Global structure of static spherically symmetric solutions surrounded by quintessence
Miguel Cruz, Apratim Ganguly, Radouane Gannouji, Genly Leon, Emmanuel N. Saridakis
aa r X i v : . [ g r- q c ] M a y Global structure of static spherically symmetricsolutions surrounded by quintessence
Miguel Cruz , Apratim Ganguly , , Radouane Gannouji ,Genly Leon , Emmanuel N. Saridakis , Facultad de F´ısica, Universidad Veracruzana 91000, Xalapa, Veracruz, M´exico Department of Mathematics, Rhodes University, 6140 Grahamstown, South Africa Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics andComputer Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban 4000,South Africa Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4950,Valpara´ıso, Chile Physics Division, National Technical University of Athens, 15780 Zografou Campus,Athens, Greece CASPER, Physics Department, Baylor University, Waco, TX 76798-7310, USA
Abstract.
We investigate all static spherically symmetric solutions in the contextof general relativity surrounded by a minimally-coupled quintessence field, usingdynamical system analysis. Applying the 1 + 1 + 2 formalism and introducing suitablenormalized variables involving the Gaussian curvature, we were able to reformulatethe field equations as first order differential equations. In the case of a masslesscanonical scalar field we recovered all known black hole results, such as the Fishersolution, and we found that apart from the Schwarzschild solution all other solutionsare naked singularities. Additionally, we identified the symmetric phase space whichcorresponds to the white hole part of the solution and in the case of a phantom field,we were able to extract the conditions for the existence of wormholes and define allpossible class of solutions such as Cold Black holes, singular spacetimes and wormholeslike Ellis wormhole, for example. For an exponential potential, we found that theblack hole solution which is asymptotically flat is unique and it is the Schwarzschildspacetime, while all other solutions are naked singularities. Furthermore, we foundsolutions connecting to a white hole through a maximum radius, and not a minimumradius (throat) such as wormhole solutions, therefore violating the flare-out condition.Finally, we have found a necessary and sufficient condition on the form of the potentialto have an asymptotically AdS spacetime along with a necessary condition for theexistence of asymptotically flat black holes. lobal structure of static spherically symmetric solutions surrounded by quintessence
1. Introduction
In general relativity singularities are formed through the collapse of massive objects,nevertheless under certain conditions these singularities are hidden behind a horizon,forming black holes. Is this simple picture universal?During the last decades, various situations have been studied in order to extend thisanalysis, from a canonical minimally coupled scalar field to higher spin fields. This workbecomes more pertinent if we consider that any quantum theory of gravity introducesadditional fields. Therefore, various extended versions of general relativity have beenstudied over the years.In majority of such extended versions of gravity one obtains one extra scalar degreeof freedom. For instance, in the case of f ( R ) gravity, a conformal transformation allowsto re-write the theory in the Einstein frame, i.e., to general relativity plus a scalarfield. For brane models, the brane bending mode can describe the embedding of thebrane in the bulk in a theory dubbed later on as Galileon [1]. In massive gravity, anadditional scalar field appears in the decoupling limit and it is responsible for the mostinteresting physics such as the vDVZ discontinuity or the Vainshtein mechanism [2].Additionally, scalar fields do also appear in various cases which include, e.g. compactifiedextra dimensions, such as the dilaton [3]. Finally, and more generally, scalar fieldsappear in every quantum description of gravity, and these models are considered asphenomenological theories which can capture some of the details of a more fundamentaltheory, namely the effective field theories [4, 5]. These scalar fields are known as dilaton,galileon, scalaron or just quintessence field. From the overstated, one understands thelarge success of studying scalar fields in various frameworks such as cosmology, blackholes, screening mechanisms, spontaneous scalarization or superradiance. These modelshave also been extended to larger group of models like Horndeski [6].Hence, there has been a great effort in order to study the spherically symmetric andblack hole solutions in the framework of extensions of general relativity, and especially inthe presence of an additional scalar field. According to a famous theorem by Bekenstein[7], a stationary canonical and minimally coupled scalar field which satisfies the condition φV ′ ( φ ) ≥ V ′′ ( φ ) > φV ′ ( φ ) ≥
0. This no-hairtheorem suggests that stationary, asymptotically flat black holes for minimally coupledscalar tensor theory will not differ from black holes in general relativity. It has beenlater generalized to different models such as Brans-Dicke [8].However, violating the assumptions of these theorems, although does not guaranteea non-trivial solution, may lead to new solutions with scalar hair. Such scalar fieldcarries a conserved Noether charge but not subject to a Gauss’s law, unlike the electriccharge, referred in the literature as hairy black holes [9]. In some cases, this definition isextended to non-trivial fields for which scalar charge is not an independent parameter.It has to be noticed that even if the no-hair theorems concerns the end point of a lobal structure of static spherically symmetric solutions surrounded by quintessence ψ , given by the action S = Z d x √− g " R − ε ∂ µ ψ∂ µ ψ − V ( ψ ) , (1)where for generality we consider both the canonical ( ε = +1) and phantom ( ε = −
2. 1+1+2 covariant approach for spherically symmetric geometry
The 1 + 1 + 2 decomposition [15, 16] follows the same strategy as the 1 + 3decomposition [17, 18], where the spacetime is decomposed into a timelike vector fieldand an orthogonal three-dimensional spacelike hypersurface. This surface is furtherdecomposed into a spacelike vector field and a 2-surface. Hence, all the informationsare embedded in a set of kinematic and dynamical variables. In cosmology, where oneusually considers Friedmann-Robertson-Walker (FRW) spacetimes, the 3-dimensionalhypersurface becomes homogeneous and isotropic. Therefore, only scalar fields remainafter the decomposition, namely the expansion scalar ( θ ), the density ( ρ ) and thepressure ( P ). lobal structure of static spherically symmetric solutions surrounded by quintessence Let us first perform the standard 1 + 3 decomposition. We define a unit timelike vector u a ( u a u a = − h ab = g ab + u a u b .Therefore, we can define two derivatives, one along the vector u a defined as˙ T a..bc..d = u e ∇ e T a..bc..d , (2)and a projected derivative defined as D e T a..bc..d = h af h pc ...h bg h qd h re ∇ r T f..gp..q . (3)Next, in order to perform the split of the 3-space, we introduce a unit spacelike vector n a , such that n a u a = 0 , n a n a = 1 , (4)along with a projection tensor on the 2-space (sheet) orthogonal to n a and u a , N ab ≡ h ab − n a n b = g ab + u a u b − n a n b , N aa = 2 . (5)Thus, we can introduce two additional derivatives in the surface orthogonal to u a , onealong the vector n a ˆ T a..bc..d ≡ n f D f T a..bc..d , (6)and the corresponding projected derivative onto the sheet δ e T a..bc..d ≡ N af ...N bg N ic ..N j d N ek D k T f..gi..j . (7) The set of variables is obtained by decomposing the various tensors along the timelikeand spacelike directions. The energy-momentum tensor T ab can be decomposed relativeto u a as T ab = ρu a u b + ph ab + q b u a + q a u b + π ab , (8)where ρ is the energy density, p the isotropic pressure, q a the energy flux and π ab thetrace-free anisotropic pressure (anisotropic stress). After the decomposition of each lobal structure of static spherically symmetric solutions surrounded by quintessence q a , π ab ) along the spacelike vector n a , the only non-zero part of the heat flux andanisotropic pressure read as q a = Qn a ,π ab = Π( n a n b − N ab ) , (9)where Q is the scalar part of the heat flux and Π is the scalar part of the anisotropicstress.Regarding the geometrical variables, the electric part of the Weyl tensor isdecomposed as E ab = E ( n a n b − N ab ), and since we focus on vorticity-free LRS-IIspacetimes, the magnetic Weyl curvature becomes H ab = 0 [19]. The additional non-zero geometrical quantities are respectively the expansionΘ = ∇ a u a , (10)the shear Σ = n