New interpretation of the Extended Geometric Deformation in Isotropic Coordinates
NNew interpretation of the Extended Geometric Deformation in Isotropic Coordinates
C. Las Heras ∗ and P. León † Departamento de Física, Universidad de Antofagasta, Aptdo 02800, Chile.
We study the particular case in which Extended Gravitational Decoupling does consist in consecutive defor-mations of temporal and spatial components of the metric, in Schwarzschild-like and isotropic coordinates. Inthe latter, we present two inequivalent ways to perform this 2-steps GD. This was done in such a way that themethod may be applied to different seed solutions. As an example, we use Tolman IV as seed solution, in orderto obtain two inequivalent physical solutions with anisotropy in the pressures in Schwarzschild like coordinates.In the isotropic sector, we obtained 4 different solutions with anisotropy in the pressures that satisfies physicalacceptability conditions, using Gold III as seed solution.
INTRODUCTION
It is known that General Relativity (GR) is the most success-ful theory describing the Gravitational interaction. This theoryexplores the relation between the geometry of space-time andthe energy-matter content of the universe. The evidence ofthis relation it is manifested in Einstein Fields equations [1],which in general is a non linear system of equations in partialderivatives. Gravitational systems, such as, Relativistic Com-pact Stars or Black Holes, are modelled by solutions to theseequations. Now, due to the complicated form of the Einsteinequations, finding analytical solutions describing general mat-ter configurations could be very difficult, or even impossiblein many cases. For this reason, Einstein’s equations are solvedusing numerical method for more complicated (and also morerealistic) systems. The known analytical solutions are obtainedby assuming some simplifications of the systems. These sim-plifications are related to restrictions of the geometry of thespace-time, or the matter distributions.A common choice for the geometry of space time is to as-sume spherical symmetric systems, which leads to huge sim-plifications of the Einstein equations. On the matter side, themajor part of the known analytical solutions are for isotropicperfect fluid, which is one of the simplest cases known. His-torically, since Tolman [2] presented his seminal work, thereare different interior solutions in terms of the isotropic perfectfluid solutions that has been found [3–5]. However, anisotropyin the pressures of interior solutions, it is a desirable propertyin order to describe realistic compact objects [6–12]. Indeed,the local anisotropy in the pressures is expected to be presentin matter distributions, since it could be caused by a series ofphenomena that could be present in compact objects, as forexample the viscosity or the intense magnetic fields.Over the last years, one popular way to obtain analyticalsolutions of Einstein equations with physical relevance, is us-ing the Minimal Geometric Deformation (MGD) approach[13, 14], which allow us to decouple gravitational sourcesin GR. MGD was first introduced in the context of Randall-Sundrum Brane World (RSBW)[14–22]. But it has been re-cently used to decouple gravitational sources in GR and tostudy compact objects [23–29], black holes [30–34], worm-holes, cosmology [35] or the coupling of Einstein equationswith matter fields [36, 37], among other possibilities. More- over, it has been studied in dimensions different that four[33, 38–41], for theories beyond GR (for example 𝑓 ( 𝑅 ) grav-ity [42], 𝑓 (G) [43, 44], 𝑓 ( 𝑅, 𝑇 ) gravity [45], Lovelock grav-ity [46] and a more recent application to RSBW [47]) andother gravitational theories as Brans-Dicke gravity [48, 49].Noticing that interior solutions related with self-gravitatingobjects obtained by MGD, must have anisotropy in the pres-sures, we can then understand MGD as a way to obtain realisticanisotropic systems [50–52]. In the context of MGD, the wordminimal is related to the fact that we are considering only thegeometric deformation of the spatial component. Other gen-eralizations of the MGD method include the study of systemswith cylindrical symmetry [53] and axial symmetry [54].In [55] we propose 2 new MGD-inspired methods, to solveEinstein’s equations in isotropic coordinates, in order to obtainnew physical anisotropic solutions. This could be interestingsince the coordinate transformation between the standard andisotropic is not always well defined (see [56]). Then, it is pos-sible to find solutions in isotropic coordinates whose transfor-mation to Schwarzschild-like coordinates is not well defined.Or equivalently, solutions that can not be written in analyticalform in Schwarzschild-like coordinates. Therefore,the MGD-inspired methods in [55] could be useful to study solutionsof Einstein’s equations in theses cases. They may also bemore useful in cosmology and astrophysics [57]. It is worthto mention that in general, there is no gravitational decouplingin isotropic coordinates. Despite the large number of applica-tions of MGD method, probably one of the biggest limitationsit is that we are restricted to study sources that only modify thespatial component of the metric. Therefore, the method can notbe applied to decouple some self gravitational configurations,as for example the Einstein-Maxwell configuration.The Extended Gravitational Decoupling (EGD) presented in[58], allow us to decouple two spherically symmetric and staticgravitational sources by geometric deformation of both, spatialand temporal components of the metric. The decoupling ofthe two sources, in this case, it is only possible when there isan interchange of energy between both sources, which is not arequirement in the MGD case. It has been recently studied in[40, 59, 60] that the extended version of the method also allowsto extend solutions to the anisotropic domain and study evenmore complicated contribution of modified theories of gravityof the GR. This shows the great potential of the gravitational a r X i v : . [ g r- q c ] J a n decoupling to study self gravitational systems. It is evidentthat the extended case contains a new unknown function to bedetermined. Therefore, we have to give more information thanstandard MGD in order to solve the system (see [61–87] formore applications of the gravitational decoupling method).In this work, we are interested in study the Extended Versionof MGD in Standard-like and isotropic coordinates. In partic-ular, we will consider consecutive geometric deformations ofradial and temporal components of the metric. We will showthat in general, these deformations does not commute, how-ever the solutions are always contained in EGD. In section 1we will briefly resume the EGD method in Schwarzschild-likecoordinates. In Section 2 we will study consecutive geometricdeformations in Schwarzschild-like coordinates and we willapply the results to the known Tolman IV solution. We willpresent in section 3, the extended version of the gravitationaldecoupling in isotropic coordinates and in section 4, we willconsider consecutive spatial and temporal deformations of themetric in isotropic coordinates. In this section we will alsopresent some examples of solutions taking as a seed the GoldIII solution for perfect fluid. Finally in Section 5, we present adiscussion of the obtained results. EXTENDED GRAVITATIONAL DECOUPLING
Let us begin by writing the line element in standard coordi-nates, also known as Schwarzschild-like coordinates 𝑑𝑠 = 𝑒 𝜈 ( 𝑟 ) 𝑑𝑡 − 𝜇 ( 𝑟 ) 𝑑𝑟 − 𝑟 (cid:16) 𝑑𝜃 + sin 𝜃 𝑑𝜙 (cid:17) . (1)It can be shown that Einstein’s equations 𝑅 𝜇𝜈 − 𝑅 𝑔 𝜇𝜈 = − 𝜋 𝑇 .𝜇𝜈 , (2)take the following form8 𝜋𝑇 = 𝑟 − 𝜇𝑟 − 𝜇 (cid:48) 𝑟 , (3) − 𝜋𝑇 = − 𝑟 + 𝜇 (cid:18) 𝑟 + 𝜈 (cid:48) 𝑟 (cid:19) , (4) − 𝜋𝑇 = 𝜇 (cid:18) 𝜈 (cid:48)(cid:48) + 𝜈 (cid:48) + 𝜈 (cid:48) 𝑟 (cid:19) + 𝜇 (cid:48) (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19) , (5)where the prime indicates derivatives respect to variable 𝑟 . Fi-nally the conservation equation, derived from the latter systemof equations is ∇ 𝜇 𝑇 𝜇𝜈 = , (6)whose radial component leads to the equilibrium equation ofthe matter distribution8 𝜋 ( 𝑇 ) (cid:48) + 𝜋𝑟 ( 𝑇 − 𝑇 ) + 𝜋𝜈 (cid:48) ( 𝑇 − 𝑇 ) = . (7)The starting point of this method is to assume that theenergy-momentum tensor has the specific form 𝑇 𝜇𝜈 = 𝑇 𝜇𝜈 + 𝛼 𝜃 𝜇𝜈 , (8) where 𝛼 is a coupling constant. In this work, for simplicity wewill assume that 𝑇 is the matter-energy content associated toa perfect fluid 𝑇 = 𝑇 ( PF ) 𝜇𝜈 = ( 𝜌 + 𝑃 ) 𝑢 𝜇 𝑢 𝜈 − 𝑃 𝑔 𝜇𝜈 , (9)with the fluid 4-velocity given by 𝑢 𝜇 = 𝑒 − 𝜈 / 𝛿 𝜇 .Now, we can consider a perfect fluid solution of Einsteinequations ( 𝛼 =
