Massive stealth scalar fields from deformation method II: charged case
Cristian Quinzacara, Paola Meza, Almeira Sampson, Mauricio Valenzuela
aa r X i v : . [ h e p - t h ] M a r Massive stealth scalar fields from deformation method II:charged case
Cristian Quinzacara ∗ , Paola Meza † , Almeira Sampson ‡ and Mauricio Valenzuela § Facultad de Ingenier´ıa y Tecnolog´ıaUniversidad San Sebasti´an, General Lagos 1163, Valdivia 5110693, Chile.
Abstract
Extending the results of our previous work we construct an uniparametric class of action princi-ples for complex scalar fields with the property that their energy momentum tensor and the electriccurrent vanish for certain massive configurations. These stealth fields do not curve the spacetimeand they do not source electromagnetic fields in spite their non-trivial degrees of freedom and U p q gauge invariance. We shall also show that the presence of these stealth fields can affect the strengthof the gravity-matter and radiation-matter coupling of other massive configurations. Indeed, theenergy momentum tensor of other massive (non-stealth) configurations and the electric current canbe rescaled (with a stealth-mass depending factor) with respect to the predictions of the standardcomplex scalar field model. Hence we argue that stealth fields could be detected indirectly by meansof their effects on standard configurations of matter. ∗ Electronic address: cristian.quinzacara at gmail.com † Electronic address: paola.meza.b at gmail.com. ‡ Electronic address: sampson.almeira at gmail.com. § Electronic address: valenzuela.u at gmail.com. ontents θ -deformation of complex scalar field theory 5 It has been shown by many authors [1–14] that scalar matter fields may not always curve the spacetime,in spite of their non-trivial degrees of freedom. Similar results were obtained also in the context ofmassive gauge theories in three dimensions [15]. This type of fields are now dubbed stealth fields . Thesecurious objects deserve some attention since, besides of being counter examples of the idea that matteralways curves the space, they could also play a role in cosmology.The existence of stealth field configurations suggest that the spacetime curvature may not reflect thepresence of matter as expected from standard field theories in curved space. Indeed, it is well knownthat the gravitational Hilbert energy momentum tensor ( i.e. the variation of their action principle withrespect to the spacetime metric) of the visible matter in galaxies fails to explain the observed spacetimecurvature, which have lead to cosmologists to propose the existence of dark matter , which have not yetbeen discovered. Alternatively, one could argue that perhaps, there may be alternative field theories inwhich the curvature induced by the visible matter is modified with respect to standard field theories.The goal of this paper is to construct a generic class of complex-scalar field theories containing stealthconfigurations. Our models suggest that the presence of these stealth configurations can also affect thestrength of the energy-momentum tensor of other (non-stealth) matter fields, modifying their effects ontheir gravity backgrounds with respect to standard field theories. This is illustrated in the case of amassive and charged Klein-Gordon field.In this work we extend the results of reference [16], for real scalar fields. This is, we