a n b ∇ a u b , (11)the sheet expansion φ = δ a n a , (12)and finally, the acceleration A = n a ˙ u a . (13) As we mentioned in the Introduction, in the present work we are interested in the staticLRS class II spacetimes for the action (1), i.e., for general relativity with a minimally-coupled scalar field ψ . The advantage of our formalism is that it allows for investigationwithout the need to consider an explicit coordinate choice for the metric. Thus, for anunspecified metric g ab , the energy-momentum tensor for the scalar field is given by T ( ψ ) ab := ε ∇ a ψ ∇ b ψ − g ab (cid:2) ε ( ∇ ψ ) + 2 V ( ψ ) (cid:3) , (14)where V ( ψ ) is the scalar field potential.Considering only static spacetimes, all time derivatives are zero, which impliesΘ = Σ = Q = 0 and ˙ ψ = 0 [14]. Hence, the 1+1+2 decomposition of (14) leads to ρ = 12 ε ˆ ψ + V ( ψ ) , (15a) p = − ε ˆ ψ − V ( ψ ) , (15b)Π = 23 ε ˆ ψ , (15c) lobal structure of static spherically symmetric solutions surrounded by quintessence ψ = n µ D µ ψ ). Fromwhich we can easily derive the propagation and constraint equations [15]ˆ φ = − φ − ρ − Π2 − E , (16a)ˆ E − ˆ ρ − φ (cid:18) E + 12 Π (cid:19) , (16b)ˆ A = − ( A + φ ) A + 12 ( ρ + 3 p ) , (16c)ˆ p + ˆΠ = − (cid:18) φ + A (cid:19) Π − ( ρ + p ) A , (16d)and 0 = −A φ + 13 ( ρ + 3 p ) − E + 12 Π , (17a) K = ρ − E − Π2 + φ , (17b)where we have defined the Gaussian curvature via the Ricci tensor on the sheet as R ab = KN ab [19]. Taking the spatial derivative of (17b), and using the previousequations, we getˆ K = − φK . (18)Note that the constraint equations, which are equivalent to the hamiltonian andmomentum constraints, include no derivatives.Introducing a new variable, Ψ = ˆ ψ , we can rewrite the system (16a)-(16d) with thehelp of (15a)-(15c) and (18), in the following formˆ ψ = Ψ , (19a)ˆ φ = − φ − (cid:2) ε Ψ + V ( ψ ) (cid:3) − E , (19b)ˆ E = ε (cid:18) A − φ (cid:19) − φ E , (19c)ˆ A = − ( A + φ ) A − V ( ψ ) (19d) ε ˆΨ = − ε ( A + φ ) Ψ + V ′ ( ψ ) , (19e)ˆ K = − φK , (19f)subject to the constraints (17a,17b) E = −A φ − V ( ψ )3 + ε Ψ , (20a) K = A φ + V ( ψ ) − ε Ψ φ . (20b) lobal structure of static spherically symmetric solutions surrounded by quintessence
3. The dynamical system
Let us now follow the standard procedure and re-write the system of equationsas an autonomous dynamical system. Similar to the cosmological case, where oneuses the Friedmann equations in order to define the auxiliary dimensionless variables(Ω m , Ω r , · · · ), in the present case we will use Eq. (20b) in order to introduce the suitabledimensionless variables x = − E K , x = φ √ K , x = A√ K , y = Ψ √ K , y = V ( ψ )3 K . (21)Therefore, the constraints equations (20a),(20b) become x + 2 x x − εy + 3 y = 1 , (22a)3 x − x x + 2 εy − y = 0 . (22b)Furthermore, we define the additional variables λ = − V ,ψ V ,
Γ =
V V ,ψψ V ,ψ , (23)and thus for a given potential the scalar field can be expressed as a function of λ ( ψ = ψ ( λ )), or equivalently of Γ = Γ( λ ). In summary, the propagation equations forthe variables (21) and λ are given by x ′ = 23 εy ( x − x ) − x x , (24a) x ′ = 16 (cid:0) x − εy − y (cid:1) , (24b) x ′ = − x ( x + x ) − y , (24c) y ′ = − y ( x + x ) − ε λy √ , (24d) y ′ = y (cid:16) x − √ λy (cid:17) , (24e) λ ′ = −√ − λ y , (24f)where primes denote the normalized spatial derivative f ′ = ˆ f √ K (where as we have saidthe hat ˆ marks the derivative along the spacelike vector (e.g. ˆ ψ = n µ D µ ψ )). In thecase where one chooses a particular coordinate system, the above derivatives becomederivatives with respect to the radial coordinate, however the present formalism helpsto handle the system in a coordinate-independent manner.From (24e) it follows that the sign of y (i.e., the sign of V ( ψ )) is invariant forthe flow. This implies that our set of variables is not suitable for models where thepotential changes sign. For this set of variables, the potential should be always positiveor always negative because of the definition of λ = − V ,ψ /V which diverges when V = 0.Furthermore, the sign of λ and Γ remain unaffected under the change V ( ψ ) → − V ( ψ )(or in other words, they remain unaffected under the change y → − y ). In the followingwe will investigate the massless case y = 0 and then, we will study the case y = 0, lobal structure of static spherically symmetric solutions surrounded by quintessence Table 1.
The function f ( λ ) for the most common quintessence potentials [25] (seereferences therein).Potential f ( λ ) V ( ψ ) = V ψ N − λ N V ( ψ ) = V e − kψ + V , V > − λ ( λ − k ) V ( ψ ) = V (cid:2) e αψ + e βψ (cid:3) − ( λ + α )( λ + β ) V ( ψ ) = V [cosh ( ξψ ) − − ( λ − ξ ) V ( ψ ) = V sinh − α ( βψ ) , α > λ α − αβ with special mention of some specific solutions for negative potential ( y < y > x and y , which leads to the reduced dynamicalsystem x ′ = x x − εy , (25a) x ′ = x + x x − x − εy − , (25b) y ′ = ε λ ( x + 2 x x − εy − √ − y ( x + x ) , (25c) λ ′ = −√ − λ y . (25d)We note that for a positive potential ( y > x ( x + 2 x ) − εy ≤ (cid:8) ( x , x , y , λ ) : x ( x + 2 x ) − εy ≤ , λ ∈ R (cid:9) . (26)On the other hand, for negative potentials, we have to consider the phase space (cid:8) ( x , x , y , λ ) : x ( x + 2 x ) − εy ≥ , λ ∈ R (cid:9) . (27)Defining f ( λ ) = (Γ( λ ) − λ , where we have assumed that Γ can be expressed asa function of λ as displayed in Table 1 [25] (see references therein), we can examinedifferent classes of potentials. In the following sections, we will investigate various casesof specific potentials separately.
4. Massless scalar field
In this section, we will study a standard solution, the Fisher solution [10], whichcorresponds to a massless canonical scalar field (i.e., with ε = 1 and V ( ψ ) = 0). Wewill see that, using dynamical-system approach, we can recover all standard results ofthis solution without explicitly solving the equations.For our auxiliary variables, a null potential V = 0 implies y = λ = 0.Moreover, from Eq. (22a), we obtain an additional constraint for this particular case, lobal structure of static spherically symmetric solutions surrounded by quintessence y = x + 2 x x −
1, which allows us to reduce the dynamical system to 2D. The phasespace is given by ( x , x ) with a constraint defined as y = x + 2 x x − >
0, whichimplies a real scalar field. In fact, y = ˆ ψ / K is positive for any real valued scalar fieldsince K > K = 1 /r in Schwarzschild coordinates).In summary, we obtain the reduced dynamical system x ′ = 1 − x ( x + x ) , (28a) x ′ = − x ( x + x ) , (28b)defined in the phase space (cid:8) ( x , x ) : x + 2 x x ≥ (cid:9) . (29)The system (28) admits two fixed points in the finite region, whose stability can beobtained by following the usual linearization procedure and examining the eigenvaluesof the involved perturbation matrix. In particular, these are: P M : ( x = 1 , x = 0). The corresponding eigenvalues are − , −
1, and thus this point isa local sink, i.e., stable point.¯ P M : ( x = − , x = 0). The corresponding eigenvalues are 2 ,
1, and thus this point is alocal source, i.e., unstable point.We can easily reconstruct the metric at these points. Specifically, following [14] we have B = x and d ln A/d ln r = 2 x /x , where ( A, B ) are the gravitational potentials definedas ds = − A ( r )dt + dr B ( r ) + r (cid:16) d θ + sin θ d φ (cid:17) . (30)We, therefore, see that the two critical points correspond to Minkowski spacetime, since, B = x = 1 and d ln A/d ln r = 2 x /x = 0, which implies A = const. and can be set to A = 1 by a time redefinition.Since the system is defined on an unbounded phase space, there may exist non-trivial behavior at the region where the variables diverge. For this reason, we need tointroduce Poincar´e variables to study the behavior of the system at infinity, such as, X = x p x + x , (31a) X = x p x + x , (31b)with inverse transformation: x = X p − X − X , (32a) x = X p − X − X . (32b)The infinite boundary x + x → + ∞ corresponds to the unitary circle X + X = 1.The propagation equations read as˜ X = [1 − X (2 X + X )] (cid:0) − X − X (cid:1) , (33a)˜ X = − X (2 X + X ) (cid:0) − X − X (cid:1) , (33b) lobal structure of static spherically symmetric solutions surrounded by quintessence Table 2.
Critical points at infinity for the Poincar´e (global) system (33), for the caseof a massless canonical scalar field.Point X X Stability Nature P H P H − P S √ − √ unstable Singularity¯ P S − √ √ stable Singularity defined on the phase space (cid:8) ( X , X ) : 2 X + 2 X X + X ≥ , X + X ≤ (cid:9) , (34)where we have rescaled the radial variable as˜ f → q − X − X f ′ . (35)In Table 2, we present the critical points at infinity for the Poincar´e (global) system(33). X X P S P S P H P H P M P M Figure 1.
