In the framework of SimpsonVisser the search for field sources that produce black bounces in alternative gravity theories has remained unresolved. In this paper the first in a series exploring sources for alternative theories of gravity we identify such a source for the 21 dimensional Kgravity black bounce. The Kgravity black hole is notable for allowing asymptotically locally flat solutions in lowerdimensional spacetime yet it possesses curvature singularities concealed within the event horizon. Using the SimpsonVisser regularization technique we eliminate this singularity constructing asymptotically locally flat blackbounce solutions in 21 dimensions. We explore the causal structure of these solutions identifying the conditions under which they describe regular black holes or wormholes. By calculating curvature invariants we confirm the absence of singularities within the event horizon. Additionally we demonstrate that beyond nonlinear electrodynamics a nonlinear scalar field is required to source the solution. Finally we investigate the geodesic structure of this spacetime analyzing the trajectories of both massive and massless particles. We also confirm the existence of circular orbits and assess their stability.
['General Relativity and Quantum Cosmology']
Field Sources for fRRmunu BlackBounce Solutions The Case of KGravity
{'Field Sources for f ( R, R µν ) Black-Bounce Solutions: The Case of K-Gravity': "G. Alencar, 1, ∗ M. Nilton, 2, † Manuel E. Rodrigues, 3, 4, ‡ and Marcos V. de S. Silva 1, §  \n1 Departamento de F'ısica, Universidade Federal do Cear'a, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Cear'a, Brazil 2 Universidade Federal do Cear'a Campus de Russas, 62900-420, Russas, Cear'a, Brazil 3 Faculdade de F'ısica, Programa de P'os-Gradua¸c˜ao em F'ısica, Universidade Federal do Par'a, 66075-110, Bel'em, Par'a, Brazill 4 Faculdade de Ciˆencias Exatas e Tecnologia, Universidade Federal do Par'a, Campus Universit'ario de Abaetetuba, 68440-000, Abaetetuba, Par'a, Brazil  \n(Dated: September 19, 2024)", 'Abstract': 'In the framework of Simpson-Visser, the search for field sources that produce black bounces in alternative gravity theories has remained unresolved. In this paper, the first in a series exploring sources for alternative theories of gravity, we identify such a source for the 2 + 1 dimensional K-gravity black bounce. The K-gravity black hole is notable for allowing asymptotically locally flat solutions in lowerdimensional spacetime, yet it possesses curvature singularities concealed within the event horizon. Using the Simpson-Visser regularization technique, we eliminate this singularity, constructing asymptotically locally flat black-bounce solutions in 2+ 1 dimensions. We explore the causal structure of these solutions, identifying the conditions under which they describe regular black holes or wormholes. By calculating curvature invariants, we confirm the absence of singularities within the event horizon. Additionally, we demonstrate that, beyond non-linear electrodynamics, a non-linear scalar field is required to source the solution. Finally, we investigate the geodesic structure of this spacetime, analyzing the trajectories of both massive and massless particles. We also confirm the existence of circular orbits and assess their stability.', 'I. INTRODUCTION': "General Relativity (GR) has garnered significant interest in recent years, driven by advancements in high-precision measurements of phenomena like black holes and cosmology. This period has witnessed milestones such as the direct detection of gravitational waves by LIGO and VIRGO, the image of supermassive black holes by the Event Horizon Telescope (EHT), and precise assessments of the Λ CDM parameters [1-7]. However, the interior of black holes presents a challenge due to spacetime singularities.  \nA valuable avenue for probing black hole characteristics is through its 2 + 1 dimensional counterpart, introduced in 1992 by Banados, Teitelboim, and Zanelli [8]. In the 2 + 1 Einstein-Hilbert theory, the classical level presents a trivial scenario devoid of gravitational waves, where any two solutions are locally equivalent. Interestingly, the 2 + 1 dimensional vacuum lacks local degrees of freedom [9], making black holes unexpected since a vacuum solution in 2 + 1 dimensions is typically flat. However, by introducing a cosmological constant with Λ < 0, Banados, Teitelboim, and Zanelli uncovered the renowned BTZ black holes [8].  \nIn the context of GR, it is possible to avoid the singularity problem through the models known as regular black hole [10] (RBH). These spacetimes do not present singularities inside them, in such a way that the geodesics are not interrupted [11]. The first model of a RBH was proposed by Bardeen and was not, until then, a solution to the field equations of gravitational theories. Beato and Garcia demonstrated that the Bardeen solution could be obtained in GR if a non-linear electrodynamics (NED) were considered as the source [12, 13]. These black holes, which undergo a regularization process that allows r = 0 to be a regular point, similar to Bardeen's model, are extensively studied in the literature [14-19]. They violate the strong energy condition in some region within the event horizon [20] and can be derived from the coupling between GR and a NED, at least in static cases [11].  \nIn another direction, Simpson and Visser proposed a different method to avoid the singularity problem [21] by introducing a regularization parameter a 2 . Depending on the value of a , the four-dimensional spacetime could represent a traversable wormhole (TWH), a one-way wormhole (OWWH) with an extremal null throat at r = 0, or a RBH. This type of solution is known as a black-bounce (BB). This method has been recently used to regularize cylindrical black holes, such as BTZ and 4 d black strings [22-27]. There are also some models of BB that are not derived from the Simpson-Visser regularization method [28-30].  \nAn important point in the solutions of BB, within the context of GR, is that for this type of  \nspacetime to be a solution to Einstein equations, it is necessary to couple the gravitational theory with a NED and a phantom scalar field. Neither the scalar field alone nor the NED by itself are capable of generating these objects in GR [24, 26, 27, 31-33]. In some cases, in alternative theories of gravity, it becomes simpler to consider anisotropic fluids rather than trying to determine the specific source of these fluids [34-37].  \nRecently, Bronnikov proposed a new regularization method. In this method, the singular region of spacetime is not replaced by a wormhole at its center. Instead, the solution is regularized by simply 'curing' the singularity [38, 39]. Depending on how the 'cure' is applied, the source of these solutions can be either NED alone or a combination of NED and a scalar field.  \nA prominent three dimensional GR modification is known as 'topologically massive gravity' (TMG). This theory extends the Einstein-Hilbert (EH) action (plus the cosmological constant) by incorporating a Lorentz-Chern-Simons (LCS) term [40], leading to the following equation:  \nσG µν + 1 µ C µν +Λ 0 g µν = 0 , (1)  \nwhere σ is a sign (plus for GR and minus for TMG). The TMG violates parity, however, recently, a novel formulation of massive gravity that is parity preserving was proposed by Bergshoeff, Hohm and Townsend (BHT). They added to the EH action a quadratic term, to obtain  \nS BHT = 1 κ 2 ∫ d 3 x { √ | g | [ R -1 m 2 K -2 λ ]} , (2)  \nwhere g is the determinant of the metric g µν , m is a 'relative' mass parameter, which could be traded for the effective dimensionless coupling constant mκ 2 , and K = R µν R µν -3 8 R 2 . This model circumvented the Boulware-Deser ghost and was shown to be unitary [41, 42]. Soon later, a unique maximally symmetric solution was found, given by [43]  \nds 2 = f ( r ) dt 2 -f ( r ) -1 dr 2 -r 2 dφ 2 , f ( r ) = -Λ r 2 + br -µ, (3)  \nwhere Λ = 2 λ and b and µ are constants. With b = 0, the solution corresponds to the static BTZ black hole with a twice mass parameter. The authors studied the AdS solution with Λ = -1 /l 2 . However, in Ref. [44], an asymptotically locally flat 1 black hole solution was found, with Λ = 0. This is remarkable since it is generally believed that only AdS solutions are allowed in three dimensions. Further properties were studied in Refs. [45, 46].  \nFor the asymptotically locally flat solution, with Λ = 0, we have  \nf ( r ) = br -µ. (4)  \nWe see that we have a black hole horizon only in the case µ > 0. Computing the mass parameter, the curvature scalar and the Kretschmann scalar, 2 we find  \nM = 1 + µ 4 G , R = 2 b r , K = 2 b 2 r 2 . (5)  \nAlthough the solution does not directly recover Minkowski at infinity, by the form of these scalars, we can see that the solution is asymptotically locally flat and has a singularity at r = 0. The singularity is hidden by the presence of the event horizon, located at r h = µ/b .  \nFor Λ = -1 /l 2 , we have an AdS solutions, with mass parameter and Ricci scalar given by  \nr ± = l 2 2 ( -b ± √ b 2 + 4 µ l 2 ) , M = 1 + µ 4 G , R = 6 l 2 + 2 b r , K = 4 l 4 + 2 r 2 ( b + 2 r l 2 ) 2 . (6)  \nWe must have µ ≥ -1 4 b 2 l 2 and since the mass is solely determined by µ , the constant b can be interpreted as a type of 'gravitational hair.' This is further supported by the fact that the black hole does not possess any additional global charges arising from asymptotic symmetries [44]. In contrast, introducing the gravitational hair parameter b causes a curvature singularity to form at the origin, as illustrated in equation (6). We point that (6) is very similar to the ReissnerNordstrom(RN) case and a Cauchy horizon can appear. However, now, b and µ can have negative ranges which results in a naked singularity, one or two horizons. We describe in detail below:  \n- 1. b > 0: For µ < 0 we have no horizon and a naked singularity. For µ ≥ 0 we have a single event horizon located at r = r + .\n- · For µ > 0, the event horizon surrounds a spacelike singularity.\n- · For µ = 0, the horizon coincides with a null singularity.\n- 2. b < 0 : The singularity is surrounded by an event horizon and there is also the possibility to a Cauchy horizon.\n- · For µ > 0, there is a single event horizon at r = r + , enclosing a spacelike singularity.\n- · For µ = 0 we have a event horizon and an inner horizon appears on top of a null singularity.  \n- · For -1 4 b 2 l 2 < µ < 0, we have an inner Cauchy horizon at r = r -, surrounding a timelike singularity, and an event horizon located at r = r + .\n- · For µ = -b 2 l 2 / 4, both horizons coincide, enclosing a timelike singularity.  \nThis way, we see that asymptotically locally flat solutions in 2+1 dimensions are possible and hold significant theoretical interest. Therefore, studying regularized versions of these solutions becomes very interesting, as they no longer exhibit pathologies such as naked singularities.  \nHowever, up to now, field sources for BB of alternative theories of gravity, such as f ( R, R µν ), has never been found. Therefore, this is the first in a series of papers where we propose a method to find such sources. The structure of this article is organized as follows: In Sec. II, we present the asymptotically locally flat solution in 2+1 dimensions and discuss the properties of the regularized spacetime. The field sources are studied in Sec. III .In Sec. IV, we examine which of the energy conditions are violated. In Sec. V, we analyze the trajectories of both massive and massless particles in the regularized spacetime. Our conclusions and perspectives are presented in Sec. VI.", 'II. THE REGULARIZED SOLUTION': 'Just as has been done with several solutions in the literature [47], we can apply the SimpsonVisser regularization method to eliminate the presence of the singularity. The regularized solution is  \nds 2 = f ( r ) dt 2 -f ( r ) -1 dr 2 -Σ 2 ( r ) dϕ 2 , (7)  \nwith  \nf ( r ) = -Λ( r 2 + a 2 ) + b √ r 2 + a 2 -µ, and Σ( r ) = √ r 2 + a 2 . (8)  \nFrom now on, we will analyze the locally flat and AdS case separately.', 'A. Asymptotically locally flat black hole': 'For Λ = 0, the regularized solution is  \nf ( r ) = b √ r 2 + a 2 -µ, and Σ( r ) = √ r 2 + a 2 . (9)  \nThe curvature invariants to the regularized spacetime are  \nR = b (3 a 2 +2 r 2 ) ( a 2 + r 2 ) 3 / 2 -2 a 2 µ ( a 2 + r 2 ) 2 , (10)  \nK = -4 a 2 µb (2 a 2 + r 2 ) ( a 2 + r 2 ) 7 / 2 + 4 a 4 µ 2 ( a 2 + r 2 ) 4 + b 2 (5 a 4 +4 a 2 r 2 +2 r 4 ) ( a 2 + r 2 ) 3 . (11)  \nThe invariants become more complex than in the singular case, but if we analyze the asymptotic behaviors, we observe that to r → 0 we find that both R and K tend to a constant. To r →∞ we find that R ∝ r -1 and K ∝ r -2 . It means that the spacetime has no curvature singularities and is asymptotically locally flat.  \nSolving f ( r ) = 0, we find that the radius of the event horizon is given by r h = √ µ 2 /b 2 -a 2 . Depending on the values chosen for the parameter a , the properties of the spacetime structure can be altered. In summary, we have that:  \n- · For a < µ/b , we have a RBH with two horizons. At r = 0, the circumference of this solution, C = 2 π √ r 2 + a 2 , has a minimum, which is precisely the BB.\n- · If a = µ/b , the event horizon and the minimum circumference are located at the same point, and in this way, we have a black throat.\n- · If a > µ/b , there are no longer any horizons, and thus we have a TWH in 2 + 1 dimensions with the throat located at r = 0.', 'B. Asymptotically AdS black hole': 'In the case we have Λ = -1 /l 2 , the regularized solution is given by  \nf ( r ) = 1 l 2 ( r 2 + a 2 ) + b √ r 2 + a 2 -µ, and Σ( r ) = √ r 2 + a 2 . (12)  \nThe curvature invariants to the regularized spacetime are  \nR = 4 a 2 +6 r 2 l 2 ( a 2 + r 2 ) + b (3 a 2 +2 r 2 ) ( a 2 + r 2 ) 3 / 2 -2 a 2 µ ( a 2 + r 2 ) 2 , (13) K = 4 (2 a 4 +4 a 2 r 2 +3 r 4 ) l 4 ( a 2 + r 2 ) 2 -8 a 2 µ l 2 ( a 2 + r 2 ) 2 + 4 b (3 a 4 +4 a 2 r 2 +2 r 4 ) l 2 ( a 2 + r 2 ) 5 / 2 -4 a 2 µb (2 a 2 + r 2 ) ( a 2 + r 2 ) 7 / 2 + 4 a 4 µ 2 ( a 2 + r 2 ) 4 + b 2 (5 a 4 +4 a 2 r 2 +2 r 4 ) ( a 2 + r 2 ) 3 . (14)  \nThe invariants become more complex than in the singular case, but if we analyze the asymptotic behaviors, we observe that to r → 0 we find that both R and K tend to a constant. To r → ∞  \nwe find that R ∝ 6 l 2 + 2 br -1 and K ∝ 12 l 4 + 8 b l 2 r . It means that the spacetime has no curvature singularities and is asymptotically AdS.  \nSolving f ( r ) = 0, we find four possibilities that are given by  \nr = ± √ √ √ √ ( bl 2 2 ± √ b 2 l 4 4 + l 2 µ ) 2 -a 2 . (15)  \nAs said above, if b and µ are positive, the structure is identical to that of charged BB spacetime studied in Ref. [47]. However, the fact that the parameters can be negative will give rise to new possibilities, described below:  \n- 1. b > 0 and µ ≤ 0: There is no event horizon and we have a TWH with a throat at r = 0.\n- 2. b > 0 and µ > 0: This analysis is identical to the charged BB, but we repeat here for completes.\n- · For a > bl 2 / 2 ± √ b 2 l 4 / 4 + l 2 µ , we have no horizon. However, since there is no singularity, we have a TWH.\n- · For a = bl 2 / 2 ± √ b 2 l 4 / 4 + l 2 µ , the event horizon and the minimum circumference are located at the same point, and in this way, we have a black throat.\n- · For a < bl 2 / 2 ± √ b 2 l 4 / 4 + l 2 µ , we have a wormhole (WH) throat inside a event horizon.\n- 3. b < 0 and µ = 0: We have a event horizon located at r = ± √ b 2 l 4 -a 2 .\n- 4. b < 0 and µ > 0: Depending on the value of a , we have one or no event horizon.\n- 5. b < 0 and -b 2 l 2 / 4 < µ < 0: The WH throat is surrounded by a Cauchy horizon and we also have a event horizon.\n- 6. b < 0 and -b 2 l 2 / 4 = µ : The WH throat is surrounded by an extremal horizon.\n- 7. b < 0 and µ < -b 2 l 2 / 4: There is no horizon and we have a TWH with a throat at r = 0.', 'III. FIELD SOURCES': "As the singular solution arise in the context of a alternative theory of gravity, we will consider the field equations from the quadratic gravity. The regular spacetime is proposed in a generic manner without considering the gravitational theory that generates it. Let's consider that this  \nmetric is a solution of the K -gravity in 2 + 1 dimensions and determine the material content that can generate it. To do this, we will consider the theory described by the action:  \nS = ∫ √ | g | d 3 x [ K -2 h ( φ ) g µν ∂ µ φ∂ ν φ +2 V ( φ ) + L ( F )] , (16)  \nwhere φ is the scalar field, V ( φ ) is its potential, L ( F ) is the NED Lagrangian, F = F µν F µν is the electromagnetic scalar, and F µν = ∂ µ A ν -∂ ν A µ is the electromagnetic field tensor. The function h ( φ ) will determine if the scalar field is phantom, h ( φ ) < 0, or standard, h ( φ ) > 0.  \nVarying the action (16) with respect to φ , A µ , and g µν , we obtain the field equations  \n∇ µ [ L F F µν ] = 1 | g | ∂ µ [ √ | g | L F F µν ] = 0 , (17)  \n√ 2 h ( φ ) ∇ µ ∇ µ φ + dh ( φ ) dφ ∂ µ φ∂ µ φ = -dV ( φ ) dφ , (18)  \nK µν = 2 /square R µν -1 2 ∇ µ ∇ ν R -1 2 g µν /square R +4 R µρνσ R ρσ -g µν R ρσ R ρσ -3 2 RR µν + 3 8 g µν R 2 = T [ φ ] µν + T [ F ] µν , (19)  \nwhere L F = dL/dF , T [ φ ] µν and T [ F ] µν are the stress-energy tensors of the scalar and electromagnetic fields, respectively,  \nT [ F ] µν = 1 2 g µν L ( F ) -2 L F F ν α F µα , (20)  \nT [ φ ] µν = 2 h ( φ ) ∂ ν φ∂ µ φ -g µν ( h ( φ ) ∂ α φ∂ α φ -V ( φ ) ) . (21)  \nF 10 = qL -1 F √ r 2 + a 2 , (22)  \nF = -2 q 2 L 2 F ( r 2 + a 2 ) . (23)  \nUsing the 1 -1 and 2 -2 components of the gravitational field equations, we can obtain that the electromagnetic quantities are given by  \nL ( r ) = -1 4 f '' 2 + f ' ( f (3) -4 f ' Σ '' Σ ) + f ( 8 f ' Σ ' Σ '' +Σ ( f (4) ( r )Σ -4 f '' Σ '' -8Σ (3) f ' -2Σ hφ ' 2 )) Σ 2 + f 2 ( Σ ( 3Σ '' 2 -2ΣΣ (4) ) +4ΣΣ (3) Σ ' -4Σ ' 2 Σ '' ) Σ 3 -2 V, (24)  \nL F ( r ) = 8 q 2 Σ { Σ 2 ( Σ f '' 2 +12 f ' 2 Σ '' -f ' ( 3 f (3) Σ+ f '' Σ ' )) + 2 f Σ ( Σ ( -f (4) Σ+ f (3) Σ ' +3 f '' Σ '' +9Σ (3) f ' +4Σ hφ ' 2 ) -12 f ' Σ ' Σ '' ) + 4 f 2 ( Σ ( ΣΣ (4) -Σ '' 2 ) -3ΣΣ (3) Σ ' +3Σ ' 2 Σ '' )} -1 . (25)  \nSolving the Maxwell equations, considering an electrically charged solution, we find that the electric field has the form  \nand the scalar F is  \nFigure 1. Behavior of the function h ( φ ) as a function of the radial coordinate for different values of the regularization parameter.  \n<!-- image -->  \nTo obtain the electromagnetic expressions, we must first derive the quantities associated with the scalar field. Solving the component 0 -0 of the gravitational field equations, it's possible to find h ( φ ) φ ' 2 in terms of the metric functions. Following the procedure developed by [48], we will choose the scalar field as a monotonic function  \nφ = arctan ( r a ) , (26)  \nand then we solve the component 0 -0 of the gravitational field equations to find h ( φ ), that is given by  \nh ( φ ) = a 2 ( 5 µ -b √ a 2 + r 2 ) -12 µr 2 2 ( a 2 + r 2 ) 2 . (27)  \nIn Fig. 1, we observe the behavior of the function h ( φ ) and notice that it changes sign depending on the region. As a result, the scalar field is canonical in some regions and a phantom in others, making it a partly phantom scalar field.  \nWe still need to determine V ( r ). To do this, we must solve the equation (18), which results in  \nV ( r ) = a 2 bµ (197 a 2 -168 r 2 ) 70 ( a 2 + r 2 ) 7 / 2 -a 2 (2 a 4 b 2 + a 2 (2 b 2 r 2 +19 µ 2 ) -32 µ 2 r 2 ) 8 ( a 2 + r 2 ) 4 . (28)  \nOnce we have the expressions related to the scalar field, we can obtain the electromagnetic quan-  \ntities, which are  \nL ( r ) = 264 a 2 bµ (7 r 2 -3 a 2 ) 35 ( a 2 + r 2 ) 7 / 2 + a 2 (15 µ 2 (5 a 2 -16 r 2 ) + b 2 (13 a 4 + a 2 r 2 -12 r 4 )) 4 ( a 2 + r 2 ) 4 , (29)  \nL F ( r ) = -8 q 2 ( a 2 + r 2 ) 4 a 2 ( b 2 (7 a 2 -8 r 2 ) ( a 2 + r 2 ) + 36 µ 2 ( a 2 -4 r 2 )) -44 a 2 bµ ( a 2 -3 r 2 ) √ a 2 + r 2 . (30)  \nDespite being obtained as independent functions, the electromagnetic quantities are related to each other and must satisfy the consistency relation  \nL F -dL dr ( dF dr ) -1 = 0 , (31)  \nwhich is indeed satisfied.  \nSince we are dealing with electrically charged sources, it is not possible to write the expression for L ( F ) analytically. However, we can still explicitly express V ( φ ) and h ( φ ), which are given by:  \nV ( φ ) = a 4 bµ (197 -168 tan 2 φ ) 70 ( a 2 sec 2 φ ) 7 / 2 -cos 6 φ (4 a 2 b 2 +51 µ 2 cos 2 φ -13 µ 2 ) 16 a 4 , (32)  \nh ( φ ) = -cos 2 φ ( 2 b cos 2 φ √ a 2 sec 2 φ -17 µ cos 2 φ +7 µ ) 4 a 2 . (33)  \nIn this way, we were able to find the quantities associated with the matter fields of the source such that all the field equations are satisfied.", 'IV. ENERGY CONDITIONS': 'In this section, we will analyze the energy conditions of the regularized asymptotically locally flat solution. The field equations of the purely quadratic gravity can be rewritten as  \n2 /square R µν -1 2 ∇ µ ∇ ν R -1 2 g µν /square R +4 R µρνσ R ρσ -g µν R ρσ R ρσ -3 2 RR µν + 3 8 g µν R 2 = T µν , (34)  \nwhere T µν is the stress-energy tensor.  \nIn order to analyze the energy conditions associated with the regularized solution, we rewrite the stress-energy tensor considering that the matter is described by an anisotropic fluid. In regions where f ( r ) > 0, we can write the stress-energy tensor as  \nT µ ν = diag [ ρ, -p r , -p t ] , (35)  \nwhere ρ is the energy density, p r is the radial pressure, and p t is the tangential pressure. In regions where f ( r ) < 0, the stress-energy tensor is written as  \nT µ ν = diag [ -p r , ρ, -p t ] . (36)  \nFrom the field equations (34), we can immediately obtain the expressions for the fluid quantities. We have for f ( r ) > 0:  \nρ = ( 4 a 2 ( b 2 r 4 √ a 2 + r 2 +32 µ 2 r 2 √ a 2 + r 2 -31 bµr 4 ) -a 6 b ( 11 b √ a 2 + r 2 -48 µ ) -a 4 ( 7 b 2 r 2 √ a 2 + r 2 +36 µ 2 √ a 2 + r 2 +72 bµr 2 ) -4 bµr 6 ) ( 4 -1 ( a 2 + r 2 ) -9 / 2 ) , (37)  \n-p r = ( 4 a 2 ( 11 b 2 r 4 √ a 2 + r 2 +40 µ 2 r 2 √ a 2 + r 2 -49 bµr 4 ) +4 br 6 ( 2 b √ a 2 + r 2 -3 µ ) + a 6 b ( 56 µ -27 b √ a 2 + r 2 ) + a 4 ( 9 b 2 r 2 √ a 2 + r 2 -28 µ 2 √ a 2 + r 2 -128 bµr 2 )) × ( 4 -1 ( a 2 + r 2 ) -9 / 2 ) , (38)  \np t = ( 4 a 2 ( 5 b 2 r 4 √ a 2 + r 2 +8 µ 2 r 2 √ a 2 + r 2 -10 bµr 4 ) +8 b 2 r 6 √ a 2 + r 2 + a 6 b ( b √ a 2 + r 2 +4 µ ) + a 4 ( 13 b 2 r 2 √ a 2 + r 2 -4 µ 2 √ a 2 + r 2 -36 bµr 2 )) ( 4 -1 ( a 2 + r 2 ) -9 / 2 ) . (39)  \nThe above expressions for energy density and radial pressure must be exchanged if f ( r ) < 0. The analysis will become more clearer if we introduce a new radial coordinate ˜ r 2 = r 2 + a 2 . The fluid quantities written in terms of ˜ r are given by:  \nρ (˜ r ) = 1 4˜ r 8 [ -4 b ˜ r 5 µ +4 a 2 b 2 ˜ r 4 -112 a 2 b ˜ r 3 µ + ˜ r 2 ( -15 a 4 b 2 +128 a 2 µ 2 ) + 164 a 4 b ˜ rµ -164 a 4 µ 2 ] , (40)  \np r (˜ r ) = 1 4˜ r 8 [ -8 b 2 ˜ r 6 +12 b ˜ r 5 µ -20 a 2 b 2 ˜ r 4 +160 a 2 b ˜ r 3 µ + ˜ r 2 (55 a 4 b 2 -160 a 2 µ 2 ) -228 a 4 b ˜ rµ +188 a 4 µ 2 ] , (41)  \np t (˜ r ) = 1 4˜ r 8 [ 8 b 2 ˜ r 6 -4 a 2 b 2 ˜ r 4 -40 a 2 b ˜ r 3 µ + ˜ r 2 ( -3 a 4 b 2 +32 a 2 µ 2 ) + 44 a 4 b ˜ rµ -36 a 4 µ 2 ] . (42)  \nOnce we have identified the fluid quantities, the energy conditions associated with the stressenergy tensor are given by  \nNEC 1 , 2 = WEC 1 , 2 = SEC 1 , 2 ⇐⇒ ρ + p r,t ≥ 0 , (43)  \nSEC 3 ⇐⇒ ρ + p r + p t ≥ 0 , (44)  \nDEC 1 , 2 ⇐⇒ ρ -| p r,t | ≥ 0 ⇐⇒ ( ρ + p r,t ≥ 0) and ( ρ -p r,t ≥ 0) , (45)  \nDEC 3 = WEC 3 ⇐⇒ ρ ≥ 0 . (46)  \nAs part of the dominant energy condition is already in the null energy condition, we will consider only DEC 1 , 2 = ⇒ ρ -p r,t ≥ 0. In the notation we are following, the conditions NEC , WEC , DEC , and SEC are the null, weak, dominant, and strong energy conditions, respectively. The subscript 1 is used when the energy density with radial pressure is considered, subscript 2 when the energy density with tangential pressure is used. The subscript 3 is used when only the energy density is considered, as in the case of WEC 3 , or the energy density with both pressures, as in the case of SEC 3 .  \nFrom the equations (40), (41), and (42), we can analyze the validity of each of energy conditions mentioned above. The WEC 3 is given by the following restriction:  \nρ (˜ r ) = 1 4˜ r 8 [ -4 b ˜ r 5 µ +4 a 2 b 2 ˜ r 4 -112 a 2 b ˜ r 3 µ -˜ r 2 (15 a 4 b 2 -128 a 2 µ 2 ) + 164 a 4 b ˜ rµ -164 a 4 µ 2 ] ≥ 0 . (47)  \nAs we can see, the energy density is an eighth-degree polynomial in the denominator and a fifthdegree polynomial in the numerator. Since the denominator is always positive, it will be the numerator that will determine the sign of ρ (˜ r ). However, the positivity of this function will depend on the relationship between these parameters a , b , and µ . Therefore, we will consider the cases a < µ/b , where the metric (7) will describe a RBH with two horizons, a = µ/b , describing a RBH with a black throat, and finally the case a > µ/b , describing a WH with throat located in r = 0 (˜ r = a ). The behavior of the polynomial in the numerator of the expression for the energy density for different values of the parameter a can be seen in the Fig. 2. As we can note, the energy density will be negative when we have a RBH with two horizons. Furthermore, although there may be regions where ρ ≥ 0 when the metric describes a WH, these regions are limited to be near the throat of the WH. Therefore, we can conclude that the WEC 3 is not satisfied. A source of energy density originates from a type of matter known as exotic matter. Exotic matter is particularly important in the context of WHs, as according to the general relativity, Morris-Thorne WHs would require this type of matter to exist. We can observe that exotic matter capable of supporting such spacetimes can also appear in K -gravity.  \nNext, we will check the validity of the NEC 1 , 2 = WEC 1 , 2 = SEC 1 , 2 . We must have for the WEC 1 :  \nρ + p r = 1 ˜ r 8 [ -2 b 2 ˜ r 6 +2 b ˜ r 5 µ -4 a 2 b 2 ˜ r 4 +12 a 2 b ˜ r 3 µ -2˜ r 2 a 2 (5 a 2 b 2 -4 µ 2 ) -16 a 4 b ˜ rµ +6 a 4 µ 2 ] ≥ 0 , (48)  \nand for WEC 2 :  \nρ + p l = 1 2˜ r 8 [ 4 b 2 ˜ r 6 -2 b ˜ r 5 µ -76 a 2 b ˜ r 3 µ + ˜ r 2 a 2 ( -9 a 2 b 2 +80 µ 2 ) + 104 a 4 b ˜ rµ -100 a 4 µ 2 ] ≥ 0 . (49)  \nFigure 2. The numerator of the energy density considering different values of the parameter a .  \n<!-- image -->  \nAgain, we see that the sign of each of these expressions will be determined by the polynomial of their denominators, which, in turn, will depend on the relationship between the parameters a , b and µ . As we can see by the Fig. 3, the conclusions will be essentially the same as for WEC 3 , that is, the WEC 1 will not be satisfied when the metric describe a RBH, and is satisfied only in regions near the throat when the metric describes a WH. Therefore, we can conclude that the WEC 1 is not also satisfied. However, as seen in Fig. 4, a very interesting result is that WEC 2 is always satisfied for ab < µ , and although is not satisfied for regions close to the throat when we have a WH, it will be satisfied as we move away from the WH.  \nNow, we will check if the SEC 3 is satisfied. The strong energy condition is satisfied if we have  \nρ + p r + p t = 1 4˜ r 8 [ 8 b ˜ r 5 µ -20 a 2 b 2 ˜ r 4 +8 a 2 b ˜ r 3 µ +37 a 4 b 2 ˜ r 2 -20 a 4 b ˜ rµ -12 a 4 µ 2 ] ≥ 0 . (50)  \nWe can see the validity zone of SEC 3 in Fig. 5 for different values of the parameter a . This result is particularly interesting, because when we have a RBH, a < µ/b , the strong energy condition is always satisfied, but, for the case where ab ≥ µ , we can see that the SEC 3 is satisfied only when the parameter a does not reach a limit value, when a exceeds a certain value the strong condition is no longer satisfied. Therefore, we can conclude that the strong energy condition is validy when the metric describes a RBH or a BH with black throat ( ab = µ ). But for WHs ( ab > µ ) it will only be satisfied when parameter a is not greater than its limiting value.  \nFigure 3. Validity of the WEC 1 considering different values of the parameter a .  \n<!-- image -->  \nFigure 4. Validity of the WEC 2 considering different values of the parameter a .  \n<!-- image -->  \nFinally, we will consider the dominant energy conditions DEC 1 and DEC 2 . We can immediately notice that the DEC 1 , 2 encompasses the WEC 1 , 2 , once for the inequality ρ -| p r,t | ≥ 0 to be true we must have ρ + p r,t ≥ 0 which is precisely the WEC 1 , 2 . As we have shown previously, the WEC 1 is not satisfied, and therefore the DEC 1 will not be valid either. Then, let us focus just on DEC 2 . Since the inequality ρ + p t ≥ 0 is already satisfied, we just need check the validity of ρ -p t ≥ 0, which is written by:  \nρ -p t = 1 ˜ r 8 [ -2 b 2 ˜ r 6 -b ˜ r 5 µ +2 a 2 b 2 ˜ r 4 -18 a 2 b ˜ r 3 µ -˜ r 2 a 2 (3 a 2 b 2 -24 µ 2 ) + 30 a 4 b ˜ rµ -32 a 4 µ 2 ] ≥ 0 . (51)  \nThe Fig. 6 shows the behavior of the polynomial in the numerator of the expression ρ -p t .  \nFigure 5. Validity of the SEC 3 considering different values of the parameter a .  \n<!-- image -->  \nFigure 6. Validity of the DEC 2 considering different values of the parameter a .  \n<!