consider ageneric action principle for a complex scalar field coupled to gravity and electromagnetism, referred asto original action , and according to a determined procedure of “deformation”, a new action principle isproduced and dubbed deformed action , which is an uni-parametric extension of the original action. Thelatter contains stealth field configurations which satisfy the minimally coupled Klein-Gordon equationin curved space. The stealth fields occur as result of the deformation procedure, independently ofthe details of the original action. The respective equations of motion, which are in general of higher2rder in derivatives, contain stealth and non-stealth solutions. In some cases, the solutions of theundeformed theory are preserved in the deformed model, so that only stealth solutions are added tothe spectrum of the theory. For the stealth configurations both, the energy momentum tensor and theelectric current vector vanish. Surprisingly enough, the Hilbert energy momentum tensor of the non-stealth configurations in the deformed theory consists of the original energy momentum tensor times aconstant factor depending on the deformation parameter.For example, for the minimally coupled Klein-Gordon complex field, of mass M , our procedure yieldsa 6th order in derivatives action principle, which contains as solution the stealth configurations of mass θ ´ (being θ the deformation parameter) and it preserves the solution of the original model of mass M .When the Hilbert energy momentum tensor is calculated, we observe that it vanishes for the stealthconfiguration, and the same happens for the electric current (so that the magnetic field equations are thesame than in the vacuum). However for the mass M configuration the energy momentum tensor and theelectric current are rescaled with respect to the original Klein-Gordon action by a factor 1 ´p M θ q . Hencestealth field are invisible to gravity and electromagnetism but they affect the non-stealth configurationswith respect to their original theories. This is, the effects of massive scalar fields on their gravitybackgrounds can be fine-tuned using the θ parameter.This paper is organized as follows. In section 2 we introduce our notation and define what a stealthfield is. In section 3 we define the deformation of the action principle and obtain the respective equationsof motion, as deformations of the original equations of motion. We prove that the deformed theoriescontain massive fields with mass inversely proportional to the deformation parameter. In section 4 weconstruct some examples and characterize their solutions, and in section 5 we present some conclusions.Though this paper is self-contained, it is meant to be considered the second part of reference [16], wherethe reader can find more details and references. In this section we shall follow closely reference [16]. This is, in D -dimensional spacetime dimensions,with background metric g µν with signature p´ , ` , ` , ... q , consider the action principle, S r g, A, φ, s φ s “ S G r g s ` S A r g, A s ` S M r g, A, φ, s φ s , (2.1)where S G r g s , S A r g, A s and S M r g, A, φ, s φ s represent respectively the gravitational sector, the U p q gaugefield sector, and the complex scalar matter field sector. Here s φ “ φ ˚ is the complex conjugated of φ .We do not specify the details of each sector.Under gauge transformation, with parameter α p x q , the fields must transform as, g “ exp p iα p x qq P U p q : φ Ñ gφ, s φ Ñ s φg ˚ , A µ Ñ A µ ` B µ g . (2.2)The variation of this action with respect to the (inverse) metric tensor g µν yields, δS r g, A, φ, s φ s δg µν “ ?