Global phase space for the system (33), corresponding to the case of amassless canonical scalar field. The blue region is forbidden since it leads to theviolation of the reality condition y ≥ Additionally, in Fig. 1, we depict the corresponding (global) phase space behaviorfor the system (33), i.e., the global behavior of the spherically symmetric solutions inthe case of a massless canonical scalar field. This phase space is 2D, therefore anytrajectory (or any solution) demands two initial conditions, which can be related totwo integration constants of the system, which we can call the mass and the scalarcharge. The light blue region is forbidden since it leads to the violation of thereality condition y ≥
0. The critical points ( P H , ¯ P H ) correspond to horizons, sincetheir X coordinate is X = 0 (see [14] for the definition and Appendix A.1 for the lobal structure of static spherically symmetric solutions surrounded by quintessence P S and ¯ P S correspond to singularity, since from the coordinates d ln A/d ln r = 2 x /x = 2 X /X = −
1, which reduces to A ∝ /r . The same behaviorappears for B, since for the critical point, B = x = ∞ , but after the linearization of thesystem following [14], it appears that B diverges as 1 /r when r → P H , we find X + X ≃
1. This relation doesn’t have a free parameter and henceit is unique, which implies that there is only one trajectory in the phase space from P H . It is then easy to conclude that we have only one trajectory and, therefore, onlyone solution connecting the horizon P H to Minkowski point P M asymptotically. This isillustrated in Fig. 1, where we see only one trajectory connecting P H and P M . This isthe Schwarzschild solution, as it was shown in [14]. Any other trajectory starts froma singularity and not from the horizon, and therefore it describes a naked singularity.In summary, the only solution that describes a black hole is the Schwarzschild solution.This is one of the most important properties of the Fisher solution: the unique solutiondescribing a black hole is the trivial case where the scalar field is constant and the metricis Schwarzschild. Any other situation corresponds to a naked singularity.Let us now estimate how the metric diverges near the critical surface in the Poincar´ecoordinates. The solutions at infinity correspond to x = ∞ and leads to a divergenceof the gravitational potential B . More precisely, we have B = x = X / (1 − X − X ),hence, all points at infinity which correspond in Poincar´e coordinates to the points onthe circle X + X = 1, have a divergent potential. To estimate how this divergencebehaves, we can linearize the equations (33) as X = ¯ X − ε and X = ¯ X − ε where¯ X + ¯ X = 1, which gives for the region X > ^ (cid:0) ¯ X ε + ¯ X ε (cid:1) = 2 (cid:16) ¯ X + q − ¯ X (cid:17)(cid:16) ¯ X ε + ¯ X ε (cid:17) . (36)And since ˜ X = X dX/d ln r (see [14]), we have d ln (cid:16) ¯ X ε + ¯ X ε (cid:17) d ln r = 2 (cid:16) X p − ¯ X (cid:17) , (37)from which we can conclude that B ≃ r − (cid:16) ¯ X √ − ¯ X (cid:17) for − √ < ¯ X < , (38)or equivalently B ≃ r α where α ≤ −
1. Finally, we have d ln A/d ln r = 2 x /x =2 X /X = 2 ¯ X / p − ¯ X , which implies A ≃ r X / √ − ¯ X , or equivalently A ≃ r β with β ≥ − lobal structure of static spherically symmetric solutions surrounded by quintessence ds = − F S dt + dr F S + r F − S d Ω ψ = r − S F, F = 1 − r S r , (39)then the forbidden region corresponds to S > x = 0 is an invariant submanifold, andtherefore it is interesting to study this case in particular. The equations reduce to x ′ = 1 − x . (40)Knowing that the derivative ′ is related to the radial coordinate in Schwarzschildcoordinates as x ′ ≡ dxdξ (where ξ is an affine parameter) (41)= x dxd ln r (see [14]) , (42)it implies x dξ = d ln r . Hence, from (40) we have x = ( e ξ + α ) / ( e ξ − α ), where α isan integration constant, which using (42) implies that r = e ξ − αe − ξ and therefore B = x = 1 + 4 αr . (43)Furthermore, since x = 0 we deduce that A = const. , which can be set to A = 1 by atime redefinition, and this leads to the metric ds = − dt + dr α/r + r d Ω . (44)Finally, since y = 4 α/r >
0, we have α > S = 0. As shown previously, all non trivial solutions describe a nakedsingularity.In conclusion, we see that the phase space of Fig. 1 represents completely thespectrum of solutions in the case of a massless scalar field. We have obtained thesolution connecting the horizon to an asymptotic Minkowski spacetime ( P H → P M ),which corresponds to the Schwarzschild black hole solution. Furthermore, we havethe solution connecting a singularity to an asymptotically flat solution ( P S → P M ),corresponding to the Schwarzschild solution with negative mass (naked singularity, see[14]). All other solutions connect a singularity to the Minkowski spacetime. Therefore,all solutions are asymptotically flat, nevertheless they all include naked singularities,apart from the Schwarzschild solution.Finally, we can easily see from Fig. 1 that the phase space is divided in two parts.In the above, we have analyzed only the region x >
0, since we can easily see that thephase space region x < x < lobal structure of static spherically symmetric solutions surrounded by quintessence For completeness, let us now investigate the case of a massless phantom scalar field,which has a negative kinetic term, i.e., taking ε = − x = 0, where x ′ = y >
0, satisfy the flare-out condition (see Appendix B), whichimplies a minimal size of the radius, namely a throat at x = 0. (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:2) (cid:3) Figure 2.
Global phase space for a massless ghost scalar field. The blue regionrepresents the forbidden part of the phase space. The orange region is defined by anegative Misner-Sharp mass.
As we observe in Fig. 2, the phase space has a rich structure, and we have threetypes of solutions: • The red curve represents a solution for which one asymptotic is Minkowski, whilethe other starts from the region x < x >
0. We see that asymptotically(near P M ) the Misner-Sharp mass is positive (see Appendix C). These solutions areknown in the literature as Cold Black Holes [22]. • The green curve represents solutions which are asymptotically Minkowski but startfrom the region x > x <
0. We can see that these solutions have a negativeMisner-Sharp mass and are very similar to the Fisher solution. They exhibit anaked singularity. lobal structure of static spherically symmetric solutions surrounded by quintessence • The blue curves represent solutions where both asymptotics are Minkowski, with aminimum radius when x = 0 (a throat). These are wormhole solutions [24]. Theparticular case where x = 0 gives the solution (44), however now with α <
5. Exponential potential: V = V e − λψ In this section we will investigate the case of an exponential potential V = V e − λψ ,where we have chosen V ≥
0, focusing on the standard case where the scalar fieldis non phantom, i.e where ε = +1. In the exponential potential case we have that λ = − V ,ψ /V = const. and hence Γ = 1. Thus, the last equation of (25) is triviallysatisfied, and hence we can study the reduced 3D system for ( x , x , y ) where λ is aconstant.The fixed points at the finite region of the phase space are: P M : ( x = 1 , x = 0 , y = 0). The corresponding eigenvalues are 2 , − , −
1, and thereforethis point is saddle.¯ P M : ( x = − , x = 0 , y = 0). The corresponding eigenvalues are − , ,
1, and thusthis point is saddle.These two critical points represent Minkowski spacetime. However, and in significantcontrast with the massless scalar case, these points are now saddle. Therefore it is nottrivial to have an asymptotically flat spacetime (which should be an attractor). Onlya reduced phase space will give rise to an asymptotically flat solution. And we will seethat it corresponds to Schwarzschild spacetime.Similarly to the previous section, we introduce the Poincar´e variables X = x p x + x + y , (45a) X = x p x + x + y , (45b) Y = y p x + x + y , (45c)in order to study the phase space behavior at infinity. The infinity boundary x + x + y → + ∞ corresponds to the unitary circle X + X + Y = 1. lobal structure of static spherically symmetric solutions surrounded by quintessence X = − λ √ X Y (cid:2) X + 2 X X + X − (cid:3) + Y [ X (2 X + X ) − − X X [ X (3 X + X ) − , (46a)˜ X = − λ √ X Y (cid:2) X + 2 X X + X − (cid:3) − X X + 2 X − X X + X Y (2 X + X ) + X X + X − , (46b)˜ Y = λ √ (cid:2) − Y (cid:3) (cid:2) X + 2 X X + X − (cid:3) − X Y (cid:2) X X + X + 1 (cid:3) + Y [2 X + X ] . (46c)defined on the phase space (cid:8) ( X , X , Y ) : 2 X + 2 X X + X ≤ , X + X + Y ≤ (cid:9) , (47)where we have rescaled the radial variable through˜ X → q − X − X − Y X ′ . (48)Defining I ( X , X , Y ) = 2 X + 2 X X + X −
1, it follows˜ I = I h −√ λY (cid:0) X + 2 X X + X (cid:1) − X (cid:0) X X + X − (cid:1) +2 Y (2 X + X ) + 2 X i . (49)Thus, 2 X + 2 X X + X = 1 ( I = 0) defines an invariant submanifold, which impliesthat any trajectory along this surface remains on the surface. These surfaces are denoted M and ¯ M in the Fig. 3.The analysis of the above system shows that we have the same critical pointsthan the massless case representing the horizon and the singularity (see Appendix A.3and Appendix A.4 for complete analysis). Furthermore, we have an additional criticalcurve (i.e. curve of critical points), which is defined by the boundary of the phasespace 2 X + 2 X X + X = 1 and X + X + Y = 1 (marked by the red curve inFig. 3 for x > x < X + 2 X X + X = 1.Hence, we can analyze two cases: • The trajectory is on the critical surface ( M or ¯ M ). This case reduces to a 2Dsystem studied below. • The trajectory flows from one surface to the other ( M to ¯ M ).For the first case, and having in mind that ¯ M is just the inverse of M , we needto study only the sub-system projected onto the surface M . For this we define anew set of two variables induced on M . In particular, this surface is defined by2 X + 2 X X + X = 1, hence defining X = cos θ and X = sin θ − cos θ we have lobal structure of static spherically symmetric solutions surrounded by quintessence P M P S P H P M P S P H MM P λ P λ P M P S P H P M P S P H MM X X Y Figure 3.
