Braneworld-Klein-Gordon system in the framework of gravitational decoupling
BBraneworld-Klein-Gordon system in the framework ofgravitational decoupling
P. Le´on a, ∗ , A. Sotomayor b a Departamento de F´ısica, Universidad de Antofagasta, Antofagasta, Chile b Departamento de Matem´aticas, Universidad de Antofagasta, Antofagasta, Chile
Abstract
We analyze the Randall-Sundrum braneworld effective equations coupled with aKlein-Gordon scalar field by minimal geometric deformation decoupling method(MGD-decoupling). We introduce two different ways to apply MGD-decouplingmethod to obtain new solutions for this enlarged system. We also compare thebehavior of the new solutions with those without coupling the scalar field.
Keywords:
BraneWorld, Gravitational Decoupling, Scalar field, Black Holes
1. Introduction
The theory of General Relativity (GR) is both the simplest and most suc-cessful theory for describing the gravitational interaction. Over the last yearshad been a plenty of experimental data which agree in a very good way withpredictions of GR. However, despite the great success of GR, the theory cannotexplain satisfactorily some phenomena such as dark matter and dark energyproblems. This is one of the reasons that leads physicists to consider theoriesbeyond GR that could explain some of these problems. These kind of theoriesmay include different types of scalar fields to model dark energy and dark matter(see [1] and references therein) which can be considered a fundamental aspectof the theory or an effective description of other fundamental fields. Besides the ∗ Corresponding author
Email addresses: [email protected] (P. Le´on ), [email protected] (A.Sotomayor )
Preprint submitted to Journal of L A TEX Templates March 1, 2021 a r X i v : . [ h e p - t h ] F e b osmological applications of the scalar fields, they also can have, for example, aparticular interest in the study of the no-hair conjecture (see [2]).In addition to dark matter and dark energy problems, there is also a funda-mental problem in physics, known as the hierarchy problem, which correspondsto the huge scale difference between the gravitational and weak interactions.In [3, 4], Randall and Sundrum proposed a theory known as Randall-SundrumBraneworld (RSBW) which can explain this problem. In the simplest models ofRSBW all the gauge interactions described by the standard model of particles,together with all our observable universe, are confined to live in a 3-brane em-bedded in a five dimensional space-time called the bulk. In contrast with gaugeinteraction, the gravitational one is not restricted to live in the 3-brane and canspread into the bulk. This mean that in our observable universe (the 3-brane) wecan only see a fraction of the gravitational interaction, therefore, since the extradimension can be very large, this can explain why the gravitational interactionis so weak compared to the Planck scale.Currently, there is not direct experimental data that could support theRSBW theory. However, since this theory can explain the hierarchy problem,the study of the Einstein’s field equations including the contribution whichcomes from RSBW model is justified and is interesting by itself. It is importantto mention that there are other realizations, in higher dimensions, of RSBW(see for example [5]). In the context of RSBW, there are also some cosmologi-cal models which consider the existence of an scalar field to study perturbationof the RSBW scenario and which could give some information about the darkmatter and dark energy problems.Now, even when RSBW based models are very promising and a covariantformulation of the theory in five dimensions is known, there are some openproblems associated with these models, mainly due to the lack of solutions as-sociated with the equations of motion in the complete five dimensional theory,which still are not clear [6, 7, 8, 9]. One convenient way to clarify these prob-lems and give some information of the impact of RSBW in gravity is basedin the study of the effective field equations in four dimensions (our observable2niverse). In this case the contribution which arises from the five dimensioncan be interpreted as a contribution to the energy momentum tensor in the fourdimensional Einstein’s equations of GR.In recent years was proposed a new method, known as minimal geometricdeformation decoupling method (MGD-decoupling), which among many of itapplications allows us to find new physically relevant solutions of Einstein’sequations through a very simple and elegant way [10, 11].Specifically, the method allows us to study gravitational systems with anenergy momentum tensor which can be decomposed into a sum of differentsources T µν = T µν + n (cid:88) i =1 α i θ µνi . (1)The sources θ µνi could have different interpretations, as for example anothermatter fluid (see for example [12, 13, 14, 15, 16, 17, 18]), the coupling of Ein-stein’s equations with other fields like a Klein-Gordon scalar field (see [19]),the corrections of the GR coming from gravitational theories beyond GR (forexample [20, 21, 22, 23, 24, 25]).Originally the method was restricted to spatial metric deformation in spher-ically symmetric systems. However, in the last years the method has beenextended to include the formulation in alternative coordinates [26], spatial andtemporal deformation of the metric components [27], cylindrical or axially sym-metric matter distributions [28, 29] (for more applications of the gravitationaldecoupling method see [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64] ).In this paper we study the RSBW effective Einstein’s equations coupled toa Klein-Gordon (KG) scalar field using the MGD-decoupling method. In thiskind of systems, the coupling with the KG scalar field can be interpreted asa field restricted to live in the observable brane or the effective contributionof a bulk scalar field (see [65]). In both cases the energy-momentum tensor ofeffective field equations can be decomposed in two sources, one associated with3he corrections of the RSBW and the other with the KG scalar field. Now, in thecontext of GR the order in which the MGD-decoupling is applied to decouplethe different sources θ µνi is not relevant, but when the sources are contributionscoming from theories beyond GR this is not so clear. In fact we will showthat the MGD method leave us with two inequivalent possibilities for study thisenlarged system, which are related with the order in which decoupling of thesource is made.The paper is organized as follows: in section 2 we summarize the MGDmethod. In section 3 we present the effective RSBW four dimensional Einstein’sequations and in section 4 we introduce two ways in which the MGD methodcan be applied to analyze RSBW model coupled with a Klein Gordon scalarfield. In section 5 we give our conclusions.
