1 Introduction
Abstract

We discuss charged and static solutions in a shift-symmetric scalar-tensor gravity model including a negative cosmological constant. The solutions are only approximately Anti-de Sitter (AdS) asymptotically, i.e. contain a correction of quadratic order in the scalar-tensor coupling. While spherically symmetric black holes with scalar-tensor hair do exist in our model, the uncharged spherically symmetric scalar-tensor solitons constructed recently cannot be generalised to include charge. We point out that this is due to the divergence of the electric monopole at the origin of the coordinate system, while higher order multipoles are well-behaved. We also demonstrate that black holes with scalar hair exist only for horizon value larger than that of the corresponding extremal Reissner-Nordström-AdS (RNAdS) solution, i.e. that we cannot construct solutions with arbitrarily small horizon radius. We demonstrate that for fixed an horizon radius exists at which the specific heat diverges - signalling a transition from thermodynamically unstable to stable black holes. In contrast to the RNAdS case, however, we have only been able to construct a stable phase of large horizon black holes, while a stable phase of small horizon black holes does not (seem to) exist.

Charged scalar-tensor solitons and black holes

with (approximate) Anti-de Sitter asymptotics

Yves Brihaye and Betti Hartmann

Physique-Mathématique, Université de Mons-Hainaut, 7000 Mons, Belgium

Instituto de Física de São Carlos (IFSC), Universidade de São Paulo (USP), CP 369,

13560-970 , São Carlos, SP, Brazil

1 Introduction

Scalar-tensor gravity models have gained lots of interest in recent years (for reviews see e.g. [27, 28]), in particular in the context of so-called Horndeski theories [29] which – like General Relativity – lead to field equations that are maximally second order in derivatives. Asymptotically flat black holes have been discussed extensively [31] and a explicit example of a black hole with scalar-tensor hair has been presented [30]. Unfortunately, a big part of these models are now ruled out by the recent gravitational wave observations [32]. However, this does not exclude the use of these gravity models in the context of the gauge-gravity duality. This is the approach we are taking in this paper. We discuss charged and static black hole and soliton solutions of a scalar-tensor gravity model in (3+1) space-time dimensions including a negative cosmological constant. Our paper is organised as follows : in Section 2, we give the model and Ansatz. In Section 3, we discuss soliton solutions and demonstrate that static, spherically symmetric solutions do not exist. In Section 4, we discuss charged black holes with scalar hair and point out that black holes can only be constructed for a sufficiently large value of the horizon radius. Section 5 contains our conclusions.

2 The model

The model we are studying in this paper is a Horndeski scalar-tensor model coupled to a gauge field. Its action reads :

 S=∫d4x√−g[R−2Λ+γ2ϕG−∂μϕ∂μϕ−14FμνFμν] , (1)

where the Gauss-Bonnet term is given by

 G=RμνρσRμνρσ−4RμνRμν+R2 , (2)

and is the field strength tensor of the gauge field . is the scalar-tensor coupling and is the negative valued cosmological constant. Units are chosen such that .

Varying the action (1) with respect to the metric, the scalar field and the gauge field gives the following coupled system of non-linear differential equations :

 Gμν+Λgμν−∂μϕ∂νϕ+12gμν∂αϕ∂αϕ+γ2(gρμgσν+gρνgσμ)∇λ(∂γϕϵγσαβϵδηRδηαβ)=0 , (3)
 □ϕ=−γ2G ,  ∂μ(√−gFμν)=0 . (4)

For the metric and the scalar, we choose the following Ansatz :

 ds2=−N(r)σ(r)2dt2+1N(r)dr2+r2(dθ2+sin2θdφ2)  ,  ϕ=ϕ(r)  . (5)

We will discuss the Ansatz for the U(1) gauge field separtely in the case of solitons and black holes. In [33] the case with vanishing gauge field, i.e. the uncharged case has been studied. It was found that the metric and scalar functions asymptotically behave as follows :

 N(r)=C1−λ3r2−Mr+O(r−2)  ,  σ(r)=1+O(r−2)  ,  ϕ(r)=C0−C2ln(r)+O(r−2) , (6)

where determines the gravitational mass of the solution. , . , are constants that depend on and . In particular note that for , hence our solutions are not asymptotically equal to global Anti-de Sitter (AdS), but only approximately.

3 Soliton solutions

We have investigated the coupling of the static, spherically symmetric scalar-tensor solitons to a spherically symmetric, static electric field, but were unable to find a corresponding globally regular solution numerically.

Hence, we have studied the electric and magnetic multipoles of a static and axially symmetric electromagnetic field in the background of the scalar-tensor soliton solution presented in [33]. We will show in the following that the spherically symmetric monopole always diverges in this case, hence spherically symmetric, charged generalisations of the AdS scalar-tensor solitons constructed in [33] do not exist.

Following [34] we choose the static, axially symmetric separation Ansatz for the U(1) gauge field as follows

 Aμdxμ=At(r,θ)dt+Aφ(r,θ)dφ . (7)

The Maxwell equations (see (4)) then read :

 Nσ∂r(r2∂rAtσ)+1sinθ∂θ(sinθ∂θAt)=0   ,   r2σ∂r(r2Nσ∂rAφ)+sinθ∂θ(∂θAφsinθ)=0 . (8)

These equations have been studied in great detail for , i.e. in global AdS space-time in [34], where and . In this latter case, analytical solutions are possible, which is not the case when extending the model to include a non-minimally coupled scalar field.

In the following, we will treat the electric and magnetic multipoles separately. Of course, solutions with both an electric and a magnetic field are possible, as was shown for the case of global AdS in [34].