Phase space of the exponential potential where the blue part represents theforbidden region (phantom scalar field). M and ¯ M are the boundaries of the phasespace and represent two invariant submanifold where most of the critical points arelocalized. On the left figure, the phase space is represented with some orbits for λ = 1for which all critical points are localized on M and ¯ M , while the right figure representsthe orbits for λ = − P λ and ¯ P λ exist. our first variable, namely θ , while the second will be Y . Since on M, we have that X ≥
0, and additionally, X + X + Y ≤ θ ∈ h cos − (cid:16) √ (cid:17) , π i . The systemof equation on the surface M then becomes˜ θ = − (cid:18) cos θ − sin θ (cid:19) (cid:2) Y + cos(2 θ ) − θ ) (cid:3) (50)˜ Y = Y (cid:18) cos θ + sin θ (cid:19) (cid:2) Y + cos(2 θ ) − θ ) (cid:3) , (51)whose orbits are trivially given by Y ( θ ) = c [cos( θ ) − sin( θ )] . (52)In order to make the behavior of this case more transparent, in Fig. 4 we depict someorbits of the system (50)-(51) representing the dynamics on the invariant surface M . Aswe can see, the Minkowski point P M is the local sink on this surface, while has we havealready mentioned it, it is a saddle point in the 3D phase space. Finally, the dynamicson the invariant set ¯ M is the reversed of this figure.In summary, we deduce that we have the same conclusions previously encounteredin the massless case. The solution connecting the horizon ( P H ) to an asymptotic flatregion ( P M ) is unique and it is the Schwarzschild solution (green curve in Fig.4). Allother solutions exhibit naked singularities. This is consistent with the no-go theorem,which states that for any convex potential ( V ,ψψ >
0) , the Schwarzschild spacetime isthe unique static black hole solution which is asymptotically flat.For the second case, namely when the trajectory flows from one surface to the other lobal structure of static spherically symmetric solutions surrounded by quintessence θ (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:1) (cid:3) Figure 4.
Phase space of of the system (50)-(51) representing the dynamics on theinvariant surface M , for the case of exponential potential. ( M to ¯ M ), we define a negative definite function: h = X + X + p − X X + X − p − X , (53)and we obtain ˜ h = − Ah , where A = 2 ( Y + 1) − ( X + X ) (cid:0) X + √ λY (cid:1)p − X . (54)The function h is defined as the ratio between the equations defining ¯ M (cid:16) X + X + p − X = 0 (cid:17) and M (cid:16) X + X − p − X = 0 (cid:17) . Hence, for the points satisfying A > M to ¯ M . Using the coordinates of the critical points, it iseasy to check that A > P H , P M and P S . Therefore, any solutionstarting near one of these critical points will evolve to the surface ¯ M . The other pointslocated on the critical curve (the red curve in Fig. 3), are repelling for some values of λ and saddle elsewhere, however they are never stable. Thus, any trajectory startingnear M will end in the hypersurface ¯ M . Moreover, we found an additional critical pointwhich appears for λ > x < x > x <
0. Hence, all these solutions will cross the surface x = 0, nevertheless by violatingthe flare-out condition (˜ x = − y < x = 0.These solutions are the exact opposite of wormhole solutions encountered previously.We can now perform an expansion around each critical point, in order to revealthe behavior of the metric around it. For instance, for the two additional critical points P λ := (cid:16) √ √ λ +4 , √ √ λ +4 , λ √ λ +4 (cid:17) , ¯ P λ := (cid:16) − √ √ λ +4 , − √ √ λ +4 , − λ √ λ +4 (cid:17) , which exist only when lobal structure of static spherically symmetric solutions surrounded by quintessence λ > A ∝ r , B ∝ r − λ , (55)and since B diverges ( x → ∞ ) we approach these points in the limit r →
0. Thisis consistent with the global picture we had previously: the solution starts from aminimum radius (at r = 0), it evolves until a maximum radius ( x = 0), and then theradius decreases again to r = 0. The same analysis can be performed for the criticalcurves, for which we found A ∝ r (cid:0) − √ − X | X | (cid:1) , B ∝ r − √ − X | X | , (56)where X is the coordinate of each point on the critical curves ( − / √ < X < / √ r → B → ∞ ). Hence,each solution connects a singularity at r = 0 ( x >
0) to another singularity at r = 0( x < x = 0. Additionally, we have solutionsconnecting a non singular point, such as the horizon P H or the Minkowski space P M , toa singular space at x <
0. Finally, we have solutions connecting a singular space (redcritical curve in Fig. 3 for x >
0) to a horizon ¯ P H , which is also a naked singularity.We therefore see that, for the exponential potential, there is no other regular solutionapart from the Schwarzschild one.Finally we mention that if we consider the case where the potential is negative,then we have to reverse the allowed phase space. Specifically, only the regions in bluein Fig. 3 are accessible. Hence, we have two disconnected phase spaces. We have thesame critical points as described previously for the positive potential case (see Fig. 3),however the points P λ and ¯ P λ are now located in the blue region for λ <
6. In this newphase space the value λ = 0 brings an additional critical point (since now λ <
6) andit corresponds to a constant potential V ( φ ) = V e φ = V . According to Eq. (55) thisadditional point is an AdS solution, and it is reached when r → ∞ . But this criticalpoint corresponds to a constant potential and not exponential and therefore can’t beconsidered in this analysis. We consider for the exponential case only λ = 0. Therefore,we conclude that AdS critical point does not exist for the exponential potential, andtherefore there are no solutions with an exponential potential asymptotically AdS.
6. On some aspects of the general case
Finally, we close this work by trying to extract information for the case of a generalpotential. In this case, the only critical points at the finite region of the phase space(26), (for positive potential) correspond to Minkowski spacetime, since any other pointwill be excluded due to the reality condition of the scalar field. In particular, the criticalpoints are P M : ( x = 1 , x = 0 , y = 0 , λ = λ c ). The corresponding eigenvalues are 2 , − , − , P M : ( x = − , x = 0 , y = 0 , λ = λ c ). The corresponding eigenvalues are − , , , lobal structure of static spherically symmetric solutions surrounded by quintessence λ c is any real number). Therefore, in general,a solution with a potential will not be asymptotically flat. But the Minkowski critical linecan be stable if the phase space is reduced to a stable invariant submanifold containing it.In fact, P M has 2 attractive directions (2 negative eigenvalues) and therefore Minkowskiis an attractor on a 2 dimensional sub-space containing these two attractive directions.In the case where λ is finite, it is easy to find an invariant submanifold containingMinkowski as a critical point, namely x + 2 x x − y − . (57)According to the condition (22a) we have that y = 0, which implies that this subspacecorresponds to V = 0 which we previously studied. In summary, we conclude that for any non-negative potential which is notasymptotically zero (i.e. corresponding to finite λ ), the unique flat black hole solution isthe Schwarzschild spacetime. For completeness, we consider also negative potential for which an additional criticalpoint appears. We will study only non-phantom scalar field ( ε = +1). In this case,apart from the Minkowski points, P M , ¯ P M , we have the new points Q ( λ ⋆ ): ( x , x , y , λ ) = (cid:18) λ ⋆ √ λ ⋆ − , √ λ ⋆ − , √ λ ⋆ √ λ ⋆ − , λ ⋆ (cid:19) ,¯ Q ( λ ⋆ ): ( x , x , y , λ ) = (cid:18) − λ ⋆ √ λ ⋆ − , − √ λ ⋆ − , − √ λ ⋆ √ λ ⋆ − , λ ⋆ (cid:19) ,where λ ⋆ are the roots of the function f , f ( λ ⋆ ) = 0. These points exist only if λ ⋆ > V ( ψ ) <
0, because 3 y ≡ V ( ψ ) /K = 1 − x − x x + y < Q ( λ ⋆ ) are − λ ⋆ +2 √ λ ⋆ − , − λ ⋆ + √ − λ ⋆ +4 λ ⋆ +362 √ λ ⋆ − , − λ ⋆ − √ − λ ⋆ +4 λ ⋆ +362 √ λ ⋆ − , − λ ⋆ f ′ ( λ ⋆ ) √ λ ⋆ − . For f ′ ( λ ⋆ ) = 0 the fixed point is non-hyperbolic. Onthe other hand, for λ ⋆ f ′ ( λ ⋆ ) >