2. Gravitational decoupling method
The so-called gravitational decoupling method (MGD-decoupling) [10] hadits motivation in the context of minimal geometric deformation (MGD), appliedto the brane world model [3, 4, 66] (to read an elegant and detailed expositionrelating these aspects see [11]) .To give a concise review of the method we start with Einstein field equations(EFEs) R µν − R g µν = − k ˆ T µν , (2)where we assume that the energy momentum tensor ˆ T µν has contributions oftwo (different) gravitational sources given byˆ T µν = T µν + Θ µν . (3)We will be interested in this work in the case of spherically symmetric andstatic systems. We use the usual Schwarzschild-like coordinates with the corre-sponding line element given by 4 s = e ν ( r ) dt − e λ ( r ) dr − r (cid:0) dθ + sin θ dφ (cid:1) . (4)In this setting, Einstein’s field equations (2) can be written in the knownform k ¯ ρ = k ( T + Θ ) = 1 r − e − λ (cid:18) r − λ (cid:48) r (cid:19) , (5) k ¯ p r = − k ( T + Θ ) = − r + e − λ (cid:18) r + ν (cid:48) r (cid:19) , (6) k ¯ p t = − k ( T + Θ ) = e − λ (cid:32) ν (cid:48)(cid:48) + ν (cid:48) − λ (cid:48) ν (cid:48) + 2 ν (cid:48) − λ (cid:48) r (cid:19) . (7)We recall that prime indicates a derivative respect to r and ¯ ρ , ¯ p r and ¯ p t aredefined to be the effective energy density, the effective radial pressure and theeffective tangential pressure respectively.The conservation equation for the given system, which can be given as alinear combination of the equations defining it, is written by ∇ µ ˆ T µν = (¯ p r ) (cid:48) − ν (cid:48) ρ + ¯ p r ) − r (¯ p t − ¯ p r ) = 0 , (8)which in terms of the gravitational sources T µν and Θ µν takes the followingform (cid:0) T (cid:1) (cid:48) − ν (cid:48) (cid:0) T − T (cid:1) − r (cid:0) T − T (cid:1) + (cid:0) Θ (cid:1) (cid:48) − ν (cid:48) (cid:0) Θ − Θ (cid:1) − r (cid:0) Θ − Θ (cid:1) = 0 . (9)We notice, from equations (5)-(7), that the combination of the two sourcesin the energy-momentum tensor describes a fluid with local anisotropy on thepressures.In order to solve the complete system of equations (5)-(7), we apply theMGD-decoupling method. The first step considers only the contribution of thesource T µν , whose line element is written by5 s = e ξ ( r ) dt − µ ( r ) dr − r (cid:0) dθ + sin θ dφ (cid:1) , (10)where µ ( r ) ≡ − k r (cid:90) r x T dx = 1 − m ( r ) r , (11)is the standard definition of the mass function in General Relativity (GR).The next step includes also the contribution of Θ µν . This can be done byconsidering that the effects induced by this gravitational source are encoded bydeformations of temporal and radial components in the metric given by ν ( r ) = ξ ( r ) + αg ∗ ( r ) , (12) e − λ ( r ) = µ ( r ) + αf ∗ ( r ) , (13)were g ∗ ( r ) and f ∗ ( r ) are two functions to be determined.In this work we will be interested only in the case when g ∗ = 0 (historicallyassociated to MGD), that is, when we have only deformations for the radialcomponent of the metric in the reduced form ν ( r ) = ξ ( r ) , (14) e − λ ( r ) = µ ( r ) + αf ∗ ( r ) . (15)By using (14) and (15), Einstein’s field equations can be decoupled into twosystems of equations.The first one is given by Einstein field equations relative to the source T µν k T = 1 r − µr − µ (cid:48) r , (16) − k T = − r + µ (cid:18) r + ν (cid:48) r (cid:19) , (17) − k T = µ (cid:18) ν (cid:48)(cid:48) + ν (cid:48) + 2 ν (cid:48) r (cid:19) + µ (cid:48) (cid:18) ν (cid:48) + 2 r (cid:19) , (18)6ith associated conservation equation (cid:0) T (cid:1) (cid:48) − ν (cid:48) (cid:0) T − T (cid:1) − r (cid:0) T − T (cid:1) = 0 , (19)while the second one system being only related to the gravitational source Θ µν ,and written by k Θ = − αf ∗ r − αf ∗ (cid:48) r , (20) k Θ = − αf ∗ (cid:18) r + ν (cid:48) r (cid:19) , (21) k Θ = − α f ∗ (cid:18) ν (cid:48)(cid:48) + ν (cid:48) + 2 ν (cid:48) r (cid:19) − α f ∗ (cid:48) (cid:18) ν (cid:48) + 2 r (cid:19) , (22)with corresponding conservation equation (cid:0) Θ (cid:1) (cid:48) − ν (cid:48) (cid:0) Θ − Θ (cid:1) − r (cid:0) Θ − Θ (cid:1) = 0 . (23)To convert the system given (20)-(22) into a Einstein system for Θ µν wecan redefine the components of Θ µν in order to include the corresponding lostfactors of 1 /r . We have also, from equations (9), (21) and (22), that interactionbetween the sources T µν and Θ µν is purely gravitational.Thus, to decouple the complete Einstein system given by (5)-(7) we proceedsolving, at a first step, the Einstein system given by (16)-(19) for the originalsource T µν , determining the triple { T µν , ξ, µ } . A second step consists by solvingthe system (20)-(22) for the source Θ µν to find { Θ µν , f ∗ } . Finally, the solutionfor the complete system can be obtained by a direct combination of the resultsgiven in these two steps.This simple and systematic procedure, known as MGD-decoupling, can beused as a powerful tool in the analysis of more complicated and realistic distri-butions of matter in the context of General Relativity.