´ g ´ H µν r g s ´ Θ µν r g, A s ´ Ξ µν r g, A, φ, s φ s ¯ , (2.3)where H µν r g s : “ ?´ g δS G r g s δg µν ν, (2.4)Θ µν r g, A s : “ ´ ?´ g δS A r g, A s δg µν , (2.5)Ξ µν r g, A, φ, s φ s : “ ´ ?´ g δS M r g, A, φ, s φ s δg µν , (2.6)3re respectively the (generalized) Einstein tensor, and the U p q gauge field and the scalar field Hilbertenergy-momentum tensors.From the variation of the action with respect to the scalar fields φ and s φ we define,Υ r g, A, φ, s φ s : “ δS M r g, A, φ, s φ s δφ , (2.7) s Υ r g, A, φ, s φ s : “ δS M r g, A, φ, s φ s δ s φ , (2.8)which are functionals of the fields φ and s φ containing differential operators. Note that, since φ ( s φ )transforms under U p q from the left (right) the operator (2.7) (operator (2.8)) transforms from theopposite right (left) side: g “ exp p iα p x qq P U p q : s Υ r g, A, φ, s φ s Ñ g s Υ r g, A, φ, s φ s , Υ r g, A, φ, s φ s Ñ Υ r g, A, φ, s φ s g ˚ . (2.9)The variation of the action with respect to the gauge field A µ yields, δS r g, A, φ, s φ s δA µ “ ?´ g ´ ´ E µ r g, A s ` J µ r g, A, φ, s φ s ¯ , (2.10)where we have defined the term proportional to the variation of the pure gauge sector as E µ r g, A s : “ ´ ?´ g δS A r g, A s δA µ (2.11)and the U p q gauge current as J µ r g, A, φ, s φ s : “ ?´ g δS M r g, A, φ, s φ s δA µ . (2.12)Using these definitions, the field equations of the theory (2.1) read, H µν r g s ´ Θ µν r g, A s ´ Ξ µν r g, A, φ, s φ s “ , (2.13)Υ r g, A, φ, s φ s “ , s Υ r g, A, φ, s φ s “ , (2.14) E µ r g, A s ´ J µ r g, A, φ, s φ s “ , (2.15)obtained respectively from the variation with respect to the metric ( ?´ g ‰ r g, A, φ, s φ s “ , s Υ r g, A, φ, s φ s “ , Ξ µν r g, A, φ, s φ s “ . (2.16)Hence, from (2.13), the generalized Einstein tensor is sourced only by the U p q gauge energy-momentumtensor, H µν r g s “ Θ µν r g, A s , (2.17)so that in presence of the stealth fields φ and s φ the metric tensor must satisfy identical equations ofmotion than in the vacuum φ “ s φ “ θ -deformation of complex scalar field theory Consider now the following map: φ θ r g, A, φ s “ p ´ θ D q φ , s φ θ r g, A, s φ s “ p ´ θ s D q s φ (3.18)where θ is a real-valued parameter and D µ : “ ∇ µ ` iqA µ , s D µ : “ ∇ µ ´ iqA µ , (3.19)are the covariant derivatives, acting upon the right-module and the left-module of the U p q represen-tation respectively, ∇ µ is the Levi-Civita derivative and q is the electromagnetic coupling constant.Hence, D φ “ g µν D µ D ν φ, s D s φ : “ g µν s D µ s D ν s φ . (3.20)Note that ¯ φ θ “ p φ θ q ˚ .Given the original action principle, say S r g, A, φ, s φ s , we obtain its θ -deformed action principle, say S θ r g, A, φ, s φ s , by means of the replacement, φ : Ñ φ θ , s φ Ñ s φ θ (3.21)in the action, such that, S r g, A, φ, s φ s Ñ S r g, A, φ θ , s φ θ s . (3.22)Then, the deformed matter field action principle corresponding to the original theory S M r g, A, φ, s φ s (2.1)consists of the formal replacement of φ by φ θ (3.18) and the correspondent conjugated, S θM r g, A, φ, s φ s : “ S M “ g, A, φ θ r g, A, φ s , s φ θ r g, A, s φ s ‰ , (3.23)which leads to the following decomposition, S θ r g, A, φ, s