0, the critical point is an attractor while it is saddleotherwise. As previously noticed, ¯ Q ( λ ⋆ ) is a complementary point and therefore hasan exact opposite behavior. Notice that if λ ⋆ = ∞ , the critical point reduces to theMinkowski spacetime, otherwise it represents a non physical spacetime where the metricis defined by B = x = λ ⋆ / ( λ ⋆ −
4) and A = r /λ ⋆ .To study the behavior of the orbits at infinity, we use the Poincar´e variables (45)to examine the limit where x + x + y becomes infinity (we assume that λ is finite). lobal structure of static spherically symmetric solutions surrounded by quintessence X = − λ √ X Y (cid:2) X + 2 X X + X − (cid:3) + Y [ X (2 X + X ) − − X X [ X (3 X + X ) − , (58a)˜ X = − λ √ X Y (cid:2) X + 2 X X + X − (cid:3) − X X + 2 X − X X + X Y (2 X + X ) + X X + X − , (58b)˜ Y = λ √ (cid:2) − Y (cid:3) (cid:2) X + 2 X X + X − (cid:3) − X Y (cid:2) X X + X + 1 (cid:3) + Y [2 X + X ] , (58c)˜ λ = − √ Y f ( λ ) , (58d)where, as before, we have introduced a new radial variable through ˜ X → p − X − X − Y X ′ . The system (46) is recovered for f ( λ ) = 0 , λ = constant,as expected. In this case the dimensionality of the dynamical system reduces to 3D.For positive potentials the system (58) defines a flow on the phase space (cid:8) ( X , X , Y , λ ) : 2 X + 2 X X + X ≤ , X + X + Y ≤ , λ ∈ R (cid:9) , (59)whereas, for negative potentials V ( ψ ) <
0, the phase space is given by (cid:8) ( X , X , Y , λ ) : 2 X + 2 X X + X ≥ , X + X + Y ≤ , λ ∈ R (cid:9) . (60)In the following list are enumerated half of the critical points/lines at infinity( X , X , Y , λ ) of the system (58), since the points with barred labels and thecorresponding unbarred ones have the opposite dynamical behavior (see the summaryin table 3): P H : (0 , , , λ c ) with λ c ∈ R . The eigenvalues are 2 , , ,
0, thus the point isnonhyperbolic. The center manifold of P H is the 2-dimensional surface definedby the boundary of M ; X + X − p − X = 0 , X + X + Y = 1, λ ∈ R andthe critical point is unstable (see the full analysis in Appendix A.5). This pointcorresponds to the horizon as analyzed previously. P S : (cid:16) √ , − √ , , λ c (cid:17) with λ c ∈ R . The eigenvalues are √ , √ , ,
0, thus the point isnonhyperbolic. The center manifold of P S is the same as for P H and it is alsounstable (see the full analysis in Appendix A.6). As seen, in the massless case, thispoint corresponds to a singularity where A ( r ) = B ( r ) ≃ r − . P ( λ ⋆ ): (cid:18)q λ ⋆ +4 , q λ ⋆ +4 , λ ⋆ √ λ ⋆ +4 , λ ⋆ (cid:19) . It exists for f ( λ ⋆ ) = 0 , λ ⋆ ≥
6. The eigenvaluesare λ ⋆ − √ λ ⋆ +4) , λ ⋆ − √ λ ⋆ +4) , √ ( λ ⋆ − ) √ λ ⋆ +4 , − √ λ ⋆ f ′ ( λ ⋆ ) √ λ ⋆ +4 . It is unstable for f ′ ( λ ⋆ ) < , λ ⋆ > √
6, or f ′ ( λ ⋆ ) > , λ ⋆ < −√
6, and saddle otherwise. This point has already beenstudied in the case of an exponential potential. lobal structure of static spherically symmetric solutions surrounded by quintessence Table 3.
Critical points at infinity for the Poincar´e (global) system (58), for the caseof a canonical scalar field with arbitrary potential. We have defined the surfaces ¯ M (cid:16) X + X + p − X = 0 (cid:17) and M (cid:16) X + X − p − X = 0 (cid:17) .Point X X Y λ Existence Stability Nature P H λ c λ c ∈ R unstable Horizon¯ P H − λ c λ c ∈ R stable Horizon P S √ − √ λ c λ c ∈ R unstable Singularity¯ P S − √ √ λ c λ c ∈ R stable Singularity P M √ λ c λ c ∈ R saddle Minkowski¯ P M − √ λ c λ c ∈ R saddle Minkowski P AdS √ √ V ( ψ ) < f (0) > P AdS − √ − √ V ( ψ ) < f (0) > Q ( λ ⋆ ) λ ⋆ √ λ ⋆ (1+ λ ⋆ ) √ √ λ ⋆ (1+ λ ⋆ ) λ ⋆ √ λ ⋆ (1+ λ ⋆ ) λ ⋆ V ( ψ ) <
0, stable for λ ⋆ f ′ ( λ ⋆ ) > A ∝ r /λ ⋆ , f ( λ ⋆ ) = 0 saddle otherwise B ∝ λ ⋆ / ( λ ⋆ − λ ⋆ ≥ Q ( λ ⋆ ) − λ ⋆ √ λ ⋆ (1+ λ ⋆ ) − √ √ λ ⋆ (1+ λ ⋆ ) − λ ⋆ √ λ ⋆ (1+ λ ⋆ ) λ ⋆ V ( ψ ) <
0, unstable for λ ⋆ f ′ ( λ ⋆ ) > A ∝ r /λ ⋆ , f ( λ ⋆ ) = 0 saddle otherwise B ∝ λ ⋆ / ( λ ⋆ − λ ⋆ ≥ P ( λ ⋆ ) q λ ⋆ +4 q λ ⋆ +4 λ ⋆ √ λ ⋆ +4 λ ⋆ f ( λ ⋆ ) = 0, unstable for A ∝ r , B ∝ r − ( λ ⋆ ) λ ⋆ ≥ f ′ ( λ ⋆ ) < , λ ⋆ > √
6, singularityor f ′ ( λ ⋆ ) > , λ ⋆ < −√ P ( λ ⋆ ) − q λ ⋆ +4 − q λ ⋆ +4 − λ ⋆ √ λ ⋆ +4 λ ⋆ f ( λ ⋆ ) = 0, stable for A ∝ r , B ∝ r − ( λ ⋆ ) λ ⋆ ≥ f ′ ( λ ⋆ ) < , λ ⋆ > √
6, singularityor f ′ ( λ ⋆ ) > , λ ⋆ < −√ C ( λ ⋆ ) cos θ sin θ − cos θ p − X − X λ ⋆ f ( λ ⋆ ) = 0 M is unstable for singularity f ′ ( λ ⋆ ) < λ ⋆ < θ )+2 sin( θ ) √ − − cos(2 θ )+2 sin(2 θ ) cos θ sin θ − cos θ − p − X − X λ ⋆ f ( λ ⋆ ) = 0 M is unstable for singularity f ′ ( λ ⋆ ) > λ ⋆ > − θ )+2 sin( θ ) √ − − cos(2 θ )+2 sin(2 θ ) saddle otherwise¯ C ( λ ⋆ ) cos θ − sin θ − cos θ p − X − X λ ⋆ f ( λ ⋆ ) = 0 M is stable for singularity f ′ ( λ ⋆ ) > λ ⋆ > θ ) − θ ) √ − − cos(2 θ )+2 sin(2 θ ) cos θ − sin θ − cos θ − p − X − X λ ⋆ f ( λ ⋆ ) = 0 M is stable for singularity f ′ ( λ ⋆ ) < λ ⋆ < − θ ) − θ ) √ − − cos(2 θ )+2 sin(2 θ ) saddle otherwise lobal structure of static spherically symmetric solutions surrounded by quintessence C ( λ ⋆ ): is given by the conditions f ( λ ⋆ ) = 0, 2 X + 2 X X + X = 1, X + X + Y = 1.This line has already been studied in the exponential potential case where it wasrepresented by the red curve in Fig.3. It is the boundary of M and exists only if f ( λ )has roots. Notice that for a given root of f ( λ ), this line corresponds to the centermanifold of P H and P S . By using the same analysis than the exponential case, wefound that the line is unstable for f ′ ( λ ⋆ ) < , λ ⋆ < θ +2 sin θ √ − − cos(2 θ )+2 sin(2 θ ) and Y > f ′ ( λ ⋆ ) > , λ ⋆ > − θ +2 sin θ √ − − cos(2 θ )+2 sin(2 θ ) and Y < / √ < θ < π/ P AdS : (cid:16) √ , √ , , (cid:17) . This point belongs to the phase space (60), that is,it exists only for negative potential V ( ψ ) <
0. The eigenvalues are − √ , −√ , − √ − f (0)+32 √ , − − √ − f (0)2 √ . It is a sink for f (0) > V ( ψ ) < f (0) > V ( ψ ) = Λ h (cid:16) ψ √ (cid:17)i , Λ < f ( λ ) = 2 / − λ and therefore f (0) >
0, whichimplies that an AAdS solution exists, which is the MTZ black hole.Finally, it is interesting to expand this analysis to asymptotically flat potentials forwhich λ can diverge. λ → + ∞ In this section, we investigate only the limit λ → + ∞ , since the limit λ → −∞ can bestudied through the change λ → − λ in the analysis to follow. So far, we have shownthat in the phase space, Minkowski spacetime appears as a critical line. We have shownthat any trajectory connecting to a Minkowski critical point for any finite value of λ , istrivial (Schwarzschild) or a singular spacetime. We would like to study in this section,orbits connecting to the Minkowski critical point localized at infinity in the phase space λ = ∞ .For that, we consider the coordinate transformation v = g ( λ ) = λ − which maps theinterval [ λ , + ∞ [ onto ]0 , ǫ ], where ǫ = g ( λ ) = λ and λ is any positive real number. lobal structure of static spherically symmetric solutions surrounded by quintessence dX dς = v (cid:2) Y ( X (2 X + X ) − − X X ( X (3 X + X ) − (cid:3) − X Y (2 X + 2 X X + X − √ , (62a) dX dς = v (cid:2) X (cid:0) − X (cid:1) + X (cid:0) − X + 2 X Y + X (cid:1) + X (cid:0) Y + 1 (cid:1) − (cid:3) − X Y (2 X + 2 X X + X − √ , (62b) dY dς = v (cid:0) Y (2 X + X ) − X Y (cid:0) X X + X + 1 (cid:1)(cid:1) − ( Y −
1) (2 X + 2 X X + X − √ , (62c) dvdς = √ v Y f (cid:18) v (cid:19) , (62d)where we have introduced the derivative dfdς = g ( λ ) ˜ f .If we consider positive potentials, V ( ψ ) ≥
0, the above system defines a flow on thephase space (cid:8) ( X , X , Y , v ) : 2 X + 2 X X + X ≤ , X + X + Y ≤ , ≤ v ≤ ǫ (cid:9) , ǫ ≪ . (63)The fixed points (or fixed surfaces) of the system (62) at infinity (i.e., satisfying v = 0 and v f (1 /v ) = 0) on the phase space (63) are: M, ¯ M : 2 X +2 X X + X = 1 , v = 0. These surfaces are the two boundaries encounteredin the exponential case and they correspond to V ( ψ ) = 0. The eigenvalues are0 , , −√ Y , √ Y (3 β − β ), where β = lim v → v f (cid:0) v (cid:1) , β = lim v → vf ′ (cid:0) v (cid:1) . Thesesurfaces are saddle for 3 β − β >