0) with a line element written in standardcoordinates as (1), and define 𝜇 ( 𝑟 ) = 𝑒 − 𝜆 ≡ − 𝜋𝑟 ∫ 𝑟 𝑥 𝜌 𝑑𝑥 = − 𝑚 ( 𝑟 ) 𝑟 , (10)which is the standard expression for the mass function in GR.The next step is to take account of the anisotropy introduced bythe gravitational source 𝜃 𝜇𝜈 in our system. This will be doneby assuming that the contribution of the 𝛼 parameter, in theperfect fluid solution (1), is encoded in the deformations ℎ and 𝑓 of the temporal and radial metric components, respectively. 𝜈 → (cid:101) 𝜈 = 𝜉 + 𝛼 ℎ , (11) 𝜇 → (cid:101) 𝜇 = 𝜇 + 𝛼 𝑓 . (12)In this case is easy to check that, using (8), (11) and (12),Einstein’s equations (3)-(5) splits in two systems. The first onecoincides with Einstein’s equations system for a perfect fluid8 𝜋𝜌 = 𝑟 − 𝜇𝑟 − 𝜇 (cid:48) 𝑟 , (13)8 𝜋𝑃 = − 𝑟 + 𝜇 (cid:18) 𝑟 + 𝜉 (cid:48) 𝑟 (cid:19) , (14)8 𝜋𝑃 = 𝜇 (cid:18) 𝜉 (cid:48)(cid:48) + 𝜉 (cid:48) + 𝜉 (cid:48) 𝑟 (cid:19) + 𝜇 (cid:48) (cid:18) 𝜉 (cid:48) + 𝑟 (cid:19) , (15)with the corresponding conservation equation 𝑃 (cid:48) + 𝜉 (cid:48) ( 𝜌 + 𝑃 ) = . (16)This expression turns out to be equation (6) with the energymomentum tensor associated to a perfect fluid (9) (eq (8) with 𝛼 = 𝜋 𝜃 = − 𝑓𝑟 − 𝑓 (cid:48) 𝑟 , (17)8 𝜋 𝜃 + 𝑍 = − 𝑓 (cid:18) 𝑟 + (cid:101) 𝜈 (cid:48) 𝑟 (cid:19) , (18)8 𝜋 𝜃 + 𝑍 = − 𝑓 (cid:18) (cid:101) 𝜈 (cid:48)(cid:48) + (cid:101) 𝜈 (cid:48) + (cid:101) 𝜈 (cid:48) 𝑟 (cid:19) − 𝑓 (cid:48) (cid:18)(cid:101) 𝜈 (cid:48) + 𝑟 (cid:19) , (19)and the conservation equation associated with the source is (cid:16) 𝜃 (cid:17) (cid:48) − 𝜈 (cid:48) (cid:16) 𝜃 − 𝜃 (cid:17) − 𝑟 (cid:16) 𝜃 − 𝜃 (cid:17) = , (20)with 𝑍 = 𝜇ℎ (cid:48) 𝑟 , (21)4 𝑍 = 𝜇 (cid:18) ℎ (cid:48)(cid:48) + 𝛼ℎ (cid:48) + ℎ (cid:48) 𝑟 + 𝜉 (cid:48) ℎ (cid:48) (cid:19) + 𝜇 (cid:48) ℎ (cid:48) . (22)In order to find a solution of Einstein’s equations for an energy-momentum tensor of the form (8), we have to solve the systems(13)-(15) and (17)-(19). In the case when we start with aknown perfect fluid solution, then is only necessary to solvethe second system. Now, in both cases there are more unknownfunctions than equations, so additional information is requiredin order to solve the system. Specifically, it is necessary toimpose two conditions in order to solve the second system.This information can be given in the form of an equation ofstate or any other expression that relates the physical variablesof the system under study.Now, it is important to mention that the sources 𝑇 ( 𝑃𝐹 ) 𝜇𝜈 and 𝜃 𝜇𝜈 can be decoupled only if there exists an interchange ofenergy between then. This can be easily seen from the conser-vation equations ∇ 𝜇 ( 𝑇 𝑃𝐹 ) 𝜇𝜈 = − ℎ (cid:48) ( 𝑃 + 𝜌 ) 𝛿 𝜈 , (23)and ∇ 𝜇 𝜃 𝜇𝜈 = ℎ (cid:48) ( 𝑃 + 𝜌 ) 𝛿 𝜈 . (24)At this point is clear that EGD is a powerful tool to studymore complicated solutions of Einstein’s field equations, thanthe ones obtained with the MGD method. Nevertheless, find asolution for the equations (17)-(19) could be very complicateddepending on the system under study. In this section we will analyze a possible way to simplify theproblem of finding solutions for equations (17)-(19). This sim-plification consist in considering deformations of the metriccomponents not simultaneous, but consecutive.Let us first notice that there are two straightforward limits ofthe EGD method. The first one is obtained by taking ℎ =
0, itcorresponds with usual MGD and in this case, (17)-(19) maybe written as8 𝜋 ( 𝜃 𝑓 ) = − 𝑓𝑟 − 𝑓 (cid:48) 𝑟 , (25)8 𝜋 ( 𝜃 𝑓 ) = − 𝑓 (cid:18) 𝑟 + 𝜉 (cid:48) 𝑟 (cid:19) , (26)8 𝜋 ( 𝜃 𝑓 ) = − 𝑓 (cid:18) 𝜉 (cid:48)(cid:48) + 𝜉 (cid:48) + 𝜉 (cid:48) 𝑟 (cid:19) − 𝑓 (cid:48) (cid:18) 𝜉 (cid:48) + 𝑟 (cid:19) . (27)The second limit corresponds to the case where 𝑓 =
0, inwhich case the system of eqs (17)-(19) is given by8 𝜋 ( 𝜃 ℎ ) = , (28)8 𝜋 ( 𝜃 ℎ ) + 𝑍 = , (29)8 𝜋 ( 𝜃 ℎ ) + 𝑍 = . (30)This corresponds to a deformation purely temporal of the met-ric. Notice that this limit of the EGD can not be thought as a simple modification of the time scale due to the r-dependenceon the deformation function. In contrast with the pure spa-tial metric deformation, the ( 𝜃 ℎ ) 𝜇𝜈 can not be interpreted asa physical source of matter due to Eq (28), only the total en-ergy momentum has a physical interpretation. The “source” ( 𝜃 ℎ ) 𝜇𝜈 can be understood as a mathematical "artifact" in orderto obtain new solutions, or to connect inequivalent solutionsof Einstein’s equations. Analyzing the Ricci invariants, seeAppendix , it can be shown that all the solutions obtained byEGD (and therefore by any of their limits) are different fromthe seed solutions. Therefore, geometric geformation of thespatial or temporal component of the metric, or both of them,leads in general, to inequivalent solutions. For example, itcan be check that Einstein Metric for an isotropic perfect fluidand Schwarzschild interior solution can be related with a puretemporal deformation.We have then that both limits can be used as a method tofind solutions of Einstein field equations, by deforming onlythe spatial or temporal component of the metric, respectively.Then, instead solving directly the system (17)-(19) we canobtain solutions by taking consecutive deformations of themetric. Specifically, we can choose a seed solution and ob-tain a new one with a deformation in the spatial (temporal)component of the metric. We may now use this result as seedsolution to find another one by performing a deformation ofthe temporal (spatial) component of the metric. In this way, weare able to obtain solutions with deformations in both, spatialand temporal components of the metric. Therefore, these aresolutions of (17)-(19), which is a more general system and itmay be more difficult to solve. Let us named this particularprocess 2-steps gravitational decoupling (2-steps GD).It is evident that there are two different ways to perform the2-steps GD. For simplicity, we will refer to them as the Leftand Right path, corresponding to the case where the spatial andtemporal deformation is made first, respectively. We representthis in the figure. It is easy to verify that solutions obtainedby the left and right path, separately, are not only inequivalentbut they correspond to different set of solutions. Regularityconditions ensures that solutions obtained by the left path, cannot be obtained by the right path, and viceverse. For this reasonit is interesting to study both, left and right path. Moreover, itcan be shown (see Appendix ), that from the left and right pathsof the 2-step GD, the first four theorems presented [88], whichcorresponds to transformations between perfect fluid spheresto perfect fluid spheres (different from the original ones), maybe obtained. This gives a simple example of the inequivalencebetween the Right and Left paths.We have already mentioned that 2-steps GD is a particularcase of EGD, in fact it corresponds to solutions with a sourceterm of the form 𝜃 𝜇𝜈 = ( 𝜃 𝑓 ) 𝜇𝜈 + 𝛽𝛼 ( 𝜃 ℎ ) 𝜇𝜈 , (31)where ( 𝜃 𝑓 ) 𝜇𝜈 and ( 𝜃 ℎ ) 𝜇𝜈 are responsible of deformations ofthe spatial and temporal component of the metric, respectively(See figure 1). This particular decomposition of the source Set of seed solutionsSet of solutions with f ≠ 0 and g=0 Set of solutions with f=0 and g≠0
Set of solutions with f ≠ 0 and g≠0 L e f t P a t h R i g h t P a t h FIG. 1. The 2-steps GD diagram 𝜃 𝜇𝜈 is not required for EGD. Thus, there are solutions thatcould be obtained by the EGD and not by the 2-step GD.In order to give an example let us consider Tolman IV solu-tion of Einstein equations as seed solution 𝑒 𝜉 = 𝐵 (cid:32) + 𝑟 𝐴 (cid:33) , (32) 𝜇 = (cid:16) − 𝑟 𝐶 (cid:17) (cid:16) + 𝑟 𝐴 (cid:17)(cid:16) + 𝑟 𝐴 (cid:17) , (33) 𝜌 = 𝐴 + 𝐴 ( 𝐶 + 𝑟 ) + 𝑟 ( 𝐶 + 𝑟 ) 𝜋𝐶 ( 𝐴 + 𝑟 ) , (34) 𝑃 = 𝐶 − 𝐴 − 𝑟 𝜋𝐶 ( 𝐴 + 𝑟 ) , (35)where 𝐴 , 𝐵 and 𝐶 are constants that can be determine bythe matching conditions. In this work we will match all theinternal solutions with the Schwarzschild vacuum solution. The Left Path
If we consider first the geometric deformation of the radialcomponent ( ℎ 𝐿 =
0) of Tolman IV solution, which is theknown MGD, we have to solve the following system8 𝜋 ( 𝜃 𝑓𝐿 ) = − 𝑓 𝐿 𝑟 − 𝑓 (cid:48) 𝐿 𝑟 , (36)8 𝜋 ( 𝜃 𝑓𝐿 ) = − 𝑓 𝐿 (cid:18) 𝑟 + 𝜉 (cid:48) 𝑟 (cid:19) , (37)8 𝜋 ( 𝜃 𝑓𝐿 ) = − 𝑓 𝐿 (cid:18) 𝜉 (cid:48)(cid:48) + 𝜉 (cid:48) + 𝜉 (cid:48) 𝑟 (cid:19) − 𝑓 (cid:48) 𝐿 (cid:18) 𝜉 (cid:48) + 𝑟 (cid:19) . (38)Taking now the mimic constrain ( 𝜃 𝑓𝐿 ) = 𝑝 we obtain from(37) that 𝑓 𝐿 = ( 𝐴 + 𝑟 − 𝐶 ) 𝑟 ( 𝐴 + 𝑟 ) 𝐶 ( 𝐴 + 𝑟 )( 𝐴 + 𝑟 ) . (39) Is easy to see that, once we obtain the deformation function 𝑓 𝐿 , we are able to compute ( 𝜃 𝑓𝐿 ) and ( 𝜃 𝑓𝐿 ) . Therefore, thenew anisotropic solution is characterized by the metric 𝑒 𝜈 = 𝑒 𝜉 , (40)¯ 𝜇 = 𝜇 + 𝛼 𝑓 𝐿 , (41)¯ 𝜌 = 𝜌 + 𝛼 ( 𝜃 𝑓𝐿 ) , (42)¯ 𝑃 𝑟 = ( − 𝛼 ) 𝑝, (43)¯ 𝑃 𝑡 = 𝑝 − 𝛼 ( 𝜃 𝑓𝐿 ) (44)which is the anisotropic solutions of Einstein’s equations ob-tained in [89].Now, we can choose this result as a seed solution and per-form a deformation in the temporal component of the metric.Then we have to solve the following system of equations8 𝜋 ( 𝜃 ℎ𝐿 ) = , (45)8 𝜋 ( 𝜃 ℎ𝐿 ) = − ¯ 𝜇ℎ (cid:48) 𝐿 𝑟 , (46)8 𝜋 ( 𝜃 ℎ𝐿 ) = − ¯ 𝜇 (cid:18) ℎ (cid:48)(cid:48) 𝐿 + 𝛽ℎ (cid:48) 𝐿 + ℎ (cid:48) 𝐿 𝑟 + 𝜉 (cid:48) ℎ (cid:48) 𝐿 (cid:19) − ¯ 𝜇 (cid:48) ℎ (cid:48) 𝐿 . (47)Imposing now the constraint ( (cid:101) 𝜃 ℎ ) = ¯ 𝜇 ¯ 𝑝 𝑟 , it can be checkfrom (46) that ℎ 𝐿 = ( − 𝛼 ) (cid:18) 𝑟 𝐶 − 𝑙𝑛 ( 𝐴 + 𝑟 )− 𝑙𝑛 ( 𝐴 + 𝑟 ) 𝐴 𝐶 (cid:33) . (48)As before, once we have obtained the deformation function ℎ 𝐿 , we are able to compute ( 𝜃 ℎ𝐿 ) .The final solutions with both deformations of the metriccomponent is given by 𝑒 ¯ 𝜈 = 𝐵 ( + 𝑟 𝐴 ) (cid:101) 𝐹 ( 𝑟 ) , (49) (cid:101) 𝜇 = ¯ 𝜇 = 𝜇 + 𝛼 𝑓 𝐿 , (50) (cid:101) 𝜌 = ¯ 𝜌 = 𝜌 + 𝛼 ( 𝜃 𝑓𝐿 ) , (51) (cid:101) 𝑃 𝑟 = ¯ 𝑝 𝑟 − 𝛽 ( 𝜃 ℎ𝐿 ) = ( − 𝛼 )( − 𝛽 ¯ 𝜇 ) 𝑝, (52) (cid:101) 𝑃 𝑡 = ¯ 𝑝 𝑡 − 𝛽 ( 𝜃 ℎ𝐿 ) = 𝑝 − 𝛼 (cid:18) ( 𝜃 𝑓𝐿 ) + 𝛽𝛼 ( 𝜃 ℎ𝐿 ) (cid:19) , (53)where (cid:101) 𝐹 = 𝑒 𝛽ℎ 𝐿 . From the matching conditions we obtain 𝐴 = 𝑅𝑀 √︁ ( 𝑅 − 𝑀 ) 𝑀, (54) 𝐶 = √︃ 𝐴 + 𝑅 , (55) 𝐵 = (cid:18) − 𝑀𝑅 (cid:19) (cid:32) + 𝑅 𝐴 (cid:33) − 𝑒 − 𝛽𝑔 𝐿 ( 𝑅 ) . (56)Now, to present an example of a physically acceptable solutionwe choose the following values for the free parameters 𝑅 = FIG. 2. Radial and tangential pressures for the left path inSchwarzschild like vs the radial coordinateFIG. 3. Energy density for the left path in Schwarzschild like vs theradial coordinate 𝑀 = 𝛼 = . 𝛽 = .