φ s : “ S “ g, A, φ θ r g, A, φ s , s φ θ r g, A, s φ s ‰ “ S G r g s ` S A r g, A s ` S θM r g, A, φ, s φ s , (3.24)where we notice that only the pure-matter sector S M r g s is affected by the deformation, because S G r g s and S A r g, A s are not functionals of φ and ¯ φ . We note also that the resulting action principle will be ingeneral of higher order, since the map (3.21) increases the order of the derivatives with respect to theoriginal action.Observe that the deformed action principle takes the same value S M “ g, A, φ θ r g, A, φ s , s φ θ r g, A, s φ s ‰ˇˇˇˇˇ φ “ “ S M “ g, A, φ θ r g, A, φ s , s φ θ r g, A, s φ s ‰ˇˇˇˇˇ φ θ “ , (3.25)for φ “ φ “ φ m satisfying s φ θ “ i.e. for the solution of the minimally coupledKlein-Gordon equation, φ θ r g, A, φ m s “ ´ θ p D ´ θ ´ q φ m “ , (3.26)(analogously for the complex conjugated s φ ). Assuming that the trivial vacuum φ “ cf. [16]) that the non-trivial solution of (3.26) must be also a saddle point, where theaction takes the same value. This suggest that the massive charged scalar field φ “ φ m ( s φ “ p φ m q ˚ ) ofmass m “ θ ´ is at the same foot than the vacuum φ “
0. We can ask whether these two configurationshave also the same effects in their gravitational/electromagnetic backgrounds, i.e. null effects in theircorrespondent curvatures. Though this is less straightforward, we shall see that this is indeed the case.5 .1 Field equations
The equations of motion of deformed theory, i.e. of the class (3.24) are given by: δS θ r g, A, φ, s φ s δg µν “ δS θ r g, A, φ, s φ s δA µ “ , δS θ r g, A, φ, s φ s δφ “ , δS θ r g, A, φ, s φ s δ s φ “ H µν r g s ´ Θ µν r g, A s ´ r Ξ µν r g, A, φ, s φ s “ , E µ r g, A s ´ r J µ r g, A, φ, s φ s “ , (3.28) r Υ r g, A, φ, s φ s “ , , rs Υ r g, A, φ, s φ s “ , (3.29)with the definitions (2.4), (2.5), (2.11), and r Ξ µν r g, A, φ, s φ s : “ ´ ?´ g δS θM r g, A, φ, s φ s δg µν , (3.30) r J µ r g, A, φ, s φ s : “ ?´ g δS θM r g, A, φ, s φ s δA µ , (3.31) r Υ µν r g, A, φ, s φ s : “ δS θM r g, A, φ, s φ s δφ , rs Υ µν r g, A, φ, s φ s : “ δS θM r g, A, φ, s φ s δ s φ . (3.32)Note that in (3.28) H µν r g s , Θ µν r g, A s and E µ r g, A s remain undeformed with respect to the definitions(2.4), (2.5) and (2.11), since field redefinitions (3.18) do not affect neither the metric tensor nor the U p q gauge field. The tildes on r Ξ, r J and r Υ and rs Υ indicate that these magnitudes are analogous to theoriginal variables (2.6), (2.7), (2.8) and (2.12), in the deformed theory (3.23).The results of the variation of the deformed action principle (3.24) with respect to the matter fieldsand the spacetime metric can be related to the results of the undeformed action principle (2.1) usingthe chain-rule of functionals, i.e. reminding that the deformed action depends implicitly on g µν and φ by means of φ θ r g, φ s and analogously for the conjugated field s φ . The chain rule for functional derivationreads, δF r G r f ss δf p y q “ ż d D z δF r G r f ss δG r f sp z q δG r f sp z q δf p y q , (3.33)where F r f s and G r f s are two functionals of the function class element f and F r G r f ss is the compositionof functionals. We shall declare the dependence on of functionals, say F r f s , on the point x by attachingthe symbol p x q , say F r f sp x q , whenever is necessary. Applying this in the computation of the equationsof motion (3.32), in the point y , we