0. For 3 β − β < Y , the surface will be a 2D unstable manifold (respectively, stable manifold) for Y < Y > P ( ∞ ): X = 0 , X = 0 , Y = 1 , v = 0. It is the endpoint of the line P ( λ ) := (cid:16) √ √ λ +4 , √ √ λ +4 , λ √ λ +4 , λ (cid:17) as λ → ∞ . The eigenvalues are √ , √ , √ , √ β − β ) . This point is unstable if 3 β − β >
0, or saddle otherwise.¯ P ( ∞ ): X = 0 , X = 0 , Y = − , v = 0. It is the endpoint of the line¯ P ( λ ) := (cid:16) − √ √ λ +4 , − √ √ λ +4 , − λ √ λ +4 , λ (cid:17) as λ → ∞ . The eigenvalues are −√ , − √ , − √ , −√ β − β ) . This point is stable if 3 β − β >
0, or saddleotherwise.It is worth noticing that the stability behavior of the points for λ → + ∞ dependscrucially on the limitlim λ →∞ f ( λ ) λ − f ′ ( λ ) λ = lim λ →∞ { (Γ( λ ) − − λ Γ ′ ( λ ) } ≡ β − β lobal structure of static spherically symmetric solutions surrounded by quintessence P ( ∞ ) and ¯ P ( ∞ ) correspond to singular points and therefore are notphysically interesting, the points on the critical surfaces M and ¯ M with coordinates( X = ± √ , X = 0 , Y = 0 , v = 0) correspond to Minkowski spacetime. Thus, tohave a black hole solution asymptotically flat, an orbit should connect to one of thesepoints corresponding to Minkowski spacetime. In conclusion, any asymptotically flatblack hole solution needs to connect the horizon located at ( X = 0 , X = 1 , Y = 0 , λ ),where λ is any real number, to the Minkowski spacetime ( X = √ , X = 0 , Y =0 , λ = ∞ ). Therefore, two variables diverge along this orbit, x near the horizon and λ asymptotically.For example, in [29] the authors found a potential which gives an asymptoticallyflat solution. We found for this potential that the orbits connect the horizon to theMinkowski spacetime at infinity in the phase space λ = ∞ . Along this orbit, we haveverified that the two variables ( x , λ ) diverge in accordance with our result.Additionally, considering the potential studied numerically in [27] V ( ψ ) = α h ( ψ − a ) − η + η )3 ( ψ − a ) + 2 η η i ( ψ − a ) , (64)we found the same result. Therefore, we see that it is fundamental to study completelythe phase space at infinity. A simple Poincar´e compactification gives non-hyperboliccritical points and therefore their study by standard tools is impossible. We postponethis analysis to a future project where the goal will be to find suitable variables in orderto have hyperbolic critical points.We would like to end this section with a general comment on asymptotically flatspacetimes. In fact, we have seen in this section that we can have asymptotically flatspacetimes “localized” at finite λ or at λ = ∞ . We see therefore that we can have e.g. aMinkowski spacetime with finite λ and therefore with a particular value of the scalar fieldand another Minkowski spacetime with infinite λ which would correspond to anothervalue of the scalar field. If they usually correspond to two different solutions, they canbe connected in the phantom case through a throat at x = 0 which will correspond toa wormhole solution connecting two asymptotically flat spacetimes but correspondingto two different values of the scalar field. For example, this could happen when thescalar field has a symmetry breaking potential with two minima. One minimum willcorrespond to finite potential and V ′ ( φ ) = 0 (extremum of the potential) which wouldimply λ = 0 and another minimum corresponding to V ( φ ) = 0 and therefore λ = ∞ .These situations can be encountered in a Higgs-like potential V ( φ ) = a ( φ − b ) . Thesesolutions are very interesting because they connect two vacuum states correspondingto the minima of potential which corresponds to a kink-like configuration of the scalarfield where the scalar field is equal to some value in one asymptotically flat region andanother one in the other region, varying substantially near the throat. An interestingexample, of such case is described by the Kodama’s solution [28]. lobal structure of static spherically symmetric solutions surrounded by quintessence
7. Conclusions
In this work, we reformulated some of the important results on black holes in the presenceof a minimally coupled scalar field, by using the dynamical system analysis. Usingthe 1 + 1 + 2 formalism in the case of spherically symmetric spacetimes and suitablenormalized variables with the help of the Gaussian curvature, we could reformulatethe Einstein equations as first order differential equations. These equations can beinvestigated using the techniques of dynamical systems.We mention that, although in the majority of works in the literature one suitablyreconstructs the scalar potential in order to find a black hole solution, in the presentwork we follow the more physical approach to define the potential from the start sincein any realistic theory the potential is given by the theory a priori.As a first case, we examined the zero potential, i.e. the case of a massless scalarfield. We were able to reduce the phase space to 2D, and then to study its globalbehavior. We recovered all known black hole results, and we found that apart fromthe Schwarzschild solution all other solutions are naked singularities. This is the wellknown Fisher solution. Additionally, we identified the symmetric phase space whichcorresponds to the white hole part of the solution. Finally, in the case of a phantomfield, we were able to extract the conditions for the existence of wormholes and analyzethe full spectrum of solutions, by defining three types of orbits, which represent threeclass of solutions, known as Cold Black holes, singular spacetimes and wormholes. TheMisner-Sharp mass turns-out to be important in order to distinguish these solutions.The very simple case where x = 0 could be integrated exactly and gives the Elliswormhole.In the case of an exponential potential we found that black hole solution whichis asymptotically flat is unique and it is the Schwarzschild spacetime, while all othersolutions are naked singularities. Moreover, we found other solution subclasses, whichconnect the two regions of the phase space through x = 0 as a wormhole, howeverby violating the flare-out condition, which implies a maximum radius instead of athroat. Nevertheless, all these solutions involve naked singularities since they connecttwo singularities or a non-singular to a singular spacetime. Various types of solutionswere discussed depending on the value of the parameter λ .Finally, generic results have been derived. First we found that in order to havean AAdS spacetime, the potential should be negative and f (0) >
0. Also, for anypotential which is not asymptotically zero, λ finite, the unique black hole solution is theSchwarzschild spacetime. Expanding the analysis to potentials for which λ can diverge,we have shown that the only possibility to have a non trivial black hole solution isto have an orbit in the phase space connecting two points at infinity: the horizon( x = 0 , x = ∞ , y = 0 , λ ), where λ is any real number, to the Minkowski spacetime( x = 1 , x = 0 , y = 0 , λ = ∞ ). The full analysis of this phase space at infinity will beperformed in a future project where the objective is to find conditions on the function f ( λ ) and therefore on the form of the potential in order to have a non trivial black hole lobal structure of static spherically symmetric solutions surrounded by quintessence Acknowledgments
R. Gannouji would like to thank Julio Oliva for helpful discussions. M. Cruzacknowledges CONACyT-M´exico for support through the grants: Repatriaciones 2015-04 and SNI. R. Gannouji thanks DII-PUCV for support through the project No.039.370/2016. A. Ganguly wants to thank NRF (South Africa) and Claude LeonFoundation for financial support. This article is based upon work from COST Action“Cosmology and Astrophysics Network for Theoretical Advances and Training Actions”,supported by COST (European Cooperation in Science and Technology). G. Leon waspartially supported by FONDECYT No. 3140244 and acknowldeges DI-VRIEA forfinancial support through Proyectos VRIEA Investigador Joven 2016 and InvestigadorJoven 2017. lobal structure of static spherically symmetric solutions surrounded by quintessence Appendix A. Center Manifold Theory