1. Well-behaviour of this solutionis evident from the figures (2) and (3), where pressures andenergy density were plotted, respectively
The Right Path
Let us now consider first the deformation of the temporalcomponent ( 𝑓 =
0) of Tolman IV solutions. In this case, thesystem that we must solve is8 𝜋 ( 𝜃 ℎ𝑅 ) = , (57)8 𝜋 ( 𝜃 ℎ𝑅 ) = − 𝜇ℎ (cid:48) 𝑟 , (58)8 𝜋 ( 𝜃 ℎ𝑅 ) = − 𝜇 (cid:18) ℎ (cid:48)(cid:48) + 𝛼ℎ (cid:48) + ℎ (cid:48) 𝑟 + 𝜉 (cid:48) ℎ (cid:48) (cid:19) − 𝜇 (cid:48) ℎ (cid:48) . (59)In order to solve it, we will impose a constraint of the sameform that the one in left path ( 𝜃 ℎ𝑅 ) = 𝑝𝜇 . Then, is easy to seethat ℎ 𝑅 = 𝑟 𝐶 − 𝑙𝑛 ( 𝐴 + 𝑟 ) − 𝑙𝑛 ( 𝐴 + 𝑟 ) 𝐴 𝐶 . (60)Thus, the new anisotropic solution is given by 𝑒 ¯ 𝜈 = 𝑒 𝜉 ( 𝑟 )+ 𝛼ℎ ( 𝑟 ) = 𝐵 ( + 𝑟 𝐴 ) 𝐹 ( 𝑟 ) , (61)¯ 𝜇 = 𝜇, (62)¯ 𝜌 = 𝜌, (63)¯ 𝑃 𝑟 = ( − 𝛼𝜇 ) 𝑃, (64)¯ 𝑃 𝑡 = 𝑃 − 𝛼 ( 𝜃 ℎ𝑅 ) (65)with 𝐹 ( 𝑟 ) = 𝑒 𝛼ℎ .As before, we can take this result as seed solution, and thenmake a deformation of the spatial component of the metric. Inthis case, the system of equations that we have to solve is givenby 8 𝜋 ( 𝜃 𝑓𝑅 ) = − 𝑓 𝑅 𝑟 − 𝑓 (cid:48) 𝑅 𝑟 , (66)8 𝜋 ( 𝜃 𝑓𝑅 ) = − 𝑓 𝑅 (cid:18) 𝑟 + ¯ 𝜈 (cid:48) 𝑟 (cid:19) , (67)8 𝜋 ( 𝜃 𝑓𝑅 ) = − 𝑓 𝑅 (cid:18) 𝜈 (cid:48)(cid:48) + ¯ 𝜈 (cid:48) + 𝜈 (cid:48) 𝑟 (cid:19) − 𝑓 (cid:48) 𝑅 (cid:18) ¯ 𝜈 (cid:48) + 𝑟 (cid:19) , (68)Now, considering the mimic constraint, ( 𝜃 𝑓𝑅 ) = ¯ 𝑃 𝑟 as in theleft path (61) is easy to see that 𝑓 𝑅 = − ( 𝐴 − 𝐶 + 𝑟 ) 𝑟 ( 𝐴 + 𝑟 ) 𝐶 ( 𝐴 + 𝑟 ) 𝐽 ( 𝑟 ) 𝐽 ( 𝑟 ) , (69)with 𝐽 ( 𝑟 ) = 𝐴 𝐶 𝛼 − 𝐴 𝛼𝑟 + 𝐶 𝛼𝑟 − 𝛼𝑟 − 𝐴 𝐶 − 𝐶 𝑟 , (70) 𝐽 ( 𝑟 ) = 𝐴 𝛼𝑟 − 𝐴 𝐶 𝛼𝑟 + 𝐴 𝛼𝑟 − 𝐶 𝛼𝑟 , + 𝛼𝑟 + 𝐴 𝐶 + 𝐴 𝐶 𝑟 + 𝐶 𝑟 (71)Therefore, the final solution with both deformations in themetric is given by 𝑒 (cid:101) 𝜈 = 𝑒 ¯ 𝜈 , (72) (cid:101) 𝜇 = 𝜇 + 𝛽 𝑓 𝑅 , (73)˜ 𝜌 = 𝜌 + 𝛽 ( 𝜃 𝑓𝑅 ) , (74) (cid:101) 𝑃 𝑟 = ¯ 𝑃 𝑟 − 𝛽 ( 𝜃 𝑓𝑅 ) = ( − 𝛽 )( − 𝛼𝜇 ) 𝑃, (75) (cid:101) 𝑃 𝑡 = ¯ 𝑃 𝑡 − 𝛽 ( 𝜃 𝑓𝑅 ) = 𝑃 − 𝛼 (cid:18) ( 𝜃 ℎ𝑅 ) + 𝛽𝛼 ( 𝜃 𝑓𝑅 ) (cid:19) (76) FIG. 4. Radial and tangential pressures for the right path inSchwarzschild like vs the radial coordinate where 𝐹 ( 𝑟 ) = 𝑒 𝛼𝑔 𝑅 . The matching conditions for this caseleads to 𝐴 = 𝑅𝑀 √︁ ( 𝑅 − 𝑀 ) 𝑀, (77) 𝐶 = √︁ 𝐴 + 𝑅 , (78) 𝐵 = (cid:18) − 𝑀𝑅 (cid:19) (cid:18) + 𝑅 𝐴 (cid:19) − 𝑒 − 𝛼𝑔 𝑅 ( 𝑅 ) . (79)Finally, as before, to give an example of a physically acceptablesolution we choose the following values 𝑅 = 𝑀 = 𝛼 = . 𝛽 = . ?? ), respectively.This anisotropic solution was obtained after two consecu-tive geometric deformations of the temporal and radial compo-nents of Tolman IV. It can be check that turning off any of thecoupling constants 𝛼 or 𝛽 , related to the first and second defor-mation, respectively, results in the expected sector of solutionsas it is shown in the diagram. EXTENDED GEOMETRIC DEFORMATION IN ISOTROPICCOORDINATES
Let us consider the interior of static and spherical sym-metric matter distributions using the line element in isotropiccoordinates 𝑑𝑠 = 𝑒 𝜈 ( 𝑟 ) 𝑑𝑡 − 𝜔 ( 𝑑𝑟 + 𝑟 𝑑 Ω ) , (80) Be aware that, for simplicity, from now on we will use 𝑟 for the isotropiccoordinates and 𝑟 for the Schwarzschild coordinates FIG. 5. Energy density for the right path in Schwarzschild like vs theradial coordinateFIG. 6. Radial pressures obtained by the left and right path.FIG. 7. Energy densities obtained by the left and right path. in which Einstein’s equations take the following form8 𝜋𝑇 = 𝜔 (cid:48)(cid:48) − 𝜔 (cid:48) 𝜔 + 𝑟 𝜔 (cid:48) , (81) − 𝜋𝑇 = 𝜔 (cid:48) 𝜔 − 𝜔 (cid:48) 𝜈 (cid:48) + 𝜈 (cid:48) 𝜔 − 𝜔 (cid:48) 𝑟 , (82) − 𝜋𝑇 = (cid:18) 𝜔 (cid:48) 𝜔 − 𝜔 (cid:48)(cid:48) (cid:19) + (cid:18) 𝜈 (cid:48)(cid:48) + 𝜈 (cid:48) + 𝜈 (cid:48) 𝑟 (cid:19) 𝜔 − 𝜔 (cid:48) 𝑟 . (83)This coordinates seems to be more general than theSchwarzschild like ones (1), since there is always possibleto transform the line element from the standard form to theisotropic one by 𝑟 = 𝐾 exp (cid:26)∫ 𝜇 − / 𝑑𝑟 𝑟 (cid:27) , (84)where 𝐾 , 𝜇 − and 𝑟 are the radial component and the coordi-nate associated to the line element in standard coordinates (1).However, is not always possible to perform the reverse process.Therefore, there is a chance to obtain solutions to Einstein’sequations that can not be found using the Schwarzschild likecoordinates.If we consider that 𝜔 = ( ˜ 𝐴 ( 𝑟 )) , the eqs (81)-(83) reads8 𝜋𝑇 = − ( ˜ 𝐴 (cid:48) ) + 𝐴 ˜ 𝐴 (cid:48)(cid:48) + 𝑟 ˜ 𝐴 ˜ 𝐴 (cid:48) , (85) − 𝜋𝑇 = ( ˜ 𝐴 (cid:48) ) − ˜ 𝐴 ˜ 𝐴 (cid:48) 𝜈 (cid:48) + 𝜈 (cid:48) ˜ 𝐴 𝑟 − 𝐴 ˜ 𝐴 (cid:48) 𝑟 , (86) − 𝜋𝑇 = ( ˜ 𝐴 (cid:48) ) − ˜ 𝐴 ˜ 𝐴 (cid:48)(cid:48) + (cid:18) 𝜈 (cid:48)(cid:48) + ( 𝜈 (cid:48) ) + 𝜈 (cid:48) 𝑟 (cid:19) ˜ 𝐴 − ˜ 𝐴 ˜ 𝐴 (cid:48) 𝑟 . (87)We notice from (81)-(83) (or (85)-(87)) that the system willnot decouple if we choose an energy momentum tensor ofthe form (8) and consider the particular ansatz for the metric(12) of the MGD method. We have discussed this issue on[55] and we have shown how this conditions can be usedto obtain new internal analytical and physical solutions ofEinstein’s equations in isotropic coordinates. In fact, we haveproposed two different and inequivalent algorithms that allowus to obtain new solutions in isotropic coordinates, startingfrom a seed solution. This was done considering only (12)which in Schwarzschild coordinates it is known as MGD.We will see how we can generalize the first and secondalgorithms from [55], in order to introduce both deformations. First Algorithm