3. The Braneworld context
One of the main features of the braneworld models comes to consider thatthe five dimensional gravity induces modifications in our (3 + 1) observableuniverse, called the brane, which can be expressed by7 µν = − g µν Λ − k ˆ T µν , (24)where k = 8 πG N and Λ is the cosmological constant onto the brane.The induced modifications to the Einstein field equations are given by theeffective energy-momentum tensorˆ T µν = T µν − σ S µν + 18 π E µν + 4 σ F µν , (25)which, through the inclusion of the last three terms, take into account all theeffects of the bulk onto the brane, with σ being the brane tension.The S µν high-energy term arises from the extrinsic curvature terms in theprojected Einstein tensor onto the brane and is given by S µν = 112 T T µν − T µρ T ρν + g µν (cid:2) T ρλ T ρλ − T (cid:3) , (26)where T is the trace of T µν .The E µν Kaluza-Klein corrections term represents the projection of the Weyltensor of the bulk. For the case of spherically symmetric and static distributionsof matter, which will be the case, it can by written as k E µν = 6 σ (cid:20) U (cid:18) u µ u ν − h µν (cid:19) + P µν (cid:21) , (27)with h µν = g µν − u µ u ν , (28) P µν = P (cid:18) r µ r ν − h µν (cid:19) , (29)where U , P µν , h µν , u µ and r µ are the bulk Weyl scalar, the anisotropic stress,the projection operator operator, the four velocity of the fluid element and theradial unitary vector respectively.The last correction to the effective energy-momentum tensor F µν term de-pends on all the stresses but the cosmological constant in the bulk. From now8n, we consider that F µν = 0, which means that only the cosmological con-stant is present in the bulk. In this particular case, we recover the standardconservation equation of GR ∇ ν T µν = 0 . (30)In order to study the effects of the RSBW on anisotropic fluids we use thatthe energy-momentum tensor T µν must be given by T µν = ( ρ + p t ) u µ u ν − p t g µν + ( p r − p t ) s µ s ν , (31)where u µ = exp ν/ δ µ , s µ = exp λ/ ρ , p r , p t being the energy density andthe radial and tangential pressures of the fluid, respectively. In this situationthe equilibrium equation leads to p (cid:48) r + ν (cid:48) ρ + p r ) − r = 0 , ∆ = p t − p r . (32)We are now ready to writte the effective Einstein’s equations (when cos-mological constant Λ = 0) for the four dimensional 3-brane. Using (10) and(25)-(31), equation (24) leads to k (cid:34) ρ + 1 σ (cid:18) ( ρ − ∆ )2 + 6 U k (cid:19) (cid:35) =1 r − e − λ (cid:18) r − λ (cid:48) r (cid:19) , (33) k (cid:34) p r + 1 σ (cid:18) ρ ρp t + p t − p r U k (cid:19) + 4 P k σ (cid:35) = − r + e − λ (cid:18) r + ν (cid:48) r (cid:19) , (34) k (cid:34) p t + 1 σ (cid:18) ρ ρ p r + p t ) + 2 U k (cid:19) − P k σ (cid:35) = e − λ (cid:18) ν (cid:48)(cid:48) + ν (cid:48) − λ (cid:48) ν (cid:48) + 2 ν (cid:48) − λ (cid:48) r (cid:19) . (35)9e see from equations (38)-(40), that we have reached to an indefinite sys-tem, in which extra information is required (related with the geometry of thebulk) to be solved.In next section we show how (through MGD-decoupling), starting from givensolutions of BRST system, we can obtain solutions of the Braneworld-Klein-Gordon system. Also we show the analogue of the above procedure startingwith solutions for the Eintein-Klein-Gordon system (see figure 1 in next page).
4. RSBW coupled to a KG scalar field
In this section we analyze the inclusion of a KG scalar field present in thevisible brane of the RS braneworld scenario using MGD-decoupling method.This leaves us with two options to construct solutions for RSBW models cou-pled to a KG scalar field in the visible brane. The first approach is startingfrom a known solution for the effective field equations of the RSBW, and thenmake use of MGD-decoupling method to obtain solutions of the system coupledwith a scalar field satisfying the Klein Gordon equation. The second approachcorresponds to select a seed solution of the Einstein-Klein-Gordon system andthen extend it to the RSBW scenario through the MGD method. Now, it isknown that MGD method can be used to construct solutions to the EinsteinKlein Gordon system or also to the effective field equation of RSBW from agiven solution of Einstein’s equations, see [19, 25]. So, solutions to the effectiveRSBW field equations coupled with a KG scalar field can be seen starting froma seed solution of GR (see figure 1) through MGD-decoupling method .We analyse in the following the two possible ways to obtain solutions for theeffective RSBW coupled to the Klein Gordon scalar field. In order to study theeffect of a KG scalar field in the external solutions of the RSBW field equationlet us consider the following energy-momentum tensor T Tµν = T µν − σ S µν + 18 π E µν + θ µν , (36)where the first term is the energy momentum tensor of an anisotropic fluid10 igure 1: Construction of solution for the coupling of RSBW with KG (given by Eq. (31)), the following two terms are the contributions of the