In this section, we summarize how to build the center manifold in order to study criticalpoints for which eigenvalues have a null real part. The procedure leads to a reductionof the dimension of the system. For that, we follow the approach in chapter 18 of [30].We consider vector fields in the form x ′ = Ax + f ( x , y ) , y ′ = By + g ( x , y ) , ( x , y ) ∈ R c × R s , (A.1) f ( , ) = , D f ( , ) = , g ( , ) = , D g ( , ) = , (A.2)where all the eigenvalues of the c × c matrix A have zero real parts and all eigenvaluesof the s × s matrix B have negative real parts. The functions f and g are C r functions( r ≥
2) with D f and D g being their Jacobian matrices respectively. Definition 1 (Center Manifold)
An invariant manifold will be called a centermanifold for (A.1) if it can locally be represented as follows W c ( ) = { ( x , y ) ∈ R c × R s : y = h ( x ) , | x | < δ } , (A.3) with h ( ) = , D h ( ) = , (A.4) for δ sufficiently small (cf. [30] p. 246, [31],p. 155). The conditions h ( ) = , D h ( ) = imply that W c ( ) is tangent to E c at( x , y ) = ( , ) , where E c is the generalized eigenspace whose corresponding eigenvalueshave zero real parts. The following three theorems (see theorems 18.1.2, 18.1.3 and18.1.4 in [30] p. 245-248) are the main results to the treatment of center manifolds. Thefirst two are existence and stability theorems of the center manifold for (A.1) at theorigin. The third theorem allows to compute the center manifold to any desired degreeof accuracy by using Taylor series to solve a quasilinear partial differential equation that h ( x ) must satisfy. The proofs of these results are given in [32]. Theorem 1 (Existence)
There exists a C r center manifold for (A.1) . The dynamicsof (A.1) restricted to the center manifold is, for u sufficiently small, given by thefollowing c-dimensional vector field u ′ = Au + f ( u , h ( u )) , u ∈ R c . (A.5)The next results implies that the dynamics of (A.5) near u = 0 determine thedynamics of (A.1) near ( x , y ) = ( , ) (see also Theorem 3.2.2 in [33]). Theorem 2 (Stability) i) Suppose the zero solution of (A.5) is stable (asymptoticallystable) (unstable); then the zero solution of (A.1) is also stable (asymptotically stable)lobal structure of static spherically symmetric solutions surrounded by quintessence (unstable). Then if ( x ( τ ) , y ( τ )) is a solution of (A.1) with ( x (0) , y (0)) sufficientlysmall, then there is a solution u ( τ ) of (A.5) such that, as τ → ∞ x ( τ ) = u ( τ ) + O ( e − rτ ) , (A.6) y ( τ ) = h ( u ( τ )) + O ( e − rτ ) , (A.7) where r > is a constant. This theorem says that for initial conditions of the full system sufficiently close to theorigin, trajectories through them asymptotically approach a trajectory on the centermanifold. In particular, singular points sufficiently close to the origin, sufficientlysmall amplitude periodic orbits as well as small homoclinic and heteroclinic orbits arecontained in the center manifold.To compute the center manifold we proceed as follows: suppose we have a centermanifold W c ( ) defined by (A.3); using the invariance of W c ( ) under the dynamics of(A.1), we derive a quasilinear partial differential equation that h ( x ) must satisfy. Thisis done as follows:(i) The ( x , y ) coordinates of any point on W c ( ) must satisfy y = h ( x ) . (A.8)(ii) Differentiating (A.8) with respect to time (or any coordinate of our dynamicalsystem) implies that the ( x ′ , y ′ ) coordinates of any point on W c ( ) must satisfy y ′ = D h ( x ) x ′ . (A.9)(iii) Any point in W c ( ) obey the dynamics generated by (A.1). Therefore substituting x ′ = Ax + f ( x , h ( x )) , (A.10) y ′ = Bh ( x ) + g ( x , h ( x )) , (A.11)into (A.9) gives N ( h ( x )) ≡ D h ( x ) [ Ax + f ( x , h ( x ))] − Bh ( x ) − g ( x , h ( x ))= 0 . (A.12)Equation (A.12) is a quasilinear partial differential that h ( x ) must satisfy in orderfor its graph to be an invariant center manifold. To find the center manifold, all weneed to do is solve (A.12).Unfortunately, it is probably more difficult to solve (A.12) than our originalproblem; however the following theorem gives us a method for computing anapproximated solution of (A.12) to any desired degree of accuracy. Theorem 3 (Approximation)
Let Φ : R c → R s be a C mapping with Φ ( ) = and D Φ ( ) = such that N ( Φ ( x )) = O ( k x k q ) as x → for some q > . Then, | h ( x ) − Φ ( x ) | = O ( k x k q ) as x → .lobal structure of static spherically symmetric solutions surrounded by quintessence Appendix A.1. Horizon ( P H ) for massless scalar field We introduce the new variables u = X + X − , (A.13a) v = X − , (A.13b)that translates the point P H to the origin. Then, using the center manifold theorem[30], we find that the center manifold of the origin is given locally by the graph n ( u, v ) ∈ R : v = 12 (cid:16) u + p − u ( u + 2) − (cid:17) , | u | ≪ o . (A.14)In the original variables, the center manifold of P H is an arc of the circle X + X = 1.Since it is an invariant set, the equation on the center manifold is ˜ u = 0, as expected.The center manifold is unstable as shown in Fig. 1. Appendix A.2. Singularity ( P S ) for massless scalar field The new variables that translates the point P S to the origin are u = 25 (cid:16) X + X − √ (cid:17) , (A.15a) v = 35 (cid:16) − X + X + √ (cid:17) . (A.15b)Hence, the center manifold of the origin is given locally by the graph n ( u, v ) ∈ R : v = 3 (cid:16) − p − u − √ u + 4 (cid:17) √ − u , | u | ≪ o . (A.16)In the original variables, the center manifold of P S is an arc of the circle X + X = 1.Since it is an invariant set, the equation on the center manifold is ˜ u = 0, as expected.The center manifold of P S is unstable as shown in Fig. 1. Appendix A.3. Horizon ( P H ) for exponential potential As before, introducing the new variables that translates the point P H to the origin u = Y − λ ( X + X − √ , (A.17a) v = λ ( X + X − √ , (A.17b) v = X − , (A.17c) lobal structure of static spherically symmetric solutions surrounded by quintessence (cid:8) ( u, v , v ) ∈ R : v = h ( u ) , v = h ( u ) , h ( u ) (cid:16) λu − √ h ( u ) (cid:17) + (cid:18) λ + 2 λ (cid:19) h ( u ) + λ (cid:0) h ( u )( h ( u ) + 1) + u (cid:1) = 0 , vuut − h ( u ) − √ h ( u ) λ ! − √ h ( u ) λ = 1 . (A.18)This result have been confirmed by using Taylor series up to order O ( | u | ) bysubstituting h = − λu √ − λu √ + λ u − λu √ + λ u − ( λ ( λ +42 )) u √ + λ u + O ( u ) and h = − u − u + λu √ − u + λu √ + ( − λ − u + λu √ + ( − λ − u + λ ( λ +218 ) u √ + O ( u ).In the original variables, the center manifold of P H is a subspace of the invariantmanifold M : X + X − p − X = 0 , X + X + Y = 1 represented by the red linein Fig. 3. Since it is an invariant set, the equation on the center manifold is ˜ u = 0,as expected. Besides, it is easy to conclude that we have only one trajectory, the line X + X − p − X = 0 , X + X = 1 , ≤ X ≤ √ , and therefore one solutionconnecting the Horizon P H to Minkowski point P M . Appendix A.4. Singularity ( P S ) for exponential potential Introducing the new variables that translates point P S to the origin u = λ (cid:0) − X − X + √ (cid:1) √ Y , (A.19a) v = 35 (cid:16) − X + X + √ (cid:17) , (A.19b) v = λ (cid:0) X + X − √ (cid:1) √ , (A.19c)we find that the center manifold of the origin is the graph (cid:8) ( u, v , v ) ∈ R : v = h ( u ) , v = h ( u ) ,h ( u ) √ h ( u ) λ − √ ! + 10 h ( u ) h ( u ) (cid:18) h ( u )5 λ + h ( u ) + 2 u (cid:19) + u = 0 , (cid:16) − s − (cid:0) − λh ( u ) + 9 √ h ( u ) + 6 √ λ (cid:1) λ +10 h ( u ) + 27 √ h ( u ) λ + 3 √ (cid:17) = 0 ) . (A.20) lobal structure of static spherically symmetric solutions surrounded by quintessence O ( | u | ) bysubstituting h = u √ + u √ − λu √ + u √ − λu √ + √ λ + 10) u − λu √ +( λ +252 ) u √ − ( λ ( λ +1962 )) u √ + O ( u ) and h = − (cid:16)q λ (cid:17) u − (cid:16)q λ (cid:17) u + λ u − (cid:16)q λ (cid:17) u + λ u − (cid:16) √ λ ( λ +378 ) (cid:17) u + λ u + O ( u ).In the original variables, the center manifold of P S is a subspace of the invariantmanifold M : X + X − p − X = 0 , X + X + Y = 1 represented by the red linein Fig. 3. Since it is an invariant set, the equation on the center manifold is ˜ u = 0,as expected. Besides, it is easy to conclude that we have only one trajectory, the line X + X − p − X = 0 , X + X = 1 , − √ < X <