As we mention before, Einstein equations in isotropic coor-dinates are not decoupled when considering a deformation ofthe metric component of the form (see [55] for details) 𝜔 ↦→ (cid:101) 𝜔 = 𝜔 + 𝛼 𝑓 , (88) 𝜈 ↦→ (cid:101) 𝜈 = 𝜈 + 𝛼ℎ. (89) However, let us notice that if we consider a specific combina-tion of equations (81)-(83) given by8 𝜋 ( 𝑃 𝑟 + 𝜌 + 𝑃 𝑡 ) = 𝜈 (cid:48) 𝜔 + 𝜈 (cid:48)(cid:48) 𝜔 + 𝜈 (cid:48) 𝜔𝑟 − 𝜔 (cid:48) 𝜈 (cid:48) , (90)is easy to verify that (8), (88) and (89) implies that equation(90) splits in two equations8 𝜋 ( 𝑃 𝑟 + 𝜌 + 𝑃 𝑡 ) = 𝜈 (cid:48) 𝜔 + 𝜈 (cid:48)(cid:48) 𝜔 + 𝜈 (cid:48) 𝜔𝑟 , − 𝜔 (cid:48) 𝜈 (cid:48) , (91)8 𝜋 (− 𝜃 + 𝜃 − 𝜃 ) + 𝑍 ( 𝑟 ) = 𝐹 ( 𝑟 ) , (92)with 𝑍 ( 𝑟 ) = ℎ (cid:48)(cid:48) 𝜔 + 𝛼ℎ (cid:48) 𝜔 + 𝜉 (cid:48) ℎ (cid:48) 𝜔 + 𝑟 ℎ (cid:48) 𝜔 − 𝜔 (cid:48) ℎ (cid:48) , (93) 𝐹 ( 𝑟 ) = (cid:26) 𝜈 (cid:48) 𝑓 + 𝜈 (cid:48)(cid:48) 𝑓 + 𝜈 (cid:48) 𝑓𝑟 − 𝑓 (cid:48) ˜ 𝜈 (cid:48) (cid:27) , (94)related to the perfect fluid and the source, respectively. At thispoint, two restrictions has to be imposed in order to obtain 𝑓 and ℎ . Moreover, let us notice that from Eqs (81)-(83) it canbe check that8 𝜋𝜃 = 𝜔 + 𝛼 𝑓 (cid:26) (cid:20) 𝜔 (cid:48)(cid:48) + 𝑟 𝜔 (cid:48) − 𝜋𝑇 (cid:21) 𝑓 + (cid:20) 𝑟 𝜔 − 𝜔 (cid:48) (cid:21) 𝑓 (cid:48) , + 𝑓 (cid:48)(cid:48) 𝜔 + 𝛼 (cid:20) 𝑟 𝑓 (cid:48) 𝑓 − 𝑓 (cid:48) + 𝑓 𝑓 (cid:48)(cid:48) (cid:21) (cid:27) , (95)8 𝜋𝜃 = − ( 𝜔 + 𝛼 𝑓 ) (cid:26) (cid:20) 𝜋𝑇 + 𝑟 ˜ 𝜈 (cid:48) 𝜔 − 𝜔 (cid:48) ˜ 𝜈 (cid:48) − 𝜔 (cid:48) 𝑟 (cid:21) 𝑓 , + (cid:20) 𝜔 (cid:48) − 𝜔 ˜ 𝜈 (cid:48) − 𝑟 𝜔 (cid:21) 𝑓 (cid:48) − (cid:20) ℎ (cid:48) 𝜔 (cid:48) − ℎ (cid:48) 𝜔𝑟 (cid:21) 𝜔 , + 𝛼 (cid:20) 𝑓 (cid:48) −
12 ˜ 𝜈 (cid:48) 𝑓 (cid:48) 𝑓 + ˜ 𝜈 (cid:48) 𝑓 − 𝑓 (cid:48) 𝑓𝑟 (cid:21) (cid:27) , (96)8 𝜋𝜃 = − ( 𝜔 + 𝛼 𝑓 ) (cid:26) (cid:20) 𝜋𝑇 − 𝜔 (cid:48)(cid:48) + 𝜔 (cid:18) ˜ 𝜈 (cid:48)(cid:48) + ( ˜ 𝜈 (cid:48) ) , + ˜ 𝜈 (cid:48) 𝑟 (cid:19) − 𝜔 (cid:48) 𝑟 (cid:21) 𝑓 + (cid:104) 𝜔 (cid:48) − 𝜔 𝑟 (cid:105) 𝑓 (cid:48) − 𝜔 𝑓 (cid:48)(cid:48) + 𝜔 (cid:20) ℎ (cid:48)(cid:48) , + 𝜈 (cid:48) ℎ (cid:48) + ℎ (cid:48) 𝑟 (cid:21) + 𝛼 (cid:20) ( 𝑓 (cid:48) ) − 𝑓 (cid:48)(cid:48) 𝑓 + 𝜔 ( ℎ (cid:48) ) , , − 𝑓 (cid:48) 𝑓 𝑟 + 𝑓 (cid:18) ˜ 𝜈 (cid:48)(cid:48) + ( ˜ 𝜈 (cid:48) ) + ˜ 𝜈 (cid:48) 𝑟 (cid:19)(cid:21) (cid:27) . (97)Despite the non decoupling of Einstein equations in isotropiccoordinates, once we know the deformations functions 𝑓 and ℎ , we can obtain 𝜃 , 𝜃 and 𝜃 . Second Algorithm
Let us consider the system of equations (85-87) and let usassume that the system with 𝛼 ≠ (cid:101) 𝐴 ( 𝑟 ) = 𝐴 ( 𝑟 ) + 𝛼 𝑓 ( 𝑟 ) , (98) 𝜈 ( 𝑟 ) = 𝜉 ( 𝑟 ) + 𝛼𝑔 ( 𝑟 ) . (99)In this algorithm there are two possible ways to generalize theprocedure presented in [55]: First let us assume that the sets { 𝜉, 𝐴, 𝑇 𝑃𝐹𝜇𝜈 } and { 𝜉, 𝑓 , 𝐻 𝜇𝜈 } are solutions of Einstein equa-tions. Then the system of equations is decomposed in8 𝜋𝜌 = − ( 𝐴 (cid:48) ) + 𝐴𝐴 (cid:48)(cid:48) + 𝑟 𝐴𝐴 (cid:48) , (100)8 𝜋𝑃 = ( 𝐴 (cid:48) ) − 𝐴𝐴 (cid:48) 𝜉 (cid:48) + 𝜉 (cid:48) 𝐴 𝑟 − 𝐴𝐴 (cid:48) 𝑟 , (101)8 𝜋𝑃 = ( 𝐴 (cid:48) ) − 𝐴𝐴 (cid:48)(cid:48) + (cid:18) 𝜉 (cid:48)(cid:48) + ( 𝜉 (cid:48) ) + 𝜉 (cid:48) 𝑟 (cid:19) 𝐴 , − 𝐴𝐴 (cid:48) 𝑟 , (102)and 8 𝜋𝐻 = − ( 𝑓 (cid:48) ) + 𝑓 𝑓 (cid:48)(cid:48) + 𝑟 𝑓 𝑓 (cid:48) , (103) − 𝜋𝐻 = ( 𝑓 (cid:48) ) − 𝑓 𝑓 (cid:48) 𝜉 (cid:48) + 𝜉 (cid:48) 𝑓 𝑟 − 𝑓 𝑓 (cid:48) 𝑟 , (104) − 𝜋𝐻 = ( 𝑓 (cid:48) ) − 𝑓 𝑓 (cid:48)(cid:48) + (cid:18) 𝜉 (cid:48)(cid:48) + ( 𝜉 (cid:48) ) + 𝜉 (cid:48) 𝑟 (cid:19) 𝑓 , − 𝑓 𝑓 (cid:48) 𝑟 , (105)which are both Einstein equations systems. There is a thirdsystem given by8 𝜋 Θ = − 𝐴 (cid:48) 𝑓 (cid:48) + ( 𝐴 𝑓 (cid:48)(cid:48) + 𝐴 (cid:48)(cid:48) 𝑓 )+ 𝑟 ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) , (106) − 𝜋 Θ + 𝑍 ( 𝑟 ) = 𝐴 (cid:48) 𝑓 (cid:48) − ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19) + 𝜈 (cid:48) 𝑓 𝐴𝑟 + 𝛼 ℎ (cid:48) (cid:18) 𝑓 𝑟 − 𝑓 𝑓 (cid:48) (cid:19) , (107) − 𝜋 Θ + 𝑍 ( 𝑟 ) = 𝐴 (cid:48) 𝑓 (cid:48) − ( 𝐴 𝑓 (cid:48)(cid:48) + 𝐴 (cid:48)(cid:48) 𝑓 )+ (cid:18) 𝜈 (cid:48)(cid:48) + ( 𝜈 (cid:48) ) + 𝜈 (cid:48) 𝑟 (cid:19) 𝐴 𝑓 , + 𝛼 𝑓 (cid:18) ℎ (cid:48)(cid:48) 𝐴 + 𝜉 (cid:48) ℎ (cid:48) 𝐴 + ℎ (cid:48) 𝐴 𝑟 (cid:19) , − ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) 𝑟 , (108)In the second possibility, we can start by assuming thatthe sets { 𝜉, 𝐴, 𝑇 𝑃𝐹𝜇𝜈 } and { 𝜈, 𝑓 , 𝐻 𝜇𝜈 } are solutions of Einsteinequations, this lead us to the following systems of equations8 𝜋𝜌 = − ( 𝐴 (cid:48) ) + 𝐴𝐴 (cid:48)(cid:48) + 𝑟 𝐴𝐴 (cid:48) , (109)8 𝜋𝑃 = ( 𝐴 (cid:48) ) − 𝐴𝐴 (cid:48) 𝜉 (cid:48) + 𝜉 (cid:48) 𝐴 𝑟 − 𝐴𝐴 (cid:48) 𝑟 , (110)8 𝜋𝑃 = ( 𝐴 (cid:48) ) − 𝐴𝐴 (cid:48)(cid:48) + (cid:18) 𝜉 (cid:48)(cid:48) + ( 𝜉 (cid:48) ) + 𝜉 (cid:48) 𝑟 (cid:19) 𝐴 , − 𝐴𝐴 (cid:48) 𝑟 , (111) and 8 𝜋𝐻 = − ( 𝑓 (cid:48) ) + 𝑓 𝑓 (cid:48)(cid:48) + 𝑟 𝑓 𝑓 (cid:48) , (112) − 𝜋𝐻 = ( 𝑓 (cid:48) ) − 𝑓 𝑓 (cid:48) 𝜈 (cid:48) + 𝜈 (cid:48) 𝑓 𝑟 − 𝑓 𝑓 (cid:48) 𝑟 , (113) − 𝜋𝐻 = ( 𝑓 (cid:48) ) − 𝑓 𝑓 (cid:48)(cid:48) + (cid:18) 𝜈 (cid:48)(cid:48) + ( 𝜈 (cid:48) ) + 𝜈 (cid:48) 𝑟 (cid:19) 𝑓 , − 𝑓 𝑓 (cid:48) 𝑟 . (114)We also have a third system of equations given by8 𝜋 Θ = − 𝐴 (cid:48) 𝑓 (cid:48) + ( 𝐴 𝑓 (cid:48)(cid:48) + 𝐴 (cid:48)(cid:48) 𝑓 ) , + 𝑟 ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) , (115) − 𝜋 Θ + 𝑍 ( 𝑟 ) = 𝐴 (cid:48) 𝑓 (cid:48) − ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19) , + 𝜈 (cid:48) 𝑓 𝐴𝑟 , (116) − 𝜋 Θ + 𝑍 ( 𝑟 ) = 𝐴 (cid:48) 𝑓 (cid:48) − ( 𝐴 𝑓 (cid:48)(cid:48) + 𝐴 (cid:48)(cid:48) 𝑓 ) , + (cid:18) 𝜈 (cid:48)(cid:48) + ( 𝜈 (cid:48) ) + 𝜈 (cid:48) 𝑟 (cid:19) 𝐴 𝑓 , − (
𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) 𝑟 , (117)where 𝑍 ( 𝑟 ) = 𝐴𝐴 (cid:48) ℎ (cid:48) + 𝑟 ℎ (cid:48) 𝐴 , (118) 𝑍 ( 𝑟 ) = − ℎ (cid:48)(cid:48) 𝐴 − 𝜈 (cid:48) ℎ (cid:48) 𝐴 − ℎ (cid:48) 𝐴 𝑟 . (119)Finally, in order to avoid the appearance of singularitieson the surface of the distribution, we must impose the wellknown matching conditions between the interior and the ex-terior space-time geometries. The inner region is defined bythe metric (80) and it could being obtained using any of thetwo inequivalent algorithms. We will consider that the outerregion is described by the vacuum Schwarzschild solution 𝑑𝑠 = (cid:18) − 𝑀𝑟 (cid:19) 𝑑𝑡 − (cid:18) − 𝑀𝑟 (cid:19) − 𝑑𝑟 + 𝑟 𝑑 Ω , (120)where M denote the total mass of the distribution.Now, using (80) and (120) the matching condition takes thefollowing form 𝑒 𝜈 Σ = (cid:18) − 𝑀𝑟 Σ (cid:19) , (121) 𝑟 Σ (cid:18) 𝑟 Σ − 𝑤 (cid:48) 𝑤 (cid:19) Σ = (cid:18) − 𝑀𝑟 Σ (cid:19) / ,𝑃 𝑟 ( 𝑟 Σ ) = , (122)were the subscript Σ indicates that the quantity is evaluated atthe boundary of the distribution. Then it is possible to obtainan expression for the total mass of the distribution given by 𝑀 = 𝑟 Σ 𝑤 / (cid:34) − 𝑟 Σ (cid:18) 𝑟 Σ − 𝑤 (cid:48) 𝑤 (cid:19) (cid:35) Σ . (123) In this section we will present a way to simplify the problemof finding solutions to Einstein equations in isotropic coordi-nates, taking into account consecutive deformations on thespatial and temporal components of the metric. Let us no-tice that from the extended version of the two algorithms inisotropic coordinates, it is possible to take the same limits thatthe EGD in Schwarzschild like coordinates. This is, we cantake 𝑓 ≠ ℎ = 𝑓 = ℎ ≠
0. These correspondto • For the first algorithm , we have that the first limitcorresponds to ℎ = 𝑓 in terms of 𝐹 ( 𝑟 ) from (92) as 𝑓 = 𝑒 𝜈 ( 𝜈 (cid:48) ) 𝑟 (cid:18) ∫ 𝐹 ( 𝑟 ) 𝑒 − 𝜈 ( 𝜈 (cid:48) ) 𝑟 𝑑𝑟 + 𝐶 (cid:19) . (124)Therefore, once we have obtained 𝑓 , for a specific func-tion 𝐹 ( 𝑟 ) , then ( 𝜃 𝑓 ) , ( 𝜃 𝑓 ) and ( 𝜃 𝑓 ) are given byexpressions (95), (96) and (97) with ℎ =
0, respectively.The second limit corresponds to the case where 𝑓 = 𝜋 ( 𝜃 ℎ ) = , (125)8 𝜋 ( 𝜃 ℎ ) = (cid:20) 𝜔 (cid:48) − 𝜔𝑟 (cid:21) ℎ (cid:48) , (126)8 𝜋 ( 𝜃 ℎ ) = − 𝜔 (cid:20) ℎ (cid:48)(cid:48) + (cid:101) 𝜈 (cid:48) ℎ (cid:48) + ℎ (cid:48) 𝑟 + 𝛼ℎ (cid:48) (cid:21) . (127)Notice that we also have to impose an additional condi-tion in order to obtain the deformation function ℎ andwith it, expressions for ( 𝜃 ℎ ) , ( 𝜃 ℎ ) , ( 𝜃 ℎ ) . • For the second algorithm, the two different possibil-ities gives the sames limits. For the first limit ℎ = 𝜋 Θ = − 𝐴 (cid:48) 𝑓 (cid:48) + ( 𝐴 𝑓 (cid:48)(cid:48) + 𝐴 (cid:48)(cid:48) 𝑓 ) , + 𝑟 ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) , (128) − 𝜋 Θ = 𝐴 (cid:48) 𝑓 (cid:48) − ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19) , + 𝜈 (cid:48) 𝑓 𝐴𝑟 (129) − 𝜋 Θ = 𝐴 (cid:48) 𝑓 (cid:48) − ( 𝐴 𝑓 (cid:48)(cid:48) + 𝐴 (cid:48)(cid:48) 𝑓 ) + 𝐴 𝑓 (cid:18) 𝜈 (cid:48)(cid:48) , + ( 𝜈 (cid:48) ) + 𝜈 (cid:48) 𝑟 (cid:19) − ( 𝐴 𝑓 (cid:48) + 𝐴 (cid:48) 𝑓 ) 𝑟 (130) The second limit corresponds to the case when 𝑓 = 𝐻 = 𝐻 = 𝐻 = , (131)and 8 𝜋 Θ = , (132) − 𝜋 Θ + 𝑍 ( 𝑟 ) = , (133) − 𝜋 Θ + 𝑍 ( 𝑟 ) = , (134)Then, we can follow the same idea mentioned before, withTolman solution in standard-like coordinates and define the2-steps GD in isotropic coordinates. Let us perform a con-secutive deformations of the temporal and radial componentsof the line element in isotropic coordinates in order to obtainsolutions with ℎ ≠ 𝑓 ≠ 𝑒 𝜈 = 𝐷 (cid:18) 𝑔 − 𝑔 + (cid:19) , (135)1 𝜔 = 𝐵 (cid:18) 𝑔 + 𝑎𝑔 (cid:19) , (136) 𝑃 ( 𝑟 ) = 𝑏 𝜋𝐵 ( 𝑔 + ) (cid:20) 𝑔 ( 𝑔 + ) − 𝑏𝑟 (cid:21) , (137) 𝜌 ( 𝑟 ) = 𝑏 𝜋𝐵 ( 𝑔 + ) (cid:20) 𝑔 ( 𝑔 − )( 𝑔 − ) / − 𝑏𝑟 ( − 𝑔 ) (cid:21) , (138) 𝑔 ( 𝑟 ) = cosh 𝑎 + 𝑏𝑟 , (139)where 𝐷 , 𝐵 , 𝑎 and 𝑏 are constants. It can be seen that thissolution will be regular at the origin if 𝐵 = ( 𝑒 𝑎 + ) ( 𝑒 𝑎 + ) . (140)As we have shown before, we found [55] two different algo-rithms to obtain solutions of Einstein equations in isotropiccoordinates considering a geometric deformation of the spa-tial component of the metric, inspired in MGD. Therefore, letus study the extended version for both cases separately. Using the first algorithm
Left Path
As we show in [55], if we choose 𝐹 ( 𝑟 ) = 𝐷𝐺𝑏 𝜈 (cid:48) 𝑟 ( 𝑔 + ) , (141)0where 𝐺 is a proportionality constant, then regularity at theorigin implies 𝐵 = + 𝛼𝐷𝑏 𝐺 ( 𝑒 𝑎 + ) ( + 𝑒 𝑎 ) , (142)and it can be seen that 𝑓 𝐿 = 𝐷𝑏 𝑟 ( 𝑔 + ) (cid:20) ˜ 𝐶 − 𝐺 𝑟 (cid:21) , (143)where ˜ 𝐶 = 𝐾 + 𝐶 and 𝐾 is an integration constant. Therefore,it can be check that (cid:101) 𝐻 = 𝑟 + 𝑏 𝑟 𝑔 (cid:48) ( − 𝑔 ) + 𝐷𝐺𝑏 𝑟 𝑓 ( 𝑔 + ) , (144) (cid:101) 𝐻 = 𝐻 + 𝐻 (cid:48) , + 𝑓 𝐿 (cid:20) 𝑟 + 𝑏 𝑟 𝑔 (cid:48) ( − 𝑔 ) − 𝑔 (cid:48) 𝑔 + (cid:21) 𝐷𝐺𝑏 ( 𝑔 + ) 𝑟 . (145)The new solution is given by 𝑒 𝜈 = 𝐷 (cid:18) 𝑔 − 𝑔 + (cid:19) , (146)¯ 𝜔 = 𝜔 + 𝛼 𝑓 𝐿 , (147)¯ 𝑃 𝑟 ( 𝑟 ) = 𝑃 ( 𝑟 ) − 𝛼 ( 𝜃 𝑓𝐿 ) , (148)¯ 𝜌 ( 𝑟 ) = 𝜌 ( 𝑟 ) + 𝛼 ( 𝜃 𝑓𝐿 ) , (149)¯ 𝑃 𝑡 ( 𝑟 ) = 𝑃 ( 𝑟 ) − ( 𝜃 𝑓𝐿 ) . (150)This is the anisotropic solution that was found by the geometri-cal deformation of the radial component of Gold III, followingthe first algorithm in [55].Now, let us consider this result as seed solution, and let usperform a geometric deformation of the temporal componentof the metric. (cid:101) 𝜈 → 𝜈 = 𝜉 + 𝛽 (cid:101) ℎ 𝐿 ( 𝑟 ) . (151)It can be seen from equations of motion (81)-(83) that8 𝜋 ( 𝜃 ℎ𝐿 ) = , (152)8 𝜋 ( 𝜃 ℎ𝐿 ) = (cid:20) (cid:101) 𝜔 (cid:48) − (cid:101) 𝜔𝑟 (cid:21) ˜ ℎ (cid:48) 𝐿 , (153)8 𝜋 ( 𝜃 ℎ𝐿 ) = − 𝜔 (cid:34) ˜ ℎ (cid:48)(cid:48) 𝐿 + (cid:101) 𝜈 (cid:48) (cid:101) ℎ (cid:48) 𝐿 + (cid:101) ℎ (cid:48) 𝐿 𝑟 + 𝛽 (cid:101) ℎ (cid:48) 𝐿 (cid:35) , (154)where there are four unknown functions. In order to solve thesystem, let us consider a constraint of the form ( 𝜃 ℎ𝐿 ) = − (cid:20) 𝜔 (cid:48) − 𝜔𝑟 (cid:21) 𝜒 ( 𝑟 ) , (155)therefore is easy to verify that ℎ 𝐿 ( 𝑟 ) = − 𝜋 (cid:98) 𝐹 ( 𝑟 ) + (cid:101) 𝐶 with (cid:98) 𝐹 (cid:48) ( 𝑟 ) = 𝜒 ( 𝑟 ) , (156) FIG. 8. Radial and tangential pressures for the left path in Isotropiccoordinates vs the radial coordinate. where (cid:101) 𝐶 is an integration constant, and ( 𝜃 ℎ𝐿 ) is given byexpression (154). The new anisotropic solution is 𝑒 (cid:101) 𝜈 = 𝑒 𝜈 + 𝛽 (cid:101) ℎ = 𝐷 (cid:18) 𝑔 − 𝑔 + (cid:19) 𝑒 𝛽 (− 𝜋 (cid:98) 𝐹 ( 𝑟 )+ (cid:101) 𝐶 ) , (157)1 (cid:101) 𝜔 = 𝜔 + 𝛼 𝑓 𝐿 , (158) (cid:101) 𝜌 ( 𝑟 ) = 𝜌 + 𝛼 ( 𝜃 𝑓𝐿 ) ( 𝑟 ) , (159) (cid:101) 𝑃 𝑟 ( 𝑟 ) = 𝑃 ( 𝑟 ) − 𝛼 ( 𝜃 𝑓𝐿 ) − 𝛽 ( 𝜃 ℎ𝐿 ) , (160) (cid:101) 𝑃 𝑡 ( 𝑟 ) = 𝑃 ( 𝑟 ) − 𝛼 ( 𝜃 𝑓𝐿 ) − 𝛽 ( 𝜃 ℎ𝐿 ) . (161)Now, choosing 𝜒 ( 𝑟 ) in such a way that ℎ 𝐿 ( 𝑟 ) = 𝐾𝑟 𝑛 , (162)and imposing 𝑎 = 𝑏 =
1, ˜ 𝛼 = 𝐷𝛼 = .