BWsector and θ µν is a source coming from the presence of a KG scalar field fillingthe space time.Then θ µν is given by θ µν = ∇ µ Ψ ∇ ν Ψ − (cid:18) ∇ α Ψ ∇ α Ψ − V (Ψ) (cid:19) g µν , (37)where Ψ( r ) is a scalar minimally coupled and V (Ψ) is a self interaction potential.Einstein’s equations corresponding to the enlarged system (36) can be writ-ten by k (cid:34) ρ + 1 σ (cid:18) ( ρ − ∆ )2 + 6 U k (cid:19) + 12 e − λ Ψ (cid:48) + V (cid:35) =1 r − e − λ (cid:18) r − λ (cid:48) r (cid:19) , (38) k (cid:34) p r + 1 σ (cid:18) ρ ρp t + p t − p r U k (cid:19) + 4 P k σ + 12 e − λ Ψ (cid:48) − V (cid:35) = − r + e − λ (cid:18) r + ν (cid:48) r (cid:19) , (39) k (cid:34) p t + 1 σ (cid:18) ρ ρ p r + p t ) + 2 U k (cid:19) − P k σ − e − λ Ψ (cid:48) − V (cid:35) = e − λ (cid:18) ν (cid:48)(cid:48) + ν (cid:48) − λ (cid:48) ν (cid:48) + 2 ν (cid:48) − λ (cid:48) r (cid:19) . (40)11 .1. First approach In this approach we use as a seed in the implementation of MGD-decouplinga solution of the 4D effective BRSW before coupling with Klein-Gordon scalarfield. Following the procedure presented in section (2) the system of equationswe need to solve is given by k (cid:18) e − λ Ψ (cid:48) + V (cid:19) = − α (cid:32) f ∗ r + f ∗ (cid:48) r (cid:33) , (41) k (cid:18) − e − λ Ψ (cid:48) + V (cid:19) = − αf ∗ (cid:18) r + ξ (cid:48) r (cid:19) , (42) k (cid:18) e − λ Ψ (cid:48) + V (cid:19) = − αf ∗ (cid:18) ξ (cid:48)(cid:48) + ξ (cid:48) + 2 ξ (cid:48) r (cid:19) − αf ∗ (cid:48) (cid:18) ξ (cid:48) + 2 r (cid:19) , (43)where the conservation equation can be written (using Eq. (23)) asΨ (cid:48)(cid:48) + (cid:18) r + 12 ( ξ (cid:48) − λ (cid:48) ) (cid:19) Ψ (cid:48) = e λ dVd Ψ . (44)We can find a differential equation for f ∗ , using (41) and (43), which is (cid:18) ξ (cid:48) − r (cid:19) f ∗ (cid:48) + f ∗ (cid:18) ξ (cid:48)(cid:48) ξ ) (cid:48) ξ (cid:48) r − r (cid:19) = 0 , (45)whose solution is given by f ∗ ( r ) = C exp − (cid:90) (cid:18) ξ (cid:48)(cid:48) ξ (cid:48) ξ (cid:48) r − r (cid:19) (cid:18) ξ (cid:48) − r (cid:19) − dr. (46)Before give any particular example will be useful to express the scalar field Ψ (cid:48) and the potential V (Ψ) as followsΨ (cid:48) = e λ α πr ( f ∗ ξ (cid:48) − f ∗ (cid:48) ) , (47) V = − α π (cid:32) f ∗ (cid:18) r + ξ (cid:48) r (cid:19) + f ∗ (cid:48) r (cid:33) . (48)Let us discuss now two different examples of exterior solutions ( ρ = p r = p t = 0), in the first one we choose the following metric (see [67, 25, 6])12 ξ = 1 − Mr , µ = (cid:18) − Mr (cid:19) (cid:20) Dσ (2 r − M ) (cid:21) , (49)where D and M are constants.For this metric we found that f ∗ ( r ) = (cid:18) − Mr (cid:19) (cid:18) Cr − M (cid:19) , (50)and thenΨ (cid:48) = 2 αC r ( r − M ) (cid:34) Dσ (2 r − M ) + α (cid:18) Cr − M (cid:19) (cid:35) − , (51) V = αC M πr ( r − M ) . (52)From above equations we see that in the limit σ → ∞ we recover the sameexpression for Ψ (cid:48) presented in [19] (as expected).The 4 D scalar curvature of the metric is given by R = − αC r ( r − M ) . (53)Therefore this solution exhibits the same naked singularity at r = 3 M presentedin [19] and therefore this can not be interpreted as a black hole solution. How-ever, this is only a problem if the radius of the source R is less than 3 M . If R > M we would be in presence of a well behave external solution.Now, Eq. (47) in general will not have an analytical solution. Then, in orderto find an explicit one we can assume that α ≈ α and σ − . In that case Eq. (51) can beintegrated to obtainΨ ≈ α (cid:0) C r (cid:1) M ( r − M )) − C D √ r (cid:16) √ r − √ r − M arctan (cid:16) √ r √ r − M (cid:17)(cid:17) σM ( r − M ) , where it is clear that to zero order in σ − we recover the result found in [19] V ≈ αC k M Ψ K (cid:18) − K Ψ (cid:19) , (54)13here K = 8 αC k M . (55)However, in higher orders the analytical form of the functional V (Ψ) it is notclear.As a second example let us use the tidial charge black hole [68] solution ofthe effective RSBW equations given by e ξ t = 1 − Mr − Qr , (56) µ = 1 − Mr − Qr (57)(the subindex t is referred to “tidial”). Thus it can be shown that f ∗ t = D ( r ( r − M ) − Q )( r ( r − M ) − Q ) , (58)Ψ (cid:48) = − αD (cid:0) Q + r (cid:1) π ( r (3 M − r ) + 2 Q ) ( r ( αD + ( r − M ) ) + 4 Qr (3 M − r ) + 4 Q ) , (59) V t = αD (cid:0) M (cid:0) r − Q (cid:1) + 2 Qr (cid:1) πr ( r ( r − M ) − Q ) . (60)Now, for α << α to obtainΨ = 2 aD (2 Q + r ) k r ( r ( r − M ) − Q ) . (61)Thus we can find the form of the potential as function of Ψ V t (Ψ) = αD (cid:0) M (cid:0) r (Ψ) − Q (cid:1) + 2 Qr (Ψ) (cid:1) πr (Ψ)( r (Ψ)( r (Ψ) − M ) − Q ) , (62)where r (Ψ) = M + T / T + 13 / Ψ k T (Ψ) , (63)where T (Ψ) ≡ aC + Ak (cid:0) M + 2 Q (cid:1) , (64)14 (Ψ) ≡ (cid:104) Ψ k (cid:16) (cid:0) αC k ( M + 2 Q ) + Ψ k M (cid:0) M + Q (cid:1)(cid:1) − (cid:0) αC + Ψ k (cid:0) M + 2 Q (cid:1)(cid:1) (cid:17)(cid:105) / , (65) T ≡ (cid:113) √ T (Ψ) + 9 a Ψ C k ( M + 2 Q ) + 9Ψ k M ( M + Q ) . (66)The scalar curvature is given by R = 2 αD (2 Q − Qr + 6 M Qr − r ) r ( r − rM − Q ) , (67)where it is easy to see that there are two singularities at r ± = (3 M ± (cid:112) M + 8 Q ) / r + is an essential singularity. However, as before, this not a problemis we have a matter source with a radius greater that r + .In a recent paper we found a series of external solutions for the BWRSscenario taking as seed the tidial charge black hole solution.Now let us denote as { ξ t , µ i } ( i = 1 , ,