obtain from the deformed action, δS θM r g, A, φ, s φ s δφ p y q “ ż d D z δS M r g, A, φ, s φ s δφ θ p z q δφ θ p z q δφ p y q , (3.34)in account of (3.23). The latter expression is equivalent to, δS θM r g, A, φ, s φ s δφ p y q “ ż d D z Υ θ r g, A, φ, s φ sp z q δφ θ p z q δφ p y q , (3.35)where Υ θ r g, A, φ, s φ s : “ Υ r g, A, φ θ , s φ θ s , (3.36)and δS M r g, A, φ θ , s φ θ s δφ θ p z q “ δS M r g, A, φ, s φ s δφ p z q ˇˇˇˇˇ φ Ñ φ θ r g,A,φ s s φ Ñ s φ θ r g,A, s φ s , (3.37)6hich is equivalent to the original operator (2.7) valued in φ θ r g, A, φ, s φ s and s φ θ r g, A, φ, s φ s . Note that in(3.35), δφ θ p z q δφ p y q “ p ´ θ D q δ D p z ´ y q . (3.38)Integrating by parts and with the definitions (3.32) and (3.20) we obtain the equation of motion for thematter field in the deformed theory (3.28), r Υ r g, A, φ, s φ s : “ p ´ θ s D q Υ θ r g, A, φ, s φ s “ . (3.39)Note that Υ θ r g, A, φ, s φ s transforms as a left-module of the U p q group. Analogously, the result of thevariation with respect to s φ yields, rs Υ r g, A, φ, s φ s : “ p ´ θ D q s Υ θ r g, A, φ, s φ s “ , (3.40)where s Υ θ r g, A, φ, s φ s : “ s Υ r g, A, φ θ , s φ θ s . (3.41)The variation of the action with respect to g µν should be carried out taking similar considerations.For this purpose, let us introduce the Lagrangian density L M r g, A, φ, s φ s , such that, S M r g, A, φ, s φ s “ ż d D x ?´ g L M r g, A, φ, s φ s , (3.42)which with the substitution (3.23) yields, S θM r g, A, φ, s φ s “ ż d D x ?´ g L M r g, A, φ θ , s φ θ s . (3.43)From the chain rule (3.33), considering that the deformed action functional depends on g µν explicitlyand also implicitly in the deformed fields φ θ and s φ θ , the variation of (3.23) is equivalent to, δS M r g, A, φ θ , s φ θ s δg µν p y q “ δS M r g, A, φ, s φ s δg µν p y q ˇˇˇˇ φ Ñ φ θ (3.44) ` ż d D z δS M r g, A, φ θ , s φ θ s δφ θ p z q δφ θ p z q δg µν p y q ` ż d D z δS M r g, A, φ θ , s φ θ s δ s φ θ p z q δ s φ θ p z q δg µν p y q . Here the first term on the r.h.s. is provided by the explicit dependence of the action on the metrictensor, while the second and third terms are those coming from the variation with respect to the metricthrough the deformed scalar fields defined in (3.18).Considering the definitions (3.36), (3.41) and (3.43), (3.44) can be written also as, δS θM r g, A, φ, s φ s δg µν p y q “ ´?´ g Ξ θµν p y q ` ż d D z Υ θ p z q δφ θ p z q δg µν p y q ` ż d D z s Υ θ p z q δ s φ θ p z q δg µν p y q , (3.45)where we have defined Ξ θµν : “ Ξ µν r g, A, φ θ , s φ θ s , and we have omitted the arguments of the functionals Υ θ and s Υ θ for simplicity, though we declare thepoints where they are valued, i.e. y or z .From (3.45) the Hilbert energy-momentum tensor of the scalar field in deformed theory is given by, r Ξ µν r g, φ sp y q “ Ξ θµν p y q ´ ?´ g ż d D z Υ θ p z q δφ θ p z q δg µν p y q ´ ?´ g ż d D z s Υ θ p z q δ s φ θ p z q δg µν p y q . (3.46)7ere, δφ θ p z q δg µν p y q “ ´ ´ g µν g σρ p D σ φ qpB ρ δ D p z ´ y q ¯ ` ?´ g ´ B p µ ` iqA p µ ¯´ ?´ gδ D p z ´ y q D ν q φ ¯ , (3.47) δ s φ θ p z q δg µν p y q “ ˆ δφ θ p z q δg µν p y q ˙ ˚ . (3.48)Here symmetrized indices (with factor 1 {