01, ˜ 𝛽 = 𝐾 𝛽 = . 𝑛 = 𝐶 = 𝑟 Σ = . 𝑟 Σ = . 𝑀 = . 𝐾 = − . Right Path
If we consider a geometric deformation of the temporalcomponent of the metric of Gold III (cid:101) 𝜈 → 𝜈 = 𝜉 + 𝛼ℎ 𝑅 ( 𝑟 ) , (163)It can be seen from the equations of motion (81)-(83) that8 𝜋 ( 𝜃 ℎ𝑅 ) = , (164)8 𝜋 ( 𝜃 ℎ𝑅 ) = (cid:20) 𝜔 (cid:48) − 𝜔𝑟 (cid:21) ℎ (cid:48) 𝑅 , (165)8 𝜋 ( 𝜃 ℎ𝑅 ) = − 𝜔 (cid:20) ℎ (cid:48)(cid:48) 𝑅 + (cid:101) 𝜈 (cid:48) ℎ (cid:48) 𝑅 + ℎ (cid:48) 𝑅 𝑟 + 𝛼 ( ℎ (cid:48) 𝑅 ) (cid:21) , (166)1 FIG. 9. Energy density for the left path in Isotropic coordinates vsthe radial coordinate where there are four unknown functions. In order to solve thesystem, let us consider a constraint of the form ( 𝜃 ℎ𝑅 ) = (cid:20) 𝜔 (cid:48) − 𝜔𝑟 (cid:21) 𝜒 ( 𝑟 ) , (167)therefore is easy to verify that ℎ 𝑅 ( 𝑟 ) = − 𝜋𝐹 ( 𝑟 ) + (cid:101) 𝐶 with 𝐹 (cid:48) ( 𝑟 ) = 𝜒 ( 𝑟 ) , (168)where (cid:101) 𝐶 is an integration constant, and we can then determine ( 𝜃 ℎ𝑅 ) . The new anisotropic solution obtained by the geometricdeformation of temporal component of Gold III [90] is givenby 𝑒 (cid:101) 𝜈 = 𝑒 𝜈 + 𝛼ℎ = 𝐷 (cid:18) 𝑔 − 𝑔 + (cid:19) 𝑒 𝛼 (− 𝜋𝐹 ( 𝑟 )+ (cid:101) 𝐶 ) , (169)1 𝜔 = 𝐵 (cid:18) 𝑔 + 𝑔 (cid:19) , (170) (cid:101) 𝜌 ( 𝑟 ) = 𝜌 ( 𝑟 ) , (171) (cid:101) 𝑃 𝑟 ( 𝑟 ) = 𝑃 ( 𝑟 ) − 𝛼 ( 𝜃 ℎ𝑅 ) , (172) (cid:101) 𝑃 𝑡 ( 𝑟 ) = 𝑃 ( 𝑟 ) − 𝛼 ( 𝜃 ℎ𝑅 ) . (173)We may consider now this result as seed solution and letus do a geometric deformation on the radial component of theline element. Let us name 𝛼 and 𝛽 the parameters of the firstand second deformation, respectively. Therefore (cid:101) 𝜔 → (cid:101) 𝜔 + 𝛽 𝑓 𝑅 , (174)It can be shown that, following once a again the first algorithmof [55], we can impose the constraint ( 𝜃 𝑓𝑅 ) + ( 𝜃 𝑓𝑅 ) − ( 𝜃 𝑓𝑅 ) = (cid:101) 𝐹 𝜋 , (175)and we get 𝑓 𝑅 = 𝑒 ˜ 𝜈 ( ˜ 𝑛𝑢 (cid:48) ) 𝑟 (cid:32) ∫ (cid:101) 𝐹 ( 𝑟 ) 𝑒 − ˜ 𝜈 ˜ 𝜈 (cid:48) 𝑟 𝑑𝑟 + 𝐶 (cid:33) . (176) FIG. 10. Radial and tangential pressures for the right path in Isotropiccoordinates vs the radial coordinate
As before, we may choose (cid:101) 𝐹 ( 𝑟 ) = 𝐷𝐺𝑏 ˜ 𝜈 (cid:48) 𝑟 ( 𝑔 + ) , (177)with 𝐺 a proportionality constant, then regularity at the originimplies 𝐵 = + 𝛼𝐷𝑏 𝐺 ( 𝑒 𝑎 + ) ( + 𝑒 𝑎 ) , (178)and it can be seen that 𝑓 𝑅 = 𝑒 ˜ 𝜈 ( 𝜈 (cid:48) ) 𝑟 (cid:18) 𝐶 − 𝐺 𝑟 (cid:19) , (179)where 𝐶 and 𝐺 are constants. Therefore, we may obtain ( (cid:101) 𝜃 𝑓 ) , ( (cid:101) 𝜃 𝑓 ) and ( (cid:101) 𝜃 𝑓 ) as usual. The new anisotropic solution isgiven by 𝑒 (cid:101) 𝜈 = 𝑒 𝜈 + 𝛼ℎ 𝑅 = 𝐷 (cid:18) 𝑔 − 𝑔 + (cid:19) 𝑒 𝛼 (− 𝜋𝐹 ( 𝑟 )+ (cid:101) 𝐶 ) , (180) (cid:101) 𝜔 = 𝜔 + 𝛽 𝑓 𝑅 , (181) (cid:101) 𝑃 𝑟 ( 𝑟 ) = 𝑃 ( 𝑟 ) − 𝛼 ( 𝜃 ℎ𝑅 ) − 𝛽 ( 𝜃 𝑓𝑅 ) , (182) (cid:101) 𝜌 ( 𝑟 ) = 𝜌 ( 𝑟 ) + 𝛽 ( 𝜃 𝑓𝑅 ) , (183) (cid:101) 𝑃 𝑡 ( 𝑟 ) = 𝑃 ( 𝑟 ) − 𝛼 ( 𝜃 ℎ𝑅 ) − 𝛽 ( 𝜃 𝑓𝑅 ) . (184)Now choosing 𝐴 ( 𝑟 ) in such a way that ℎ ( 𝑟 ) = ˜ 𝐾𝑟 𝑛 , (185)it can be seen that imposing 𝑎 = 𝑏 = 𝑛 =
4, ˜ 𝛼 = 𝐺𝛼 = .
01, ˜ 𝛽 = 𝐷𝛼 = .
001 and 𝐶 =
1, from the matchingconditions, we found that 𝑟 Σ = . 𝑟 Σ = . 𝑀 = .
149 and ˜ 𝐾 = − .
07. The pressures, energy density andacceptability conditions are plotted in figures (10)-(11). Thissolution was obtained by 2-steps GD of Gold III perfect fluidsolution.2
FIG. 11. Energy density for the right path in Isotropic coordinates vsthe radial coordinate
Using the second algorithm
Left Path
As before, we will take a deformation of the spacial com-ponent of the metric. Then we need to solve the followingsystems of equations; the first one8 𝜋 ( 𝐻 𝑓𝐿 ) = − ( 𝑓 (cid:48) 𝐿 ) + 𝑓 𝑓 (cid:48)(cid:48) + 𝑟 𝑓 𝐿 𝑓 (cid:48) 𝐿 , (186) − 𝜋 ( 𝐻 𝑓𝐿 ) = ( 𝑓 (cid:48) 𝐿 ) − 𝑓 𝐿 𝑓 (cid:48) 𝐿 𝜈 (cid:48) + 𝜈 (cid:48) 𝑓 𝐿 𝑟 − 𝑓 𝐿 𝑓 (cid:48) 𝐿 𝑟 , (187) − 𝜋 ( 𝐻 𝑓𝐿 ) = ( 𝑓 (cid:48) 𝐿 ) − 𝑓 𝐿 𝑓 (cid:48)(cid:48) 𝐿 + (cid:18) 𝜈 (cid:48)(cid:48) + ( 𝜈 (cid:48) ) + 𝜈 (cid:48) 𝑟 (cid:19) 𝑓 𝐿 , − 𝑓 𝐿 𝑓 (cid:48) 𝐿 𝑟 , (188)the second one8 𝜋 ( Θ 𝑓𝐿 ) = − 𝐴 (cid:48) 𝑓 (cid:48) 𝐿 + ( 𝐴 𝑓 (cid:48)(cid:48) 𝐿 + 𝐴 (cid:48)(cid:48) 𝑓 𝐿 )+ 𝑟 ( 𝐴 𝑓 (cid:48) 𝐿 + 𝐴 (cid:48) 𝑓 𝐿 ) , (189) − 𝜋 ( Θ 𝑓𝐿 ) = 𝐴 (cid:48) 𝑓 (cid:48) 𝐿 − ( 𝐴 𝑓 (cid:48) 𝐿 + 𝐴 (cid:48) 𝑓 𝐿 ) (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19) , + 𝜈 (cid:48) 𝑓 𝐿 𝐴𝑟 , (190) − 𝜋 ( Θ 𝑓𝐿 ) = 𝐴 (cid:48) 𝑓 (cid:48) 𝐿 − ( 𝐴 𝑓 (cid:48)(cid:48) 𝐿 + 𝐴 (cid:48)(cid:48) 𝑓 𝐿 ) − ( 𝐴 𝑓 (cid:48) 𝐿 + 𝐴 (cid:48) 𝑓 𝐿 ) 𝑟 , + (cid:18) 𝜈 (cid:48)(cid:48) + ( 𝜈 (cid:48) ) + 𝜈 (cid:48) 𝑟 (cid:19) 𝐴 𝑓 𝐿 . (191)The last system can be solved for 𝑓 𝐿 as 𝑓 𝐿 = − 𝑟 ( 𝜈 (cid:48) ) 𝑒 𝜈 𝐴 (cid:32)∫ 𝜋 (( Θ 𝑓𝐿 ) − ( Θ 𝑓𝐿 ) − ( Θ 𝑓𝐿 ) ) 𝑟 ( 𝜈 (cid:48) ) 𝑒 𝜈 𝑑𝑟 + 𝐶 (cid:33) . (192) expression that can be simplified by imposing the constraint8 𝜋 (( Θ 𝑓𝐿 ) − ( Θ 𝑓𝐿 ) − ( Θ 𝑓𝐿 ) ) = − 𝐷 ( 𝜈 (cid:48) ) 𝑒 𝜈 𝑟 , (193)as 𝑓 𝐿 = 𝑔 ( 𝑔 + ) (cid:18) 𝑟 𝐷𝐶 √ 𝐵𝑏 − (cid:19) . (194)Then, with 𝑓 𝐿 it is easy to compute all the components of Θ 𝑓𝐿 and 𝐻 𝑓𝐿 . Now we can write the solution of the Einstein’sequation as 𝑒 ¯ 𝜈 = 𝑒 𝜈 , (195)¯ 𝐴 = 𝐴 + 𝛼 𝑓 𝐿 , (196)¯ 𝜌 = 𝜌 + 𝛼 [( Θ 𝑓𝐿 ) + 𝛼 ( 𝐻 𝑓𝐿 ) ] , (197)¯ 𝑃 𝑟 = 𝑝 + 𝛼 [( Θ 𝑓𝐿 ) + 𝛼 ( 𝐻 𝑓𝐿 ) ] , (198)¯ 𝑃 𝑡 = 𝑝 + 𝛼 [( Θ 𝑓𝐿 ) + 𝛼 ( 𝐻 𝑓𝐿 ) ] . (199)Let us take this solution and perform a deformation of thetemporal component of the metric. In this case the system thatwe need to solve is given by8 𝜋 ( Θ ℎ𝐿 ) = , (200)8 𝜋 ( Θ ℎ𝐿 ) = ¯ 𝐴 ¯ 𝐴 (cid:48) ℎ (cid:48) 𝐿 + 𝑟 ℎ (cid:48) 𝐿 ¯ 𝐴 , (201)8 𝜋 ( Θ ℎ𝐿 ) = − ℎ (cid:48)(cid:48) 𝐿 ¯ 𝐴 − 𝜈 (cid:48) ℎ (cid:48) 𝐿 ¯ 𝐴 − ℎ (cid:48) 𝐿 ¯ 𝐴 𝑟 . (202)Now, it is necessary to impose a constraint to solve this system.For simplicity, we will choose8 𝜋 ( Θ ℎ𝐿 ) = 𝐾𝑟 𝑛 (cid:18) ¯ 𝐴 ¯ 𝐴 (cid:48) + 𝑟 ¯ 𝐴 (cid:19) , (203)where 𝐾 and 𝑛 are constants. In order to avoid a singularityin the center of the distribution 𝑛 ≥
1. In this case, it is anstraightforward calculation to obtain ℎ 𝐿 = 𝐸 + 𝐾𝑟 𝑛 + . (204)Then, with ℎ 𝐿 , we can write the final solution that it doesinclude both perturbations of the Gold III metric componentsas 𝑒 ˜ 𝜈 = 𝑒 𝜈 + 𝛽ℎ 𝐿 , (205)˜ 𝐴 = 𝐴 + 𝛼 𝑓 𝐿 , (206)˜ 𝜌 = 𝜌 + 𝛼 (cid:104) ( Θ 𝑓𝐿 ) + 𝛼 ( 𝐻 𝑓𝐿 ) (cid:105) , (207)˜ 𝑃 𝑟 = 𝑝 − 𝛼 (cid:20) ( Θ 𝑓𝐿 ) + 𝛽𝛼 ( Θ ℎ𝐿 ) + 𝛼 ( 𝐻 𝑓𝐿 ) (cid:21) , (208)˜ 𝑃 𝑡 = 𝑝 − 𝛼 (cid:20) ( Θ 𝑓𝐿 ) + 𝛽𝛼 ( Θ ℎ𝐿 ) + 𝛼 ( 𝐻 𝑓𝐿 ) (cid:21) , (209)where to ensure the regularity at the center of the distribution 𝐵 = ( 𝑒 𝑎 + ) ( + 𝑒 𝑎 + 𝑒 𝑎 + ( 𝛼 + ) 𝑒 𝑎 + 𝑒 𝑎 ) . (210)3 FIG. 12. Radial and tangential pressures for the left path of the secondalgorithm in Isotropic coordinates vs the radial coordinateFIG. 13. Energy density for the left path of the second algorithm inIsotropic coordinates vs the radial coordinate
Now, to show an example of the behaviour of the obtainedsolution, we choose the following values for the constants, 𝐸 = 𝐾 = . , 𝐶 = 𝛼𝐷 = . 𝛽 = − . 𝑛 = , 𝑎 = 𝑏 = 𝑟 Σ = . 𝑀 = . 𝐷 = .