3; the subindex t is referred to “tidial”)the metric components of the solutions obtained in [25], where µ = (cid:18) − Mr − Qr (cid:19) (cid:34) β (cid:16) ra (cid:17) e qM r (cid:18) − Mr (cid:19) qM (cid:35) , (68) µ = (cid:18) − Mr − qr (cid:19) Be MArcTan (cid:104) M − r √ − M − Q (cid:105) √ − M − Q (cid:112) r (2 r − M ) − Q , (69) µ = (cid:18) − Mr − Qr (cid:19) (cid:34) d (cid:18) − Mr (cid:19) QM e QM r (cid:35) . (70)Then, the coupling of these solutions with a KG scalar field are given by˜ µ i = µ r + αf ∗ t , (71)Ψ (cid:48) i = 2 αC ( r + 2 Q ) k r ( r − M r − Q ) µ r (cid:18) − Mr − Qr (cid:19) , (72) V i = V t = αD (cid:0) M (cid:0) r − Q (cid:1) + 2 Qr (cid:1) πr ( r ( r − M ) − Q ) . (73)15hen, as in the seed solution (tidial charge black hole), the coupling with aKG scalar field through MGD-decoupling introduce a naked singularity in newsolutions if the internal matter distribution is less than r + . Now, since the formof the µ r are much more complicated that in the first and second examples, Eq(72) does not have an analytical solution for Ψ. Another way to couple the four dimensional effective RSBW field equationsto a KG scalar field using MGD-decoupling consists to taking a known solu-tion of the Einstein Klein Gordon field equations (EKG) and then analize thecontributions coming from extra dimensions using the mechanism developed in[25].To do that, we write the EKG field equations in the form given by k (cid:18) T + 12 e − λ Ψ (cid:48) + V (cid:19) = 1 r − e − λ (cid:18) r − λ (cid:48) r (cid:19) , (74) k (cid:18) T − e − λ Ψ (cid:48) + V (cid:19) = 1 r − e − λ (cid:18) r − ν (cid:48) r (cid:19) , (75) k (cid:18) T + 12 e − λ Ψ (cid:48) + V (cid:19) = − e − λ (cid:18) ν (cid:48)(cid:48) + ν (cid:48) − λ (cid:48) ν (cid:48) + 2 ν (cid:48) − λ (cid:48) r (cid:19) . (76)Now, by defining the effective pressures and energy density through ρ t = T + 12 e − λ Ψ (cid:48) + V, (77) p r = T − e − λ Ψ (cid:48) + V, (78) p t = T + 12 e − λ Ψ (cid:48) + V, (79)it is possible to write the equations for the contributions of the bulk, obtainedfrom the MGD method, in the form k (cid:18) ( ρ − ∆ )2 + 6 U k (cid:19) = − f ∗ r − f ∗ (cid:48) r , (80) k (cid:20)(cid:18) ρ ρp t + p t − p r U k (cid:19) + 4 P k (cid:21) = f ∗ (cid:18) r + ν (cid:48) r (cid:19) , (81)16 (cid:20)(cid:18) ρ ρ p r + p t ) − p r k U (cid:19) − P k (cid:21) = 14 (cid:20) f ∗ (cid:18) ξ (cid:48)(cid:48) + ξ (cid:48) + 2 ν (cid:48) r (cid:19) + f ∗ (cid:48) (cid:18) ν (cid:48) + 2 r (cid:19)(cid:21) , (82)with ∆ = p t − p r .Using these equations we can find that f ∗ ( r ) = JF ( r ) + 2 k F ( r ) (cid:90) F ( r ) rrξ (cid:48) + 4 ( ρ + ρ (2 p t + p r ) + ∆ ) dr, (83)with F ( r ) = exp (cid:18)(cid:90) (cid:18) ξ (cid:48)(cid:48) + ( ξ (cid:48) ) ξ (cid:48) r + 2 r (cid:19) (cid:46) (cid:18) ξ (cid:48) r (cid:19) dr (cid:19) . (84)Now, to show the difference of the two approaches we can select the followingsolution of the EKG field equations e ν = 1 − Mr , (85) e − λ = (cid:18) − Mr (cid:19) (cid:34) α (cid:18) Cr − M (cid:19) (cid:35) , (86)Ψ (cid:48) = αC πr ( r − M )( αC + ( r − M ) ) , (87) V = αC M πr ( r − M ) , (88)which was also obtained by using the MGD method in [19].Introducing these expressions in the Eqs. (83)-(84) and taking for simplicity J = 0 we obtain F ( r ) = r (2 r − M )(2 M − r ) , (89) f ∗ ( r ) = 1162 k M r (3 M − r )(3 M − r ) ( C (3 M (156 M r − M r + 108 M − M r + 4 r ) − r ( r − M ) log( r )+ 4 r ( r − M ) log( r − M ))) , (90)17hich again exhibits a singularity at r = 3 M . Nevertheless, as before, if wehave an internal matter distribution with a radius bigger than 3 M , this couldrepresent a well behave external solution. Conclusions
We presented the formulation of the simplest RSBW scenario model coupledwith a Klein Gordon scalar field in 4D using MGD-decoupling method. Thecoupling with a KG scalar field could be interpreted as restricted to the visiblebrane or like the effective contribution of a bulk scalar field in 5D. We foundtwo different ways in which MGD-decoupling method could be applied in orderto analyze this problem that correspond to the choice of the seed solution.One possibility, and maybe the most direct one, is take as seed a solution ofthe 4D effective RSBW equations and then, through the MGD-decoupling, tocouple the system with a Klein Gordon scalar field. In this work we use asseed the solutions of the effective RSBW equations obtained in [25]. In all ofthese cases the inclusion of the scalar field introduced a naked singularity to thefinal solution and therefore can not represent black solutions. However, theseexternal solution could be well defined if there is an internal matter distributionwith a radius greater that the radius of the singularity (see [69] for more details).On the other hand, due to the BW corrections, the exact form of the potentialas a functional of Ψ is not clear, with the exception of the tidal black holesolution with α <<