2) have been enclosed in parenthesis. Replaced in (3.46) andfollowed by an integration by parts and boundary condition,Υ θ r g, A, φ, s φ s| B M “ s Υ θ r g, A, φ, s φ s| B M “ , (3.49)we obtain the energy momentum tensor, r Ξ µν “ Ξ θµν ` θ ?´ g g µν ` p D φ q Υ θ ` p s D s φ q s Υ θ ˘ (3.50) ´ θ p δ ρµ δ σν ` δ σµ δ ρν ´ g µν g σρ q ˆ p D ρ φ q s D σ ˆ Υ θ ?´ g ˙ ` p s D ρ s φ q D σ ˆ s Υ θ ?´ g ˙˙ . The computation of the electric current (3.31) of the deformed theory (3.23) yields, r J µ r g, A, φ, s φ s “ J µ r g, A, φ θ , s φ θ s (3.51) ´ iq θ ?´ g ´ Υ θ D µ φ ´ s Υ θ s D µ s φ ´ φ s D µ Υ θ ` s φD µ s Υ θ ¯ . (3.52)The (3.51) term appears from the variation of the part of the action which depends explicitly on thegauge field A µ and it is equivalent to electric current functional of the original theory (2.12) but valuedin φ θ (3.18) instead of φ , and their complex conjugated. The terms (3.52) appear from the implicitdependence, i.e. where the functional chain rule has to be used, of φ θ and s φ θ in the A -field dependencefrom the definitions (3.18). Consider the solutions φ “ φ m and s φ “ s φ m of mass m “ θ ´ the Klein-Gordon equations (3.26), suchthat φ θ r g, A, φ m s “ s φ θ r g, A, s φ m s “ . (3.53)Using this in the field equations of the deformed theory (3.29), r Υ r g, A, φ m , s φ m s “ p ´ θ D q Υ θ r g, A, φ m , s φ m s “ p ´ θ D q Υ r g, A, , s “ , (3.54) rs Υ r g, A, φ m , s φ m s “ p ´ θ s D q s Υ θ r g, A, φ m , s φ m s “ p ´ θ s D q s Υ r g, A, , s “ , (3.55)we observe that they are satisfied, under the assumption that the field equations of the original theory(2.14) admit the trivial solutions φ “ s φ “ i.e. such thatΥ θ r g, A, φ m , s φ m s “ Υ r g, A, , s “ , s Υ θ r g, A, φ m , s φ m s “ s Υ r g, A, , s “ . (3.56)8ow, let us evaluate the energy-momentum tensor (3.50) for the same solutions, r Ξ µν r g, A, φ m , s φ m s “ Ξ θµν r g, A, φ m , s φ m s ` θ g µν ?´ g ! p D φ m q Υ θ r g, A, φ m , s φ m s (3.57) `p s D s φ m q s Υ θ r g, A, φ m , s φ m s ) ´ θ p δ ρµ δ σν ` δ σµ δ ρν ´ g µν g σρ q ! p D ρ φ m q s D σ ˆ Υ θ r g, A, φ m , s φ m s?´ g ˙ `p s D ρ s φ m q D σ ˆ s Υ θ r g, A, φ m , s φ m s?´ g ˙ ) . From (3.56) all the terms containing these operators vanish. The only potentially surviving term is thefirst in the right hand side,Ξ θµν r g, A, φ m , s φ m s “ Ξ µν r g, A, φ θ r g, A, φ m s , s φ θ r g, A, s φ m ss “ Ξ µν r g, A, , s . (3.58)The expression in the middle is the energy momentum tensor of the original theory valued φ θ and s φ θ ,but from (3.56) they vanish for stealth configurations. Hence we obtain the last term, Ξ µν r g, A, , s “ r J µ r g, A, φ m , s φ m s “ J µ r g, A, φ θ r g, A, φ m s , s φ θ r g, A, s φ m s “ J µ r g, A, , s “ , (3.59)which is just the current functional of the original theory valued at the trivial vacuum of the scalarfield. Of course, this expression must vanish since the trivial vacuum can not carry electric current or,equivalently, as a functional the current must be at least of second order in the fields.Finally, in the deformed theory (3.24) the system of equations (3.28)-(3.29), are reduced to theEinstein equations in the presence of a electromagnetic fields and the Maxwell equations in a curvedbackground with no electric-charge sources, H µν r g s ´ Θ µν r g, A s “ , E µ r g, A s “ , (3.60) i.e. the same than the equations of motion of the original theory (2.1) without matter field S M r g, A, φ, s φ s “