07. The result is showed in the figures (12)-(13)
Right Path
Performing the temporal deformation of the metric first, wehave that the system that we need to solve is8 𝜋 ( Θ ℎ𝑅 ) = , (211)8 𝜋 ( Θ ℎ𝑅 ) = 𝐴𝐴 (cid:48) ℎ (cid:48) 𝑅 + 𝑟 ℎ (cid:48) 𝑅 𝐴 , (212) 8 𝜋 ( Θ ℎ𝑅 ) = − ℎ (cid:48)(cid:48) 𝑅 𝐴 − 𝜈 (cid:48) ℎ (cid:48) 𝑅 𝐴 − ℎ (cid:48) 𝑅 𝐴 𝑟 . (213)Then, as before, if we impose the constraint8 𝜋 ( Θ ℎ𝑅 ) = 𝐾𝑟 𝑛 (cid:18) 𝐴𝐴 (cid:48) + 𝑟 𝐴 (cid:19) , (214)it is possible to solve the system of equations for ℎ 𝑅 to obtain ℎ 𝑅 = ¯ 𝐸 + ¯ 𝐾𝑟 𝑛 + . (215)In that case, the new solution can be written as 𝑒 ¯ 𝜈 = 𝑒 𝜈 + 𝛼ℎ 𝑅 , (216)¯ 𝐴 = 𝐴, (217)¯ 𝜌 = 𝜌 + 𝛼 ( Θ ℎ𝑅 ) , (218)¯ 𝑃 𝑟 = 𝑝 + 𝛼 ( Θ ℎ𝑅 ) , (219)¯ 𝑃 𝑡 = 𝑝 + 𝛼 ( Θ ℎ𝑅 ) . (220)Now, we can take this solution as the starting point to performa spatial deformation of this metric. In this case, we need tosolve the following system of equations8 𝜋 ( Θ 𝑓𝑅 ) = − 𝐴 (cid:48) 𝑓 (cid:48) 𝑅 + ( ¯ 𝐴 𝑓 (cid:48)(cid:48) 𝑅 + ¯ 𝐴 (cid:48)(cid:48) 𝑓 𝑅 ) , + 𝑟 ( ¯ 𝐴 𝑓 (cid:48) 𝑅 + ¯ 𝐴 (cid:48) 𝑓 𝑅 ) , (221) − 𝜋 ( Θ 𝑓𝑅 ) = 𝐴 (cid:48) 𝑓 (cid:48) 𝑅 − ( ¯ 𝐴 𝑓 (cid:48) 𝑅 + ¯ 𝐴 (cid:48) 𝑓 𝑅 ) (cid:18) ¯ 𝜈 (cid:48) + 𝑟 (cid:19) , + 𝜈 (cid:48) 𝑓 𝑅 ¯ 𝐴𝑟 , (222) − 𝜋 ( Θ 𝑓𝑅 ) = 𝐴 (cid:48) 𝑓 (cid:48) 𝑅 − ( ¯ 𝐴 𝑓 (cid:48)(cid:48) 𝑅 + ¯ 𝐴 (cid:48)(cid:48) 𝑓 𝑅 ) − ( ¯ 𝐴 𝑓 (cid:48) 𝑅 + ¯ 𝐴 (cid:48) 𝑓 𝑅 ) 𝑟 , + (cid:18) ¯ 𝜈 (cid:48)(cid:48) + ( ¯ 𝜈 (cid:48) ) + ¯ 𝜈 (cid:48) 𝑟 (cid:19) ¯ 𝐴 𝑓 𝑅 (223)The solution of this system for 𝑓 𝑅 has the same form of the leftpath, but substituting 𝜈 for ¯ 𝜈 . Then, if we impose a constraintwith the form8 𝜋 (( Θ 𝑓𝑅 ) − ( Θ 𝑓𝑅 ) − ( Θ 𝑓𝑅 ) ) = − ¯ 𝐷 ( ¯ 𝜈 (cid:48) ) 𝑒 ¯ 𝜈 𝑟 , (224)it not difficult to show that 𝑓 𝑅 = 𝑔 ( 𝑔 + ) (cid:18) 𝑟 𝐷𝐶 √ 𝐵𝑏 − (cid:19) 𝑒 𝛼 ¯ 𝐾𝑟 𝑛 + × (cid:32) + 𝛼 ¯ 𝐾 √︁ 𝑔 − 𝑟 𝑛 − 𝑏 (cid:33) . (225)The final solution with both deformations on the metric com-ponents is given by 𝑒 ˜ 𝜈 = 𝑒 𝜈 + 𝛼ℎ 𝑅 , (226)˜ 𝐴 = 𝐴 + 𝛽 𝑓 𝑅 , (227)˜ 𝜌 = 𝜌 + 𝛽 (cid:20) ( Θ 𝑓𝑅 ) + 𝛼𝛽 ( 𝐻 𝑓𝑅 ) (cid:21) , (228)˜ 𝑃 𝑟 = 𝑝 − 𝛽 (cid:20) ( Θ 𝑓𝑅 ) + 𝛼𝛽 ( Θ ℎ𝑅 ) + 𝛽 ( 𝐻 𝑓𝑅 ) (cid:21) , (229)˜ 𝑃 𝑡 = 𝑝 − 𝛽 (cid:20) ( Θ 𝑓𝑅 ) + 𝛼𝛽 ( Θ ℎ𝑅 ) + 𝛼 ( 𝐻 𝑓𝑅 ) (cid:21) , (230)4 FIG. 14. Radial and tangential pressures for the right path of thesecond algorithm in Isotropic coordinates vs the radial coordinateFIG. 15. Energy density for the right path of the second algorithm inIsotropic coordinates vs the radial coordinate where, to ensure the regularity at the center of the distribution 𝐵 = ( 𝑒 𝑎 + ) ( + 𝑒 𝑎 + 𝑒 𝑎 + ( 𝛽 + ) 𝑒 𝑎 + 𝑒 𝑎 ) . (231)As before, to show an example of the behavior of this solutionwe choose the following values for the constants ¯ 𝐸 = ¯ 𝐾 = . , 𝐶 = 𝛼𝐷 = − . 𝛽 = . 𝑛 = , 𝑎 = 𝑏 = 𝑟 Σ = . 𝑀 = . 𝐷 = .
07. The result is showed in the figures (14)-(15)
CONCLUSIONS
The Extended version of the gravitational decouplingmethod represents a powerful tool to study gravitational sys-tems in General Relativity, including possible corrections to more general gravitational theories. It allows us to obtainsolutions with deformations in the temporal and spatial com-ponents of the metric. However, as it is expected, to solvethe remaining system to obtain the metric deformations itis, in general, a difficult task. In this work, we have pre-sented a different approach to obtain solutions with EGD forstatic and spherical symmetric solutions in Schwarzchild-likeand isotropic coordinates. This approach is based in performconsecutive, but not simultaneous deformations of the metriccomponents. For simplicity we denote it, 2-step gravitationaldecoupling. Now, in order to ensure the decoupling of thesources, it is necessary to assume that 𝜃 𝜇𝜈 can be decomposedin two parts, where each part is responsible of only one de-formation (temporal or spatial) of the metric. For this reason,the set of solutions that can be obtained by the 2-step GD,represents a subset of the set of solutions of the EGD. On theother hand, we find that the order in which the deformationsof the metric are done, it is relevant for the final solution. Wenamed the two possible cases by left and right path. It is worthto mention that, restricting the 2-steps GD to consider onlyisotropic solutions of Einstein’s equations, we were able toreproduce the transformations presented in the theorems 1,2,3and 4 of [88].Now, while in the case of the Schwarzchild-like coordinatesthe 2-step GD is constructed as limits of the EGD, in thecase of isotropic coordinates it is more complicated. In fact,we discuss in [55] that, in general, Einstein’s equations cannot be decoupled, and we proposed two inequivalent MGDinspired algorithms to obtain solutions in this case. In thefirst algorithm, the solutions are obtained even when there isno decoupling of the sources, while in the second one, thesources can be decoupled but with some restrictions in theenergy momentum tensor. In this work we have extended bothalgorithms to include temporal and spatial deformations of themetric. Then, we construct the corresponding procedure toperform the 2-step GD. For the second algorithm we foundtwo possible extensions that include both deformations of themetric. Now, from the point of view of the 2-step GD, there isno difference between them.Among all the possible applications of the EGD method,one of particular interest is that it allow us to study gravita-tional systems, either taking into account contributions fromtheories beyond GR or the coupling of the Einstein equationswith other fields (Maxwell, Klein Gordon, etc). Due to theparticular restrictions over the energy momentum tensor, thatit is required for the 2 step GD, the application of this ap-proach for the gravitational systems mentioned before it is notclear. Thus, a more detailed analysis it is required in eachparticular case. This feature become even more evident for thealgorithms in isotropic coordinates.In order to verify the 2 step GD, we have chosen seed so-lutions of Einstein equations, which for simplicity are perfectfluid solutions. In Schwarzschild-like coordinates, this solu-tion was Tolman IV. By the 2-steps GD we were able to obtaintwo inequivalent solutions with anisotropy in the pressures byconsecutive deformation of radial and temporal components.5The inequivalence of these solutions it can be check from thefigures, therefore the deformation of both components of themetric do not commute. Moreover, solutions obtained by thisapproach are not equivalent to the simultaneous case, whichhave been presented in [89] using also Tolman IV as seed so-lution. This can be verified using the same values for the freeparameters. However, solutions obtained by 2-steps GD, as wementioned before, are contained in the simultaneous case. Inisotropic coordinates, we have chosen Gold III as seed solution.We have done the 2-steps GD for each algorithm presented in[55]. For each algorithm we obtained two solutions whichcorresponds to the left and right path. Once again, it can beseen from the figures that these solutions obtained with eachpath are not equivalent. Finally, we want to emphasize that,even when all the solutions presented in this work does satisfythe acceptability conditions, our main goal is to present the2-step GD for the Schwarzchild-like and isotropic coordinatesand not to study any particular matter distribution. APPENDIX A: PHYSICAL ACCEPTABILITY CONDITIONS
Solving Einstein’s equations does not ensure that the solu-tion will describe any physical system. Indeed, among all theknown solutions of Einstein’s equations, only a part of themfulfill the physically acceptable conditions (see for example[5]).Then, in order to ensure that the solutions of Einstein’s equa-tions are physically acceptable, we must verify if the followingconditions are satisfied • 𝑃 𝑟 , 𝑃 𝑡 and 𝜌 are positive and finite inside the distribu-tion. • 𝑑𝑃 𝑟 𝑑𝑟 , 𝑑𝑃 𝑡 𝑑𝑟 and 𝑑𝜌𝑑𝑟 are monotonically decreasing. • Dominant energy condition: 𝑃 𝑟 𝜌 ≤ 𝑃 𝑡 𝜌 ≤ • Causality condition: 0 < 𝑑𝑃 𝑟 𝑑𝜌 < < 𝑑𝑃 𝑡 𝑑𝜌 < • The local anisotropy of the distribution should be zeroat the center and increasing towards the surface.