1. Now, in comparison, the contribution of the scalar fieldare dominant for small values of α .The another possibility to be applied in the use of MGD-decoupling is tostart as seed with a solution of the Einstein-Klein Gordon system (in which thecontributions of the scalar could be reinterpreted as an anisotropic fluid) anduse results presented in [25] to include the BW effective contributions. We used,for example, as seed, the solution presented in [19]. The forms of the scalar fieldand the potential (as a function of Ψ) were easier to find, at least at lowerorders in α . However, as in previous case, the final solution exhibited the same18aked singularity of the seed solution. Now, the MGD method can be used tofind solutions of the 4 D effective RSBW field equations and the Einstein-KleinGordon system using as seed a solution of GR (see [25, 19]).Then, as is shown in figure 1, we realized the study of the RSBW coupled to aKlein Gordon scalar field through MGD-decoupling starting with a GR solution.Nevertheless, the order in which the MGD method is applied to decouple eachsource is fundamental since the effective contributions of the RSBW depends ofthe form of the energy momentum-tensor in the visible brane. References [1] T. P. Sotiriou, Gravity and Scalar Fields, Lect. Notes Phys. 892 (2015)3–24. arXiv:1404.2955 , doi:10.1007/978-3-319-10070-8_1 .[2] T. P. Sotiriou, Black Holes and Scalar Fields, Class. Quant. Grav.32 (21) (2015) 214002. arXiv:1505.00248 , doi:10.1088/0264-9381/32/21/214002 .[3] L. Randall, R. Sundrum, A Large mass hierarchy from a small extra di-mension, Phys. Rev. Lett. 83 (1999) 3370–3373. arXiv:hep-ph/9905221 , doi:10.1103/PhysRevLett.83.3370 .[4] L. Randall, R. Sundrum, An Alternative to compactification, Phys.Rev. Lett. 83 (1999) 4690–4693. arXiv:hep-th/9906064 , doi:10.1103/PhysRevLett.83.4690 .[5] J. F. G. Cascales, M. P. Garcia del Moral, F. Quevedo, A. M. Uranga,Realistic D-brane models on warped throats: Fluxes, hierarchies and mod-uli stabilization, JHEP 02 (2004) 031. arXiv:hep-th/0312051 , doi:10.1088/1126-6708/2004/02/031 .[6] C. Germani, R. Maartens, Stars in the brane world, Phys. Rev. D 64 (2001)124010. arXiv:hep-th/0107011 , doi:10.1103/PhysRevD.64.124010 .197] P. Kanti, N. Pappas, K. Zuleta, On the localization of four-dimensionalbrane-world black holes, Class. Quant. Grav. 30 (2013) 235017. arXiv:1309.7642 , doi:10.1088/0264-9381/30/23/235017 .[8] S. Abdolrahimi, C. Cattoen, D. N. Page, S. Yaghoobpour-Tari, LargeRandall-Sundrum II Black Holes, Phys. Lett. B 720 (2013) 405–409. arXiv:1206.0708 , doi:10.1016/j.physletb.2013.02.034 .[9] D.-C. Dai, D. Stojkovic, Analytic solution for a static black hole in RSIImodel, Phys. Lett. B 704 (2011) 354–359. arXiv:1004.3291 , doi:10.1016/j.physletb.2011.09.038 .[10] J. Ovalle, Decoupling gravitational sources in general relativity: from per-fect to anisotropic fluids, Phys. Rev. D95 (10) (2017) 104019. arXiv:1704.05899 , doi:10.1103/PhysRevD.95.104019 .[11] J. Ovalle, R. Casadio, Beyond Einstein Gravity, SpringerBriefs in Physics,Springer, 2020. doi:10.1007/978-3-030-39493-6 .[12] C. L. Heras, P. Leon, Using MGD gravitational decoupling to extendthe isotropic solutions of Einstein equations to the anisotropical domain,Fortsch. Phys. 66 (7) (2018) 1800036. arXiv:1804.06874 , doi:10.1002/prop.201800036 .[13] M. Estrada, F. Tello-Ortiz, A new family of analytical anisotropic solutionsby gravitational decoupling, Eur. Phys. J. Plus 133 (11) (2018) 453. arXiv:1803.02344 , doi:10.1140/epjp/i2018-12249-9 .[14] L. Gabbanelli, A. Rinc´on, C. Rubio, Gravitational decoupled anisotropiesin compact stars, Eur. Phys. J. C78 (5) (2018) 370. arXiv:1802.08000 , doi:10.1140/epjc/s10052-018-5865-2 .[15] E. Morales, F. Tello-Ortiz, Compact Anisotropic Models in General Rel-ativity by Gravitational Decoupling, Eur. Phys. J. C78 (10) (2018) 841. arXiv:1808.01699 , doi:10.1140/epjc/s10052-018-6319-6 .2016] E. Morales, F. Tello-Ortiz, Charged anisotropic compact objects by gravi-tational decoupling, Eur. Phys. J. C78 (8) (2018) 618. arXiv:1805.00592 , doi:10.1140/epjc/s10052-018-6102-8 .[17] F. Tello-Ortiz, S. Maurya, A. Errehymy, K. Singh, M. Daoud, Anisotropicrelativistic fluid spheres: an embedding class I approach, Eur. Phys. J. C79 (11) (2019) 885. doi:10.1140/epjc/s10052-019-7366-3 .[18] V. Torres-S´anchez, E. Contreras, Anisotropic neutron stars by gravitationaldecoupling, Eur. Phys. J. C 79 (10) (2019) 829. arXiv:1908.08194 , doi:10.1140/epjc/s10052-019-7341-z .