0. Therefore, the stealth configuration of mass m “ θ ´ , i.e. which satisfy theminimally coupled Klein-Gordon equation (3.26), neither source space-time curvature nor electromag-netism. However, the space of functions on the spacetime manifold has nontrivial content and it mayaffect the topology of the spacetime (for example if singular regions in the domain of φ m need to beremoved). The results presented so far are valid for any theory (accepting the trivial vacuum solution φ “ M is given by, S M r g, A, φ, s φ s “ ´ ż d D x ?´ g ` g µν D µ φ s D ν s φ ` M φ s φ ˘ , (4.61)with the covariant derivatives defined as in (3.19). The field equations are given by,Υ r g, A, φ, s φ s “ ?´ g ` s D ´ M ˘ s φ “ , s Υ r g, A, φ, s φ s “ ?´ g ` D ´ M ˘ φ “ . (4.62)9he electric current (2.12) takes the form, J µ “ iq `s φD µ φ ´ φ s D µ s φ ˘ (4.63)and the energy-momentum tensor Ξ µν of the scalar field isΞ µν “ ! g µν ` D ρ φ s D ρ s φ ` M φ s φ ˘ ´ ` D µ φ s D ν s φ ` D ν φ s D µ s φ ˘) . (4.64) When applied to the theory (4.61) the deformation procedure (3.23) produces the new action principle, S θM r g, A, φ, ¯ φ s “ ´ ż d D x ?´ g ` g µν D µ φ θ r g, A, φ s s D ν s φ θ r g, A, s φ s ` M φ θ r g, A, φ s s φ θ r g, A, s φ s ˘ . (4.65)We can expand this action in terms of the fields φ and s φ using the definitions (3.18). The equations ofmotion obtained variating with respect to s φ is, r ¯Υ r g, A, φ, s φ s “ ?´ g ` ´ θ D ˘ ` D ´ M ˘ ` ´ θ D ˘ φ “ , (4.66)which is of the form (3.39). Alternatively, r ¯Υ r g, A, φ, s φ s “ θ ?´ g ` D ´ m ˘ ` D ´ M ˘ φ “ , (4.67)where m “ θ ´ is the stealth field mass. Analogously, the equations of motion of the conjugated fieldcan be written as, r Υ r g, A, φ, s φ s “ θ ?´ g ` ¯ D ´ m ˘ ` ¯ D ´ M ˘ ¯ φ “ . (4.68)The electric current, as defined in (3.51), is given by, r J µ r g, A, φ, ¯ φ s “ iq ` s φ θ D µ φ θ ´ φ θ s D µ s φ θ ˘ ´ iqθ " D µ φ ` s D ´ M ˘ s φ θ ´ ` D ´ M ˘ φ θ s D µ s φ ´ φ s D µ ` s D ´ M ˘ s φ θ ` s φD µ ` D ´ M ˘ φ θ * . (4.69)The energy-momentum tensor (3.30) of deformed matter yields in this case, r Ξ µν r g, A, φ, ¯ φ s “ ! g µν ` D ρ φ θ ¯ D ρ ¯ φ θ ` M φ θ ¯ φ θ ˘ ´ ` D µ φ θ ¯ D ν ¯ φ θ ` D ν φ θ ¯ D µ ¯ φ θ ˘) ` θ g µν ! D φ ` ¯ D ´ M ˘ ¯ φ θ ` ` D ´ M ˘ φ θ ¯ D ¯ φ ) . (4.70) The stealth field satisfies, φ θ “ “ ` ´ θ D ˘ φ m “ ´ m ` D ´ m ˘ φ m , (4.71)and their respective complex conjugated condition. Therefore the stealth field has mass m “ θ ´ . Let usdenote these solutions as φ m and s φ m . It is straightforward to show that the field equations (4.67)-(4.68)are satisfied and that the electric current (4.69) and energy-momentum tensor (4.70) vanish for φ “ φ m .Hence, though non-trivial, φ m does not curve the spacetime and it does not source electromagneticfields. 10 .3 Non-stealth solutions The field equation (4.67) admits other solutions besides stealth solutions. Those satisfy, ` D ´ M ˘ φ “ , (4.72)as one can see from the factorization of the operators in (4.67)-(4.68). Note that (4.72) coincides with theequation of motion of the original theory (4.62). So, after deformation of the theory, this configurationstill persists.Let φ M be a solution of (4.72), which has mass M , so that the operator D can be replaced by itsvalue D φ M “ M φ M . (4.73)Hence the functional φ θ (3.18) takes the value, φ θ r g, A, φ M , ¯ φ M s “ ´ m ` D ´ m ˘ φ M “ λφ M , λ : “ ˆ ´ M m ˙ , (4.74)and analogously for the conjugated field. The electric current (4.69) takes now the form, r J µ r g, A, φ M , ¯ φ M s “ iqλ ` D µ φ M ¯ φ M ´ φ M ¯ D µ ¯ φ M ˘ “ λ J µ r