APPENDIX B: THE RICCI INVARIANTS
Let us verify that solutions obtained by EGD, and therefore,by any of its limits, are in general inequivalent to the seedsolution considered. In order to see this, let us notice that theRicci invariants can be written in terms of the trace-free Riccitensor 𝑆 𝜇𝜈 = 𝑅 𝜇𝜈 − 𝛿 𝜇𝜈 𝑅 , (232)where 𝑅 𝜇𝜈 and 𝑅 are the Ricci tensor and the scalar curvature ,respectively. Besides the scalar curvature, Ricci invariants are defined as 𝑟 = 𝑆 𝜇𝜈 𝑆 𝜈𝜇 , (233) 𝑟 = − 𝑆 𝜇𝜈 𝑆 𝜌𝜇 𝑆 𝜈𝜌 , (234) 𝑟 = 𝑆 𝜇𝜈 𝑆 𝜌𝜇 𝑆 𝜆𝜌 𝑆 𝜈𝜆 . (235)Let us assume that { ˜ 𝜈, ˜ 𝜇 } and { 𝜈, 𝜇 } represent two solutions ofthe of Einstein’s equations for an spherically symmetric fluidwith a line element of the form (1). Defining ˜ 𝜈 and ˜ 𝜇 as˜ 𝜇 = 𝜇 + 𝛼 𝑓 ( 𝑟 ) , ˜ 𝜈 = 𝜈 + 𝛽ℎ ( 𝑟 ) , (236)then it can be shown that the scalar curvature and the thetrace-free Ricci tensor satisfies the following relations˜ 𝑅 = 𝑅 + 𝛼𝜁 ( 𝑟 ) + 𝛽𝜁 ( 𝑟 ) + 𝛼𝛽𝜁 ( 𝑟 ) , (237)and˜ 𝑆 𝜇𝜈 = 𝑆 𝜇𝜈 + 𝛿 𝜇 𝛿 𝜈 (cid:20) 𝛼 (cid:18) 𝜁 − 𝑟 (cid:18) 𝑓𝑟 + 𝑓 (cid:48) (cid:19)(cid:19) + 𝛽 ( 𝜁 + 𝛼𝜁 ) (cid:21) + 𝛿 𝜇 𝛿 𝜈 (cid:20) 𝛼 (cid:18) 𝜁 − 𝑓𝑟 (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19)(cid:19) + 𝛽 (cid:18) 𝜁 − 𝜇ℎ (cid:48) 𝑟 (cid:19) + 𝛼𝛽 (cid:18) 𝜁 − 𝑓 ℎ (cid:48) 𝑟 (cid:19)(cid:21) + 𝛿 𝜇 𝛿 𝜈 (cid:20) 𝛼 (cid:18) 𝑓𝑟 (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19) + 𝑓 (cid:48) 𝑟 − 𝜁 (cid:19) + 𝛽 (cid:18) 𝜇ℎ (cid:48) 𝑟 − 𝜁 (cid:19) + 𝛼𝛽 (cid:18) 𝑓 ℎ (cid:48) 𝑟 − 𝜁 (cid:19)(cid:21) + 𝛿 𝜇 𝛿 𝜈 (cid:20) 𝛼 (cid:18) 𝑓𝑟 (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19) + 𝑓 (cid:48) 𝑟 − 𝜁 (cid:19) + 𝛽 (cid:18) 𝜇ℎ (cid:48) 𝑟 − 𝜁 (cid:19) + 𝛼𝛽 (cid:18) 𝑓 ℎ (cid:48) 𝑟 − 𝜁 (cid:19)(cid:21) (238)where 𝜁 ( 𝑟 ) = (cid:26) 𝑓 (cid:20) ( 𝜈 (cid:48) ) + 𝜈 (cid:48)(cid:48) + 𝑟 (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19)(cid:21) + 𝑓 (cid:48) (cid:18) 𝜈 (cid:48) + 𝑟 (cid:19)(cid:27) ,𝜁 ( 𝑟 ) = (cid:26) 𝜇 (cid:20) ℎ (cid:48)(cid:48) + ℎ (cid:48) 𝜈 (cid:48) + 𝛽 ( ℎ (cid:48) ) + 𝑟 ℎ (cid:48) (cid:21) + 𝜇 (cid:48) ℎ (cid:48) (cid:27) ,𝜁 ( 𝑟 ) = (cid:26) 𝑓 (cid:20) ℎ (cid:48)(cid:48) + ℎ (cid:48) 𝜈 (cid:48) + 𝛽 ( ℎ (cid:48) ) + 𝑟 ℎ (cid:48) (cid:21) + 𝑓 (cid:48) ℎ (cid:48) (cid:27) . Now, using (1), the Ricci invariants can be written as˜ 𝑟 = (cid:2) ( ˜ 𝑆 ) + ( ˜ 𝑆 ) + ( ˜ 𝑆 ) (cid:3) , (239)˜ 𝑟 = − (cid:2) ( ˜ 𝑆 ) + ( ˜ 𝑆 ) + ( ˜ 𝑆 ) (cid:3) , (240)˜ 𝑟 = ( ˜ 𝑆 ) + ( ˜ 𝑆 ) + ( ˜ 𝑆 ) . (241)Thus, from (238), it is evident that in general˜ 𝑟 ≠ 𝑟 , ˜ 𝑟 ≠ 𝑟 , ˜ 𝑟 ≠ 𝑟 , (242)and therefore the two solutions of Einstein’s equations, repre-sented by { ˜ 𝜈, ˜ 𝜇 } and { 𝜈, 𝜇 } , are inequivalent. It is easy to see,from equations (237) and (238), that the solutions are different6even when we take 𝛼 = 𝛽 =
0, which corresponds to theto two limits of the EGD method metioned in section (). Sincethe Right and Left path of the 2-step GD produce differentresults for the deformations functions 𝑓 and 𝑔 , then the Ricciinvariant of the final solutions obtained with each path will bein general different. This indicates that the Right path and Leftpaths leads, in general, to inequivalent solutions. For the caseof the isotropic coordinates the analysis is analogous. APPENDIX C: 2-STEP GD AND THE BVW THEOREMS
In this appendix we will present the unnoticed relation thatexists between the 2-step GD (and therefore, EGD) with thefirst four theorems presented by the authors in [88]. Let usconsider as seed solution a perfect fluid sphere represented bythe set of functions { 𝜈, 𝜇, 𝜌, 𝑃 } . Left path
Starting with the pure spatial deformation (MGD) and im-posing the condition ( 𝜃 𝑓𝐿 ) = ( 𝜃 𝑓𝐿 ) , implies that the system(36)-(38) leads to the following differential equation2 𝑓 𝐿 (cid:2) 𝑟 𝐵 (cid:48)(cid:48) − ( 𝑟 𝐵 ) (cid:48) (cid:3) + 𝑓 (cid:48) 𝐿 𝑟 ( 𝑟 𝐵 ) (cid:48) = , (243)where 𝐵 ( 𝑟 ) = exp 𝜈 . The solution to (243) can be written as 𝑓 𝐿 = 𝑟 [( 𝑟 𝐵 ) (cid:48) ] exp ∫ 𝐵 (cid:48) ( 𝑟 𝐵 ) (cid:48) 𝑑𝑟. (244)The new solution is given by the set of functions { 𝜈, ¯ 𝜇, ¯ 𝜌, ¯ 𝑃 } ,where ¯ 𝜇 = 𝜇 + 𝛼 𝑓 𝐿 , (245)¯ 𝜌 = 𝜌 + 𝛼 ( 𝜃 𝑓𝐿 ) , (246)¯ 𝑃 = 𝑃 − 𝛼 ( 𝜃 𝑓𝐿 ) . (247)This correspond to the transformation in the first theorem in[88]. Now, let us consider { 𝜈, ¯ 𝜇, ¯ 𝜌, ¯ 𝑃 } as a seed solutionand perform a pure temporal deformation with the constraint ( 𝜃 ℎ𝐿 ) = ( 𝜃 ℎ𝐿 ) , then the system (45)-(47) leads to the followingdifferential equation¯ 𝜇𝑟 [ 𝑟 ( 𝑍 (cid:48)(cid:48) 𝐿 + 𝑍 (cid:48) 𝐿 𝜈 ) − 𝑍 (cid:48) 𝐿 ] + ¯ 𝜇 (cid:48) 𝑍 (cid:48) 𝐿 = , (248)where 𝑍 𝐿 ( 𝑟 ) = exp 𝛽ℎ 𝐿 . The solution in this case is 𝑍 𝐿 ( 𝑟 ) = 𝐴 + 𝐵 ∫ 𝑟𝑑𝑟𝑒 𝜈 √ ¯ 𝜇 . (249)The final solution is characterized by the set { ˜ 𝜈, ˜ 𝜇, ˜ 𝜌, ˜ 𝑃 } ˜ 𝜈 = 𝜈 + 𝛽ℎ 𝐿 , (250)˜ 𝜇 = ¯ 𝜇 = 𝜇 + 𝛼 𝑓 𝐿 , (251)˜ 𝜌 = 𝜌 + 𝛼 ( 𝜃 𝑓𝐿 ) , (252)˜ 𝑃 = 𝑃 − 𝛼 (cid:18) ( 𝜃 𝑓𝐿 ) + 𝛽𝛼 ( 𝜃 ℎ𝐿 ) (cid:19) . (253) The complete Left path that goes from { 𝜈, 𝜇, 𝜌, 𝑃 } to { ˜ 𝜈, ˜ 𝜇, ˜ 𝜌, ˜ 𝑃 } corresponds to the transformation in the third the-orem in [88]. Right path
As in the Left path, let us begin by taking { 𝜈, 𝜇, 𝜌, 𝑃 } as seedsolution. Then the pure temporal deformation of the metricsubject to the constraint ( 𝜃 ℎ𝑅 ) = ( 𝜃 ℎ𝑅 ) , leads to the followingdifferential equation 𝜇𝑟 [ 𝑟 ( 𝑍 (cid:48)(cid:48) 𝑅 + 𝑍 (cid:48) 𝑅 𝜈 ) − 𝑍 (cid:48) 𝑅 ] + 𝜇 (cid:48) 𝑍 (cid:48) 𝑅 = , (254)notice that it has the same form of Eq (248) but changing ˜ 𝜇 with 𝜇 . Therefore, it can be check that 𝑍 𝑅 ( 𝑟 ) = 𝐶 + 𝐷 ∫ 𝑟𝑑𝑟𝑒 𝜈 √ 𝜇 . (255)The new solution is given by { ¯ 𝜈, 𝜇, 𝜌, ¯ 𝑃 } where¯ 𝜈 = 𝜈 + 𝛼ℎ 𝐿 , (256)¯ 𝑃 = 𝑃 − 𝛼 ( 𝜃 ℎ𝐿 ) . (257)The transformation from { 𝜈, 𝜇, 𝜌, 𝑃 } to { ¯ 𝜈, 𝜇, 𝜌, ¯ 𝑃 } , given bythe pure temporal deformation, corresponds to the second the-orem in [88]. Now we will take { ¯ 𝜈, 𝜇, 𝜌, ¯ 𝑃 } as seed solutionand perform a pure spatial deformation, subject to the con-straint ( 𝜃 𝑓𝑅 ) = ( 𝜃 𝑓𝑅 ) . In this case the system (36)-(38) leadsto the following differential equation2 𝑓 𝑅 (cid:2) 𝑟 ¯ 𝐵 (cid:48)(cid:48) − ( 𝑟 ¯ 𝐵 ) (cid:48) (cid:3) + 𝑓 (cid:48) 𝑅 𝑟 ( 𝑟 ¯ 𝐵 ) (cid:48) = , (258)where ¯ 𝐵 ( 𝑟 ) = exp ¯ 𝜈 . Then 𝑓 𝑅 = 𝑟 [( 𝑟 ¯ 𝐵 ) (cid:48) ] exp ∫ 𝐵 (cid:48) ( 𝑟 ¯ 𝐵 ) (cid:48) 𝑑𝑟. (259)and the final solution of the Right path can be written as˜ 𝜈 = ¯ 𝜈 = 𝜈 + 𝛼ℎ 𝑅 , (260)˜ 𝜇 = 𝜇 + 𝛽 𝑓 𝑅 , (261)˜ 𝜌 = 𝜌 + 𝛽 ( 𝜃 𝑓𝑅 ) , (262)˜ 𝑃 = 𝑃 − 𝛽 (cid:18) ( 𝜃 𝑓𝑅 ) + 𝛼𝛽 ( 𝜃 ℎ𝑅 ) (cid:19) . (263)The complete Right path that goes from { 𝜈, 𝜇, 𝜌, 𝑃 } to { ˜ 𝜈, ˜ 𝜇, ˜ 𝜌, ˜ 𝑃 } corresponds to the transformation in the fourththeorem in [88]. ACKNOWLEDGEMENTS
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