[19] J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, Z. Stuchlik, Einstein-Klein-Gordon system by gravitational decoupling, EPL 124 (2) (2018)20004. arXiv:1811.08559 , doi:10.1209/0295-5075/124/20004 .[20] M. Sharif, A. Waseem, Anisotropic spherical solutions by gravitationaldecoupling in f ( r ) gravity, Annals of Physics 405 (2019) 14 – 28. doi:https://doi.org/10.1016/j.aop.2019.03.003 .URL [21] M. Sharif, S. Saba, Extended gravitational decoupling approach in f ( G )gravity, International Journal of Modern Physics D 29 (06) (2020)2050041. arXiv:https://doi.org/10.1142/S0218271820500418 , doi:10.1142/S0218271820500418 .URL https://doi.org/10.1142/S0218271820500418 [22] M. Sharif, S. Saba, Gravitational decoupled anisotropic solutions in f ( G )gravity, Eur. Phys. J. C 78 (11) (2018) 921. arXiv:1811.08112 , doi:10.1140/epjc/s10052-018-6406-8 .[23] S. K. Maurya, A. Errehymy, K. N. Singh, F. Tello-Ortiz, M. Daoud,Gravitational decoupling minimal geometric deformation model in mod-21fied f ( R, T ) gravity theory, Phys. Dark Univ. 30 (2020) 100640. arXiv:2003.03720 , doi:10.1016/j.dark.2020.100640 .[24] M. Estrada, A way of decoupling gravitational sources in pure Lovelockgravity, Eur. Phys. J. C 79 (11) (2019) 918, [Erratum: Eur.Phys.J.C 80, 590(2020)]. arXiv:1905.12129 , doi:10.1140/epjc/s10052-019-7444-6 .[25] P. Le´on, A. Sotomayor, Braneworld Gravity under gravitational decoupling,Fortsch. Phys. 67 (12) (2019) 1900077. arXiv:1907.11763 , doi:10.1002/prop.201900077 .[26] C. Las Heras, P. Le´on, New algorithms to obtain analytical solutions ofEinstein’s equations in isotropic coordinates, Eur. Phys. J. C 79 (12) (2019)990. arXiv:1905.02380 , doi:10.1140/epjc/s10052-019-7507-8 .[27] J. Ovalle, Decoupling gravitational sources in general relativity: Theextended case, Phys. Lett. B 788 (2019) 213–218. arXiv:1812.03000 , doi:10.1016/j.physletb.2018.11.029 .[28] M. Sharif, S. Sadiq, Gravitational decoupled anisotropic solutions for cylin-drical geometry, Eur. Phys. J. Plus 133 (6) (2018) 245. doi:10.1140/epjp/i2018-12075-1 .[29] E. Contreras, J. Ovalle, R. Casadio, Gravitational decoupling for axiallysymmetric systems and rotating black holes arXiv:2101.08569 .[30] G. Abell´an, V. A. Torres-S´anchez, E. Fuenmayor, E. Contreras, Regularitycondition on the anisotropy induced by gravitational decoupling in theframework of MGD, Eur. Phys. J. C 80 (2) (2020) 177. arXiv:2001.08573 , doi:10.1140/epjc/s10052-020-7749-5 .[31] G. Abell´an, A. Rincon, E. Fuenmayor, E. Contreras, Beyond classicalanisotropy and a new look to relativistic stars: a gravitational decouplingapproach arXiv:2001.07961 . 2232] M. Sharif, Q. Ama-Tul-Mughani, Anisotropic Spherical Solutions throughExtended Gravitational Decoupling Approach, Annals Phys. 415 (2020)168122. arXiv:2004.07925 , doi:10.1016/j.aop.2020.168122 .[33] M. Sharif, A. Majid, Extended gravitational decoupled solutions in self-interacting Brans-Dicke theory, Phys. Dark Univ. 30 (2020) 100610. arXiv:2006.04578 , doi:10.1016/j.dark.2020.100610 .[34] M. Sharif, A. Majid, Decoupled anisotropic spheres in self-interactingBrans-Dicke gravity, Chin. J. Phys. 68 (2020) 406–418. doi:10.1016/j.cjph.2020.09.015 .[35] M. Sharif, A. Majid, Extended gravitational decoupled solutions in self-interacting Brans-Dicke theory, Phys. Dark Univ. 30 (2020) 100610. arXiv:2006.04578 , doi:10.1016/j.dark.2020.100610 .[36] R. Casadio, J. Ovalle, Brane-world stars and (microscopic) black holes,Phys. Lett. B715 (2012) 251–255. arXiv:1201.6145 , doi:10.1016/j.physletb.2012.07.041 .[37] J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, Anisotropic solutionsby gravitational decoupling, Eur. Phys. J. C 78 (2) (2018) 122. arXiv:1708.00407 , doi:10.1140/epjc/s10052-018-5606-6 .[38] L. Gabbanelli, J. Ovalle, A. Sotomayor, Z. Stuchlik, R. Casadio, A causalSchwarzschild-de Sitter interior solution by gravitational decoupling, Eur.Phys. J. C 79 (6) (2019) 486. arXiv:1905.10162 , doi:10.1140/epjc/s10052-019-7022-y .[39] R. Casadio, E. Contreras, J. Ovalle, A. Sotomayor, Z. Stuchlick, Isotropiza-tion and change of complexity by gravitational decoupling, Eur. Phys.J. C 79 (10) (2019) 826. arXiv:1909.01902 , doi:10.1140/epjc/s10052-019-7358-3 .[40] R. T. Cavalcanti, A. G. da Silva, R. da Rocha, Strong deflection limitlensing effects in the minimal geometric deformation and Casadio-Fabbri-23azzacurati solutions, Class. Quant. Grav. 33 (21) (2016) 215007. arXiv:1605.01271 , doi:10.1088/0264-9381/33/21/215007 .[41] R. da Rocha, Dark SU(N) glueball stars on fluid branes, Phys. Rev.D95 (12) (2017) 124017. arXiv:1701.00761 , doi:10.1103/PhysRevD.95.124017 .[42] R. da Rocha, Black hole acoustics in the minimal geometric deformation ofa de Laval nozzle, Eur. Phys. J. C77 (5) (2017) 355. arXiv:1703.01528 , doi:10.1140/epjc/s10052-017-4926-2 .[43] A. Fernandes-Silva, R. da Rocha, Gregory-Laflamme analysis of MGD blackstrings, Eur. Phys. J. C78 (3) (2018) 271. arXiv:1708.08686 , doi:10.1140/epjc/s10052-018-5754-8 .