0, and therefore one solutionconnecting the Singularity P S to Minkowski point P M . Appendix A.5. Horizon ( P H ) for arbitrary potential To investigate the stability of the horizon P H , we introduce the linear transformation u = 12 (2( λ − λ c ) + λ c ( X + X − , (A.21a) u = λ c ( X + X − − √ Y , (A.21b) v = 1 − X − X , (A.21c) v = 1 − X , (A.21d)that translates a point on the line P H for a given λ c to the origin. We find that thecenter manifold of the origin can be expressed as (cid:26) ( u , u , v , v ) ∈ R : v = u
32 + O (5) , v = u u
16 + O (5) , | u | < δ (cid:27) , δ ≪ . (A.22)where O (5) denotes terms of fifth order. The center manifold of P H is the surface definedby X + X − p − X = 0 , X + X + Y = 1 , λ ∈ R , which is confirmed by substitutingin these equations the following asymptotic expressions (found from A.21a) X = u u
32 + O (5) , (A.23) X = 1 − u − u
16 + O (5) , (A.24) Y = − u √ − λ c u √ O (5) , (A.25) λ = λ c + u + λ c u
64 + O (5) . (A.26)The dynamics on the center manifold is given by˜ u = u f ( λ c ) + u u f ′ ( λ c ) + 12 u u f ′′ ( λ c ) + 16 u u f (3) ( λ c )+ 132 λ c u f ( λ c ) + O (5) , (A.27a)˜ u = O (5) . (A.27b) lobal structure of static spherically symmetric solutions surrounded by quintessence f ( λ c ) = 0 and considering the change of coordinates x = u , y = f ( λ c ) u the abovesystem corresponds to the Takens-Bogdanov normal form with linear part J = ! [34, 30].Under the successive coordinate transformations( x, y ) → (cid:18) x + x f ′ ( λ c )2 f ( λ c ) , y (cid:19) (A.28a)( x, y ) → (cid:18) x + x ( f ( λ c ) f ′′ ( λ c ) + f ′ ( λ c ) )6 f ( λ c ) + x y , y + xy (cid:19) (A.28b)we obtain the normal form˜ x = y + O (4) , (A.29a)˜ y = − y + O (4) . (A.29b)where ˜ f ≡ dfdξ , ( ξ is an affine parameter), and the higher terms are (cid:18) x y ( f ( λ c ) f (3) ( λ c ) − f ′ ( λ c ) ) f ( λ c ) + λ c y f ( λ c ) , (cid:19) + O (5). It can be proven from these normal formsthe instability of the horizon critical point P H . Appendix A.6. Singularity ( P S ) for arbitrary potential To investigate the stability of the singularity P S , we introduce the linear transformation u = λ − λ c + 3 X + X − √ , (A.30a) u = λ c (cid:16) X + X − √ (cid:17) + Y , (A.30b) v = − X + X + √ , (A.30c) v = − X − X + √ , (A.30d)that translates a point on the line P S for a given λ c to the origin. The center manifoldof the origin can be expressed as ( ( u , u , v , v ) ∈ R : v = √ u √ u O (5) , v = √ u O (5) , | u | < δ ) , δ ≪ , (A.31)where O (5) denotes terms of fifth order on the vector norm. Hence, we have the followingasymptotic expressions X = 2 √ − u √ − u √ O (5) , (A.32) X = − √ u √ u √ O (5) , (A.33) Y = u + 18 √ λ c u + O (5) , (A.34) λ = λ c + u + √ u O (5) , (A.35) lobal structure of static spherically symmetric solutions surrounded by quintessence u is found by integrating (A.27a) and truncating the solution up to order u . Thecenter manifold of P S is a subspace of X + X − p − X = 0 , X + X + Y = 1 , λ ∈ R .The dynamics on the center manifold is given by˜ u = − √ u f ( λ c ) − √ u u f ′ ( λ c ) − u u f ′′ ( λ c ) √ − u u f (3) ( λ c )3 √ − r λ c u f ( λ c ) + O (5) , (A.36a)˜ u = O (5) . (A.36b)For f ( λ c ) = 0 and under the change of coordinates x = u , y = −√ u f ( λ c ) we have,as before, the Takens-Bogdanov normal form with linear part J = ! [34, 30].The analysis proceeds exactly as in Appendix A.6 to obtain the normal form (A.29). Appendix A.7. Anti-de Sitter point for arbitrary potentials
We know from our previous analysis that the Anti-de Sitter point ( P AdS ) exists for thechoice V ( ψ ) < f (0) >
0. But this condition is only a sufficientcondition, it is not a necessary condition. Now we will clarify what happens at thebifurcation value f (0) = 0.Under the linear transformation u = λ, (A.37a) v = Y − λ , (A.37b) v = − X − X √ , (A.37c) v = 32 (cid:16) X + X − √ (cid:17) , (A.37d)and using the center manifold theorem [30] we find that the center manifold of the originis given approximately by the graph (cid:8) ( u, v , v , v ) ∈ R : v = a u + a u + a u + O (cid:0) u (cid:1) ,v = b u + b u + b u + O (cid:0) u (cid:1) ,v = c u + c u + c u + O (cid:0) u (cid:1)(cid:9) . (A.38)where a = f ′ (0)6 , a = (4 f ′′ (0) + 8 f ′ (0) − , a = f (3) (0)+112 f ′ (0) +3 f ′ (0)(28 f ′′ (0) − ,b = √ , b = f ′ (0)6 √ , b = f ′′ (0)+112 f ′ (0) − √ , c = − √ , c = − f ′ (0)4 √ , c = − f ′′ (0)+112 f ′ (0) − √ . In the original variables, the center manifold of P AdS is ( ( X , X , Y ) ∈ R : X − p − Y √ , X − X = 0 ) . (A.39) lobal structure of static spherically symmetric solutions surrounded by quintessence u ′ = − u f ′ (0) √ − u (3 f ′′ (0) + 2 f ′ (0) )6 √ u (cid:0) − f (3) (0) − f ′ (0) + f ′ (0) (3 − f ′′ (0)) (cid:1) √ O (cid:0) u (cid:1) (A.40)For any potential satisfying f ′ (0) = ± √ , f ′′ (0) = − , f (3) = 0, the evolution equation(A.40) simplifies to u ′ = ∓ u O (cid:0) u (cid:1) . (A.41)The equilibrium point u = 0, satisfy f ′ (0) = 0. For the positive sign in the aboveequation, the solutions with u (0) < u (0) > ∞ in finite time). Concerning the equationwith negative sign, if we take the time reversal t → − t , we end up with the same caseand have the same results upon a time reversal. Such an equilibrium with one-sidedstability is sometimes said to be semi-stable. (Example 2.3 in [35]). Now, for the choice f ( λ ) = − N λ (power-law potential ψ N ), the evolution equation (A.40) simplifies to u ′ = u √ N + O (cid:0) u (cid:1) . (A.42)The equilibrium u = 0 is asymptotically stable for N <
0, while for
N > f ′ (0) = 0 in both cases. Furthermore, perturbations from the equilibriumgrow or decay algebraically in time, not exponentially as in the linear stability analysis(Example 2.4 in [35]). Finally, in the general case, and if we neglect the fifth orderterms, we obtain a 1-dimensional gradient differential equation u ′ = − dU ( u ) du , (A.43) U ( u ) = u f ′ (0)3 √ u (3 f ′′ (0) + 2 f ′ (0) )24 √ u (cid:0) f (3) (0) + 8 f ′ (0) + f ′ (0) (8 f ′′ (0) − (cid:1) √ , (A.44)where U is the effective potential. Depending on whether u = 0 is (i) an inflectionpoint of U , (ii) a (possible degenerated) minimum of U , or (iii) a (possible degenerated)maximum of U , we will have (i) semi-stability (i.e., some orbits approach the equilibriumas time passes, while other solutions departs from it, or blows up in finite time), (ii)stability or (iii) instability for the fixed point u = 0. The previous examples capturesthe main features of this classification. Appendix B. Flare-out condition
In this Appendix we examine the requirements for the satisfaction of the flare-outcondition. Considering a metric of the following form ds = − A ( r ) dt + dr B ( r ) + r dθ + r sin ( θ ) dφ , (B.1) lobal structure of static spherically symmetric solutions surrounded by quintessence t =const and θ = π/ ds = dr B ( r ) + r dφ , (B.2)we can elaborate the flare-out condition by considering the embedding geometry. In factthis spacetime can be embedded in a 3D flat space with cylindrical coordinates ( r, φ, z ),namely ds = dr B ( r ) + r dφ = dr + dz + r dφ = h (cid:16) dzdr (cid:17) i dr + r dφ , (B.3)where dzdr = ± s − B ( r ) B ( r ) . (B.4)The space which is located at B = 0 is a throat (minimum radius) if the flare-outcondition is satisfied, namely if d rdz > . (B.5)However, since d rdz = x ′ r (1 − x ) , (B.6)we conclude that the flare-out condition is satisfied if and only if x ′ > x = 0 . (B.7)In Fig. B1 we present the throat of the wormhole. Appendix C. Misner-Sharp mass
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