. In this direction, an example of a field redefinition approach was found in 2`1 dimensions [16], where it was shown that the correspondent (deformed) gauge theory contains self-dual fields in 2`1 dimensions [17] which are stealth. It is worth mentioning that further analogies between stealth matter sources and self-duality can be found in [18].
As for the stability of stealth fields, in our framework, since they do not depend on any specific background, they must be robust under background perturbations, so we expect they must be stable. This should be corroborated by means of the appropriated methods (see e.g. [19]).
Let us comment that though stealth fields do not curve space, they may give rise to new cosmological effects, for example by means of the energy-momentum tensor rescaling of regular matter fields, which depends on the stealth field mass parameter observed in (4.53). It would be interesting to check whether this may help with the cosmological constant problem or the amplification of the gravitational effects of the visible matter in galaxies. See e.g. [20,21] for other possible cosmological implications of stealth fields.
Finally, as the reader must have noticed, the theories obtained by means of our method are in general of higher order (see examples (4.36) and (4.42)). The consistency of these theories in the quantum level requires the analysis of specific models in deeper detail, using similar techniques than in [22][23][24] and reference therein. This problem should be studied elsewhere.