[44] A. Fernandes-Silva, A. J. Ferreira-Martins, R. Da Rocha, The extendedminimal geometric deformation of SU( N ) dark glueball condensates, Eur.Phys. J. C 78 (8) (2018) 631. arXiv:1803.03336 , doi:10.1140/epjc/s10052-018-6123-3 .[45] R. Da Rocha, A. A. Tomaz, Holographic entanglement entropy un-der the minimal geometric deformation and extensions, Eur. Phys.J. C 79 (12) (2019) 1035. arXiv:1905.01548 , doi:10.1140/epjc/s10052-019-7558-x .[46] R. a. da Rocha, MGD Dirac stars, Symmetry 12 (4) (2020) 508. arXiv:2002.10972 , doi:10.3390/sym12040508 .[47] R. da Rocha, Minimal geometric deformation of Yang-Mills-Dirac stellarconfigurations, Phys. Rev. D 102 (2) (2020) 024011. arXiv:2003.12852 , doi:10.1103/PhysRevD.102.024011 .[48] R. a. da Rocha, A. A. Tomaz, MGD-decoupled black holes, anisotropicfluids and holographic entanglement entropy, Eur. Phys. J. C 80 (9) (2020)857. arXiv:2005.02980 , doi:10.1140/epjc/s10052-020-8414-8 .2449] P. Meert, R. da Rocha, Probing the minimal geometric deformation withtrace and Weyl anomalies arXiv:2006.02564 .[50] R. Casadio, P. Nicolini, R. da Rocha, Generalised uncertainty principleHawking fermions from minimally geometric deformed black holes, Class.Quant. Grav. 35 (18) (2018) 185001. arXiv:1709.09704 , doi:10.1088/1361-6382/aad664 .[51] E. Contreras, P. Bargue˜no, Extended gravitational decoupling in 2 +1 dimensional space-times, Class. Quant. Grav. 36 (21) (2019) 215009. arXiv:1902.09495 , doi:10.1088/1361-6382/ab47e2 .[52] E. Contreras, Minimal Geometric Deformation: the inverse problem, Eur.Phys. J. C78 (8) (2018) 678. arXiv:1807.03252 , doi:10.1140/epjc/s10052-018-6168-3 .[53] E. Contreras, F. Tello-Ortiz, S. K. Maurya, Regular decoupling sector andexterior solutions in the context of MGD, Class. Quant. Grav. 37 (15)(2020) 155002. arXiv:2002.12444 , doi:10.1088/1361-6382/ab9c6d .[54] C. Arias, F. Tello-Ortiz, E. Contreras, Extra packing of mass of anisotropicinteriors induced by MGD, Eur. Phys. J. C 80 (5) (2020) 463. arXiv:2003.00256 , doi:10.1140/epjc/s10052-020-8042-3 .[55] G. Panotopoulos, A. Rinc´on, Minimal Geometric Deformation in a cloudof strings, Eur. Phys. J. C 78 (10) (2018) 851. arXiv:1810.08830 , doi:10.1140/epjc/s10052-018-6321-z .[56] L. Gabbanelli, A. Rinc´on, C. Rubio, Gravitational decoupled anisotropiesin compact stars, Eur. Phys. J. C 78 (5) (2018) 370. arXiv:1802.08000 , doi:10.1140/epjc/s10052-018-5865-2 .[57] S. K. Maurya, F. Tello-Ortiz, Generalized relativistic anisotropic compactstar models by gravitational decoupling, Eur. Phys. J. C 79 (1) (2019) 85. doi:10.1140/epjc/s10052-019-6602-1 .2558] S. K. Maurya, F. Tello-Ortiz, Charged anisotropic compact star in f ( R, T )gravity: A minimal geometric deformation gravitational decoupling ap-proach, Phys. Dark Univ. 27 (2020) 100442. arXiv:1905.13519 , doi:10.1016/j.dark.2019.100442 .[59] S. Hensh, Z. Stuchl´ı k, Anisotropic Tolman VII solution by gravitationaldecoupling, Eur. Phys. J. C 79 (10) (2019) 834. arXiv:1906.08368 , doi:10.1140/epjc/s10052-019-7360-9 .[60] K. N. Singh, S. K. Maurya, M. K. Jasim, F. Rahaman, Minimally deformedanisotropic model of class one space-time by gravitational decoupling, Eur.Phys. J. C 79 (10) (2019) 851. doi:10.1140/epjc/s10052-019-7377-0 .[61] S. K. Maurya, A completely deformed anisotropic class one solutionfor charged compact star: a gravitational decoupling approach, Eu-ropean Physical Journal C 79 (11) (2019) 958. doi:10.1140/epjc/s10052-019-7458-0 .[62] S. K. Maurya, Extended gravitational decoupling (GD) solution for chargedcompact star model, Eur. Phys. J. C 80 (5) (2020) 429. doi:10.1140/epjc/s10052-020-7993-8 .[63] S. K. Maurya, Non-singular solution for anisotropic model by gravi-tational decoupling in the framework of complete geometric deforma-tion (CGD), Eur. Phys. J. C 80 (5) (2020) 448. doi:10.1140/epjc/s10052-020-8005-8 .[64] M. Zubair, H. Azmat, Anisotropic Tolman V Solution by Minimal Grav-itational Decoupling Approach, Annals Phys. 420 (2020) 168248. arXiv:2005.06955 , doi:10.1016/j.aop.2020.168248 .[65] P. Brax, C. van de Bruck, A. C. Davis, Brane world cosmology, bulk scalarsand perturbations, JHEP 10 (2001) 026. arXiv:hep-th/0108215 , doi:10.1088/1126-6708/2001/10/026 .2666] J. Ovalle, R. Casadio, A. Sotomayor, The minimal Geometric DeformationApproach, Advances in High Energy Physics 2017 (2017) 1–10. arXiv:1612.07926 , doi:10.1155/2017/9756914 .[67] R. Casadio, J. Ovalle, R. da Rocha, Classical Tests of General Relativ-ity: Brane-World Sun from Minimal Geometric Deformation, EPL 110 (4)(2015) 40003. arXiv:1503.02316 , doi:10.1209/0295-5075/110/40003 .[68] N. Dadhich, R. Maartens, P. Papadopoulos, V. Rezania, Black holes onthe brane, Phys. Lett. B 487 (2000) 1–6. arXiv:hep-th/0003061 , doi:10.1016/S0370-2693(00)00798-X .[69] J. Ovalle, R. Casadio, R. d. Rocha, A. Sotomayor, Z. Stuchlik, Black holesby gravitational decoupling, Eur. Phys. J. C78 (11) (2018) 960. arXiv:1804.03468 , doi:10.1140/epjc/s10052-018-6450-4doi:10.1140/epjc/s10052-018-6450-4