# Massive stealth scalar fields from deformation method

## Abstract

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.

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. In the references [1, 2, 3, 4, 5, 6, 7, 8, 9] 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 non-trivial effect of matter on the gravitational background is also suggested by the observation on galaxies, which have led to the conjecture on the existence of “dark matter”.

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 Sect. 1 we introduce the notation and define what a stealth field is. In Sect. 2 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 Sect. 3 we construct some examples and characterize their solutions, and in Sect. 4 we present some conclusions.

## 1 Stealth scalar field definition

*generalized Einstein tensor*, \( H_{\mu \nu }[g]\), and the Hilbert energy-momentum tensor, \({\varXi }_{\mu \nu }[g,\phi ]\), up to constant coefficients.

*F*[

*f*] of a function

*f*(

*x*) valued in the point

*x*, we shall declare the dependence on this point as

*F*[

*f*](

*x*), whenever is necessary, as in e.g. \({\delta }/{\delta g^{\mu \nu }(x)}\), \({\delta }/{\delta \phi (x)}\), \(H_{\mu \nu }[g](x)\), \({\varXi }_{\mu \nu }[g](x)\).

## 2 \(\theta \)-deformation of scalar field theories

The idea to be developed here can be spelled as follows. Given an action principle with a saddle point with vanishing expectation value of a scalar field (trivial vacua) we can construct a new (deformed-) one, with saddle points at the trivial and at a massive configurations. As a consequence of the construction, the deformed action principle, which is an extension of the original one, will have also vanishing energy momentum tensor. Surprisingly enough, the details of the original theory are not important, so that we can construct full class of models sharing these features.

*f*(

*x*) be any function of \(x\;\in \;\mathbb {R}\) with a saddle point at \(x=0\),

*y*(

*x*) another function which possess, for definiteness, two zeros at \(x=0\) and \(x=1\),

*f*and

*y*the properties,

*dF*/

*dx*at \(x=0,1\),

*y*takes zero-value and from (8) the derivative of

*f*vanishes when the argument is zero. Hence from an arbitrary function

*f*(

*x*) with saddle point at \(x=0\) (8) we can construct another arbitrary function

*F*(

*x*) with saddle points at the kernel of the map \(y:\mathbb {R}\rightarrow \mathbb {R}\), in this example \(x=0,1\).

We can promote the latter statements to functionals, *i.e.* to action principles and the solutions of the equations of motion. *f* is the analogous of an action principle for a scalar field represented by the variable *x*, of which \(x=0\) represents its trivial vacua and \(x=1\) will represent a non-trivial (massive) configuration. *F* is the analogous of a new action constructed from a transformed scalar field, *y*(*x*), *i.e.* which is a map from *x* to *y* in the class of differentiable functions. Now if \(x=0\) is a saddle point of the action *f*, then the new action principle *F* must have saddle points at the trivial vacuum (\(x=0\)) and at the non-trivial configuration (represented by \(x=1\)). Note that *F*(*x*) takes the same values at \(x=0\) and \(x=1\) since \(F(1)=F(0)=f(0)\). Hence both configurations, trivial and non-trivial, are at the same foot.

*g*, such that now \(y=y(x,g)\) is two-parametric and, consequently \(F:=f(y(x,g))\) is also two-parametric. We can easily prove that,

*g*is. Continuing our analogy,

*g*will represent a metric tensor, and the implication of (11) is that the energy momentum tensor provided by the new action principle will vanish for either configuration.

We shall apply now the same logic in the language of functional calculus in order to construct new classes of (generic) action principles which have saddle points at massive configurations and which are also stealth.

*y*(

*x*,

*g*)) which will be used to map a generic action principle of a scalar field into a new action principle with saddle points at \(\phi =0,\phi _m\). The latter will posses vanishing energy momentum tensor. Though we shall prove this in full generality, for the convenience of the reader, we shall advance here one example.

### 2.1 Quick example

*M*is the mass-coupling constant. The equation of motion for the scalar field is,

### 2.2 General models with stealth configurations

In this subsection we shall prove that the results of the example (2.1) can be extended to a wide class of action principles, *i.e.* which will possess solutions of the equations of motion with non-trivial mass and with vanishing energy momentum tensor.

*-deformed*action principle,

*i.e.*for solutions of (13).

Morally speaking, this means that the massive stealth field \(\phi =\phi _{m}\) (13) and the vacuum \(\phi =0\) are at the same foot.

### 2.3 Variation of the action with respect to \(\phi \)

*F*[

*f*] and

*G*[

*f*], and its composition

*F*[

*G*[

*f*]], the generalized chain rule reads,

*y*, we obtain the variation of the deformed action,

### 2.4 Variation of the action with respect to \(g^{\mu \nu }\)

### 2.5 Stealth theorem

## 3 Examples

Here we shall present two more examples. The reader may analyze these cases in independence of the general results of Sect. 2.

### 3.1 Deformation of the massless field action

*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 (41). 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.

### 3.2 Deformation of the massive field action

*M*the matter action principle is given by,

*i.e.*respectively

*M*. As we observe, the massive solutions of the undeformed theory remain after the deformation.

Hence, the space of solutions for the scalar field in the undeformed theory consist of the trivial solution \(\phi =0\) and the massive solution \(\phi _M\). In the deformed theory this space is extended by the massive stealth fields \(\phi _m\). The energy-momentum tensor (54) vanishes for the trivial solution and for the massive stealth fields but for the solutions \(\phi =\phi _M\) the deformed theory yields a rescaled energy-momentum tensor. Note that this can be interpreted also as a rescaling of the Newton coupling constant by a factor, \(G_\texttt {N}\rightarrow \lambda ^2 G_\texttt {N}\), 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.

## 4 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 transformation (12) 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 was found in \(2+1\) dimensions [17], where it was shown that the correspondent (deformed) gauge theory contains self-dual fields in \(2+1\) dimensions [18] which are stealth. It is worth mentioning that further analogies between stealth matter sources and self-duality can be found in [19].

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. [20]).

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 (54). 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. [21, 22] 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 (16) and (43)). The consistency of these theories in the quantum level requires the analysis of specific models in deeper detail, using similar techniques than in [15, 23, 24] and reference therein. This problem should be studied elsewhere.

## Notes

### Acknowledgements

We thank Eloy Ayón-Beato and Julio Oliva for valuable discussions. A.S. thanks the FAPEMAT postdoc grant Edital 039/2016. We thank the support of Facultad de Ingeniería y Tecnología USS for its “Núcleo de Física Teórica” initiative which facilitated the production of this article.

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