-- image -->  \nAs we can see, the value of the parameter a does not significantly affect the result. Furthermore, since the polynomial will be negative, we observe that the DEC 2 is also not satisfied  \nIn this section, we have examined the energy conditions of the regularized asymptotically locally flat solution in the context of purely quadratic gravity. The analysis revealed that the WEC 1 is not satisfied when the metric describes an RBH. However, the WEC 2 is always satisfied for ab < µ and becomes valid as one moves away from the WH throat. Also, the WEC 3 will not be valid once the energy density is not universally positive. For a RBH with two horizons, the energy density can be negative, and while it may be non-negative near the throat of a WH, it does not satisfy WEC 3 in general. On the other hand, we obtain the interesting result that the SEC 3 is valid when the metric describes an RBH only from a certain value of a , while always holds in the limit of a black hole or a WH where ab ≥ µ . Finally, both DEC 1 , 2 fails to be satisfied. Overall, the analysis emphasizes that the regularized solution requires exotic matter to meet certain energy conditions, especially in the context of WHs. This exotic matter is crucial for supporting such spacetimes, as indicated by the conditions being satisfied only under specific circumstances or parameter ranges.', 'V. GEODESICS': 'In order to study the trajectories of particles in the regularized spacetime, we can rewrite the line element of our regularized solution as  \nds 2 = A ( r ) dt 2 -C ( r ) dr 2 -D ( r ) dϕ 2 . (52)  \nThe trajectories of particles in this spacetime can be obtained through the Euler-Lagrange equations, where the Lagrangian associated with the line element is given by [49]:  \n2 L = ˙ s 2 = A ˙ t 2 -C ˙ r 2 -D ˙ ϕ 2 , (53)  \nwhere the dot represents the derivative with respect to the affine parameter λ . The equations of motion are  \nA ˙ t = E, (54)  \nA ˙ t 2 -C ˙ r 2 -D ˙ ϕ 2 = δ, (55)  \nD ˙ ϕ = /lscript, (56)  \nwhere E and /lscript are constants of motion that represent the energy and the angular momentum of the particle [49]. To massive particles we have δ = 1, and δ = 0 to massless particles.', 'A. Massless particles': "For massless particles, we can consider the initially presented line element, equation (7). Using the equations of motion to A ( r ) = C ( r ) -1 = f ( r ) = b √ r 2 + a 2 -µ and D ( r ) = Σ 2 ( r ) = r 2 + a 2 , we obtain  \nE 2 = ˙ r 2 + ( b √ r 2 + a 2 -µ ) /lscript 2 r 2 + a 2 . (57)  \nThis is a type of energy conservation equation. It is important to note that ˙ r represents the radial velocity of the particle. We may rewrite this equation as  \n˙ r 2 = E 2 -V eff . (58)  \nThe effective potential to massless particles is given by  \nV eff = l 2 ( b √ a 2 + r 2 -µ ) a 2 + r 2 . (59)  \n<!-- image -->  \nFigure 7. Effective potential for massless particles considering different values of the parameter a .  \n<!-- image -->  \nThrough the first and second derivatives, we can assess whether this potential allows for the existence of stable or unstable orbits. From the condition that V ' ( r ) = 0, we obtain that the potential has three extrema, which are given by  \nr 1 = √ 4 µ 2 -a 2 b 2 b , r 2 = -√ 4 µ 2 -a 2 b 2 b , and r 3 = 0 . (60)  \nIf a < 2 µ/b , we will have three possible orbits. For a ≥ 2 µ/b , we have only one orbit at r = 0.  \nIn Fig. 7, we show the effective potential for massless particles. If ba < µ the maximum is located at r = r 1 . If µ ≤ ba < 2 µ , the maximum are located at r = r 1 and r = r 2 , and the minimum at r = r 3 . If ba ≥ 2 µ , there is only one unstable orbit at r = r 3 .  \nWe can also calculate the radial acceleration, that is  \nr = -1 2 dV eff dr = /lscript 2 r ( a 2 b -2 µ √ a 2 + r 2 + br 2 ) 2 ( a 2 + r 2 ) 5 / 2 . (61)  \nIf we calculate the radial acceleration in the extrema points of the effective potential, we find r ( r 1 , r 2 , r 3 ) = 0. At these points, the radial velocity of the particle is also zero. This means that the particle follows a circular orbit around the black hole.  \nIf we consider that the particle has an energy E c , which is equal to the maximum of the effective potential, and using the equations of motion, we can obtain the angular velocity of the particle, given by  \n˙ ϕ 2 = ± E c f ( r 1 , 2 , 3 ) 1 / 2 Σ( r 1 , 2 , 3 ) , (62)  \nwhich is nonzero. Since angular velocity is related to the angular momentum /lscript , we can use that to obtain the expression for the critical impact parameter, which is given by [49]  \nB c = /lscript c E c = ( r 2 c + a 2 ) 1 / 2 ( b √ r 2 c + a 2 -µ ) -1 / 2 , (63)  \nwhere E c and /lscript c are the total energy and angular momentum of the particle in a circular orbit, and r c is the radius of the circular orbit that the particle describes. Here, r c = r 1 to 0 ≤ ba < 2 µ and r c = r 3 if ba ≥ 2 µ . Substituting r 1 and r 3 we find  \nB c = a √ ab -µ , if ba ≥ 2 µ. (65)  \nB c = 2 √ µ b , if ba < 2 µ, (64)  \nThe impact parameter will tell us what will happen to the particle. If B > B c , the particle will be scattered; if B < B c , the particle will be absorbed; and for B = B c , the particle will perform circular motion around the black hole. In 3 + 1 dimensions, we have the absorption cross-section. In the case of 2 + 1 dimensions, we have the absorption length, which is defined as σ = 2 B c [50]. Any particle emitted from infinity and within this length interval will be absorbed by the black hole. For the spacetime we are considering, the absorption length is given by  \nσ = 4 √ µ b , if ba < 2 µ, (66)  \nσ = 2 a √ ab -µ , if ba ≥ 2 µ. (67)  \nFor ba < 2 µ , we have the same absorption length from the singular case, which does not depend explicitly of a . For ba ≥ 2 µ the absorption length will grow with a .  \nThrough equation (58) and making the change of variables u = 1 / √ r 2 + a 2 , we obtain  \n( du dϕ ) 2 = ( 1 -a 2 u 2 ) ( 1 B 2 + u ( µu -b ) ) . (68)  \nTaking the derivation of the equation above with respect to ϕ we find  \nd 2 u dϕ 2 -3 2 a 2 bu 2 + a 2 u B 2 +2 a 2 µu 3 + b 2 -µu = 0 . (69)  \nTo solve the equation (69), we need two boundary conditions that are  \nu | ϕ =0 = 0 , and du dϕ ∣ ∣ ∣ ϕ =0 = 1 B . (70)  \n∣ In Fig. 8, for the case in which we vary the parameter a while fixing the critical impact parameter, the radius of the unstable orbit decreases as the parameter a increases. This also occurs in solutions like Reissner-Nordstrom. In the case where the impact parameter is varied, there will be values where the particle is absorbed by the black hole, scattered, or remains in an unstable circular orbit. It's interesting to note that, in the scattering case, the particle undergoes very intense scattering.  \nFigure 8. Trajectories of massless particles considering µ = 1. In (a) we fixed the value of the impact parameter, bB = bB c = 2, and varied the value of a . We place the event horizon only for the case of the largest radius, a = 0. The other values have horizons with smaller radii, and therefore, we choose not to insert them. In (b) we fix the value ba = 0 . 3 and change the value of the impact parameter.  \n<!-- image -->  \n-6  \n-4  \n-2  \n0  \n2  \n4  \n6  \nbx  \n(b)", 'B. Geodesics of massive particles': "For massive particles, we consider the equations of motion with δ = 1,  \nf ( r ) ˙ t 2 -f ( r ) -1 ˙ r 2 -Σ( r ) 2 ˙ ϕ 2 = 1 . (71)  \nUsing (54)-(56), we find  \nE 2 = ˙ r 2 + V eff . (72)  \nThe structure of the equation above is similar to the massless case. The differences are incorporated within the effective potential, which is given by:  \nV eff = f ( 1 + /lscript 2 Σ 2 ) = ( b √ r 2 + a 2 -µ ) ( 1 + /lscript 2 a 2 + r 2 ) . (73)  \nIn Fig. 9 we show the behavior of the effective potential for different values of a and /lscript . Both the parameters a and /lscript can alter the structure of the effective potential. For small values of a , there are two extrema in the potential, one maximum and one minimum. This means that we can have both stable and unstable orbits outside the event horizon. For ba > µ , we no longer have an event horizon, and as a result, the orbit at r = 0 also becomes accessible, potentially being either  \nby  \n6  \n4  \n2  \n0  \n-2  \n-4  \n-6  \nb B  \n= 1  \nb B  \n= 1.5  \nb B  \n=2  \nb B  \n= 2.5  \nHorizon  \nFigure 9. Effective potential to massive particles to different values of a and /lscript . In (a) we fix b/lscript = 6 and change the a . As to ba < m there is an event horizon, we consider only the region r > 0. In (b) the configuration is similar of (a), however in this case there is no event horizon and we consider -∞ < r < ∞ . In (c) and (d) we fix the charge, ba = 0 . 3 µ and ba = 1 . 5 µ respectively, and change /lscript .  \n<!-- image -->  \na maximum or minimum point, depending on the value of a . The change in angular momentum of the particle does not affect the presence of the event horizon but does alter the structure of the effective potential. In the case with horizons, particles with low angular momentum are necessarily absorbed by the black hole, whereas particles with high angular momentum can remain in a stable or unstable circular orbit.  \nTo obtain the extremal points, we must solve V ' eff = 0. However, the analytical expressions are quite complex, so we will analyze them graphically.  \nIn Fig. 10 we analyze the radius of the circular orbits. We observe that there are three possible orbits. However, as long as there is an event horizon, only two orbits are accessible, one stable and  \nFigure 12 illustrates the behavior of the function G ( r ) for different cases. In the case with energy E 2 = 10 µ , for some values of the parameter a , every particle emitted from distant points of the black hole reaches a maximum approach point and then returns. The higher the value of a , the closer to the black hole or throat the particle can get. As expected from the shape of the  \n<!-- image -->  \n<!-- image -->  \n10  \nFigure 10. Radius of the stable and the unstable orbit to massive particles as functions of the a and /lscript . In (a), we fix the value of angular momentum at b/lscript = 6 and vary the value of a . In Figure (b), we fix ba = 0 . 3 µ and vary the value of angular momentum.  \none unstable. As the a parameter increases, the horizon ceases to exist, and the orbit at r 0 = 0 also becomes possible. Since the solution is symmetric for r → -r , the orbits also exist for negative r . For great values of a , only orbits at r = 0 exist. As we change the angular momentum of the particle, the radius of the orbits are also altered. There is a value of /lscript such that the stable and unstable orbit tend to the same point, this is the innermost stable circular orbit (ISCO). For values of /lscript smaller than l ISCO = 3 √ 3 µ/b , only the orbit at r = 0 exists, if there is no event horizon. For /lscript = /lscript ISCO , we find the value of the ISCO radius, which is given by  \nr ISCO = √ 9 µ 2 b 2 -a 2 . (74)  \nIn Fig. 11,we analyze the change in the ISCO radius for different values of the regularization parameter. If a = 0 we have r ISCO = 3 µ/b , and if a = 3 µ/b we have r ISCO → r 0 = 0.  \nFrom equation (72), we can define the following function G ( r ):  \nG ( r ) = ( dr dφ ) 2 = ( a 2 + r 2 ) 2 /lscript 2 ( E 2 -( /lscript 2 a 2 + r 2 +1 ) ( b √ a 2 + r 2 -µ ) ) . (75)  \nThe zeros of this function represent the particle's return points.  \nFigure 11. Radius of ISCO as functions of the parameter a .  \n<!-- image -->  \neffective potential, massive particles cannot be emitted from infinity, once the radial velocity must be real and not imaginary. Depending on the energy value, the particle may reach a return point before reaching the black hole/WH or it may reach it. The higher the particle's energy, the farther it can be emitted from. For a particle to be emitted from infinity, it would need to have infinite energy. In the case of a WH, a particle emitted from distant points can only traverse it if it has a minimum value; otherwise, it will encounter a return point.", 'VI. CONCLUSIONS AND DISCUSSION': "In this work, we apply the Simpson-Visser regularization method to an asymptotically locally flat solution in 2 + 1 dimensions in the context of a purely quadratic gravity, K gravity. Once we have the regularized solution, we analyze properties and extract information from this regularized spacetime.  \nThe regularized solution has a minimum circumference, C = 2 πa , and thus we have a kind of WH hidden inside a black hole, which characterizes a BB. The BB will only be present while 0 < a < µ/b . If the regularization parameter is null, we recover the singular solution and if a = µ/b , the WH's throat coincides with the event horizon and, with this, we have a black throat, which is the throat of a WH that is traversable in one direction. If a > µ/b , there will no longer be an event horizon and, with this, we will have a WH that is traversable in both directions. For the case with a negative cosmological constant, the causal structure becomes richer by allowing the presence of event horizons even in cases where the mass or the parameter b are negative. This brings more possibilities for the causal structure of spacetime.  \nFigure 12. Behavior of the function G for massive particles as a function of the radial coordinate for different values of the particle's energy and the parameter a . In all panels, the particle's angular momentum is fixed at b/lscript = 6 µ . In panels (a) and (b), we fix the energy value at E 2 = 10 µ and vary the parameter a . In panel (a), cases of a RBH are represented, as the event horizon still exists. In panel (b), we have the case of a black throat and a WH. In panels (c) and (d), we fix ba = 0 . 3 µ and ba = 1 . 5 µ , respectively, and vary the value of the particle's energy.  \n<!-- image -->  \nWe were able to identify the source fields that generate the regular solution in quadratic theory. The source is described by a combination of a partially phantom scalar field along with NED. The analysis of the nature of the scalar field is conducted through the function h ( φ ), which accompanies the kinetic term of the scalar field. We found that the function h ( φ ) has positive values near the center of the solution and negative values at more distant points. This implies that our scalar field will be canonical in regions close to the throat and phantom in regions further outside the WH throat.  \nThe analysis of the energy conditions for the regularized asymptotically locally flat solution reveals notable behaviors based on the interplay between the parameters a , b , and µ . Specifically,  \nthe weak energy condition WEC 3 does not hold for RBHs with two horizons, though it may be met near the throat of a WH. The condition WEC 2 is particularly noteworthy as it is satisfied when ab < µ , although it fails close to the WH throat but is valid further away. The strong energy condition SEC 3 is shown to be valid only for metrics describing either a RBH with two horizons or a black hole with black throat. However, when the metric describes a WH, the SEC 3 will be satisfied only when the parameter a does not reach a certain limit value, highlighting the significant role of the parameter a in its validity. Lastly, the dominant energy condition DEC 1 is not met, whereas DEC 2 is valid under specific conditions. This implies that the presence of exotic matter, which is crucial for supporting such spacetimes, is determined by the fluid quantities and their relationships, especially in the context of WHs. This indicates that the problem of exotic matter persists for both RBHs and WHs in K -gravity theories.  \nThrough the analysis of geodesics, we find that there is only one unstable orbit for massless particles in cases where there is an event horizon and two unstable orbits and one stable for some cases of WHs. For cases with an event horizon, the critical impact parameter does not depend on the regularization parameter, while the radius of the orbit of massless particles depends on this parameter. In fact, the radius of the unstable orbit decreases as the value of the regularization parameter increases. There some values of the impact parameter where the particles are strongly scattered. Just like the critical impact parameter, the absorption length, in cases where event horizons are present, does not depend on the regularization parameter. For cases where the absorption length depends on the regularization parameter, the length will increase as the value of a increases.  \nFor massive particles, it is possible to have stable and unstable orbits, both for the case with event horizons and for the WH case. As the regularization parameter increases, the event horizon ceases to exist and another stable orbit arises. If the particles have sufficient angular momentum, it is possible to have both stable and unstable orbits. However, for particles with low angular momentum, there will only be stable orbits and this only in the case without horizons.  \nFinally, we point out that this is a series of papers where we will apply the method used here to find sources for alternative theories of gravity.", 'ACKNOWLEDGMENTS': "We thank Professor Kirill Bronnikov for his comments and suggestions. G.A. and M.E.R. would like to thank Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico - CNPq, Brazil for  \npartial financial support. G.A. and M.S. would like to thank Funda¸c˜ao Cearense de Apoio ao Desenvolvimento Cient'ıfico e Tecnol'ogico (FUNCAP) for partial financial support.  \n- [1] B. P. Abbott et al. (LIGO Scientific, Virgo), 'Observation of Gravitational Waves from a Binary Black Hole Merger,' Phys. Rev. Lett. 116 , 061102 (2016), arXiv:1602.03837 [gr-qc].\n- [2] Kazunori Akiyama et al. (Event Horizon Telescope), 'First Sagittarius A* Event Horizon Telescope Results. I. 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