g, A, φ M , ¯ φ M s . (4.75)As we see, in the deformed theory the electric current is rescaled with respect to its counterpart inthe original theory (4.63). The same behavior is observed in the energy-momentum tensor of deformedtheory, it is also rescaled with respect to the undeformed theory, as we can see replacing φ M ( s φ M ) in(4.70), r Ξ µν r g, A, φ M , ¯ φ M s “ λ Ξ µν r g, A, φ M , ¯ φ M s . (4.76)Note that a general solution of the deformed theory consists of a linear combination of the solution of theoriginal theory and the stealth field, since the differential operators p D ´ M q and p D ´ m q commuteeach other, so that each operator annihilates its respective field. Hence the charged Klein-Gordon fieldis preserved in the deformed theory, but its effects on the electromagnetic and gravity backgroundsis modified by the λ factor. As we see, though stealth fields do not have direct gravitational andelectromagnetic effects, they can modify the strength of the effects of regular massive configurations. In this paper we observed that there exist a wide class of complex scalar field theories, coupled togravity and to electromagnetism, that admit stealth configurations. These theories are constructed bymeans of a “deformation method”, which can be regarded as a θ -parametric extension of some “originaltheory”. We show that the details of the original theory are not important for the existence of stealthconfigurations in the deformed theories, except that it must accept the trivial vacuum solution.As a novel aspect, the stealth fields does not source neither gravity nor electromagnetic fields, inspite of their non-trivial coupling to electromagnetism at the level of the action principle. They can,however, modify the strength of the effects of regular (non-stealth) scalar field configurations on theirelectromagnetic/gravity background.We provide one example, the deformation of the action principle of a complex Klein-Gordon field ofmass M , and we observe that the deformed theory predicts a modification of the strength of the sourcesof gravity and electromagnetism, respectively its energy momentum tensor and its electric current, bya constant factor λ “ ´ M { m , constructed from the stealth field mass m and the non-stealth field11ass M . At the level of the equations of motion, this is equivalent to a renormalization of the Newtonconstant and the electric charge, G N Ñ λ G N , q Ñ λ q. (5.77)Hence, the mass of the stealth field m “ θ ´ (equivalently the inverse of the deformation parameter θ ) can be used to smooth or amplify the effects of the massive field of mass M on the gravitationalbackground and the electromagnetic field.We expect that our results may be useful to reduce the discrepancies between the observations andthe predictions of regular scalar field theories, which have lead for example to the proposal of darkmatter, i.e. a discrepancy between the amount of observed matter and their effects in their gravitybackgrounds. For example, one can determine the mass m of the stealth field such that the observedmatter (non-stealth) will feedback the correct curvature to the background geometry, i.e. adjusting the λ parameter in the correspondent field equations. References [1] Eloy Ayon-Beato, Cristian Martinez, and Jorge Zanelli. Stealth scalar field overflying a (2+1) blackhole.
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