3.1 Electric multipoles

Choosing , we employ the following separation of variables

 At(r,θ)=Rℓ(r)Pℓ(cosθ) (9)

where the , are the Legendre polynomials. Introducing the dimensionless quantities

 x=√−Λ3r    ,    ζ2=γ2Λ29 (10)

the equation for reads (prime denotes the derivative with respect to ) :

 R′′ℓ=ℓ(ℓ+1)Rℓx2N+R′ℓ(σ′σ−2x) (11)

with the metric functions and given by the numerical solutions presented in [33].

In the following, we will distinguish two cases:

• : this is the monopole term. In this case we find

 R0(x)∼∫σ(x)x2 dx+C , (12)

where is an integration constant and the normalisation of can be adapted to the boundary conditions. Now using the behaviour of close to given by (see [33]) we find :

 R0(x≪1)∼1x+C+O(x) . (13)

In order to make this more concrete, let us use compute the correction to the metric up to second order in . This reads :

 (14)

To second order in Eq. (11) can be solved to give

 R0(x)∼(ζ29−1)1x+ζ29arctan(x) , (15)

where the overall normalisation can be chosen appropriately. Unless we choose , this function diverges at – as confirmed by our more general numerical analysis.

We conclude that the monopole term diverges at and hence spherically symmetric, static and charged generalisations of the scalar-tensor solitons with approximate AdS asymptotics are not possible - unless we choose the absolute value of the scalar-tensor coupling exactly according to : and neglect terms higher than quadratic order in the expansions for the metric functions.

•  : In this case, the following boundary conditions can be employed :

 Rℓ(x=0)=0  ,  Rℓ(x→∞)→1  for  ℓ>0 . (16)

The equation (11) can only be solved numerically since the metric functions and are only known numerically. This contrasts with the case of pure AdS where analytical solutions have been constructed [34]. The modification of the profile of for different values of is shown in Fig.1 (left). We find that a solution for exists for arbitrary values of . For our numerical results suggest that the derivative of the function at tends to zero for .

3.2 Magnetic multipoles

Choosing , we employ the following separation of variables

 Aφ(r,θ)=Sℓ(r)Uℓ(θ) (17)

where for the source free exterior of the solution, the duality of the Maxwell equations implies

 Sℓ(r)=r2σ(r)dRℓdr  ,  Uℓ(θ)=sinθdPℓdθ=ℓ(cosθPℓ−Pℓ−1) . (18)

Note that the metric function enters into the relation between the magnetic and electric multipoles which is different to the pure AdS case. In Fig. 1 (right), we show the magnetic dipole function for different values of . Staying with the normalization used for (and not, alternatively, a normalisation where ) we find that tends to a constant value asymptotically and that this asymptotic value increases with .

4 Black holes

In the case of black holes, we have used the Ansatz (5) for the scalar-tensor field and a spherically symmetric electric field of the form

 Aμdxμ=V(r)dt . (19)

Using this Ansatz, the equations of motion (3), (4) reduce to a system of four first order, ordinary differential equations for the functions , , and , where the prime now and in the following denotes the derivative with respect to . Note that the equations do not explicitly depend on and due to the shift symmetry of the system. The Maxwell equation for can be solved explicitly giving :

 dVdr≡V′(r)=Qσ(r)r2 , (20)

where is an integration constant that can be interpreted as the electric charge of the solution. The remaining equations then depend on three additional parameters : the cosmological constant , the horizon radius and the scalar-tensor coupling . By a suitable rescaling of the fields and coordinates, respectively, we can set to a fixed value without loss of generality.

In order to understand how the presence of the scalar field changes the properties of the black holes, we have expanded the derivative of the scalar field in orders of . This reads :

 ϕ′(r)=γϕ′1(r)+O(γ2) (21)

with

 ϕ′1(r)=Q4A4+Q2A2+A04r5r2h(Q2+B0) (22)

and

 (23)
 A0=16r2r2h(r2+rrh+r2h)(29Λ2r3rh+Λ29r4h−23Λr2h+1)  ,  B0=43Λ(r3rh+r2r2h+rr3h)−4rrh . (24)

In the following, we will also be interested in the thermodynamic properties of the solutions, in particular in the heat capacity of the solutions for fixed values of given by :

 CQ=TH(∂S∂TH)Q , (25)

where the entropy and the Hawking temperature of the solutions are given by :

 S=Ah4=πr2h   ,   TH=14πσ(rh)N′(r)|r=rh . (26)

4.1 Black holes for γ=0

For the scalar field and the system of equations of motion has an analytical and well-known solution : the Reissner-Nordström-Anti-de-Sitter (RNAdS) solution which is given by

 N(r)=1−Λ3r2−2Mr+Q24r2  ,  σ(r)≡1  ,  V(r)=−Qr+c , (27)

where is a constant. This solution possesses interesting thermodynamic properties when fixing the charge of the solution [21, 22, 23] and has been used extensively in black hole chemistry (for a review see e.g. [26]). We will hence remind the reader of a number of properties of these solutions and will then use our numerical results to demonstrate how the picture changes in scalar-tensor gravity.

The mass , entropy and temperature of the RNAdS solutions read :

 M=12(rh+Q24rh−Λr3h3)   ,   S=Ah4=πr2h   ,   TH=(∂M∂S)Q=14π(1rh−Q24r3h−Λrh)   , (28)

where refers to the outer horizon of the black hole and is the area of the black hole horizon. The extremal RNAdS with fulfils

 rh,ex=√12Λ+12|Λ|√1−Q2Λ . (29)

The heat capacity at fixed charge then reads

 CQ=32π2r5hTH3Q2−4r2h−4Λr4h . (30)

As pointed out in [23], the heat capacity at fixed diverges at and since , the sign of is positive (negative, respectively) depending on whether the temperature increases (decreases) with . For two values of the horizon exist at which separating a stable phase of small horizon black holes from a stable phase of large horizon black holes. For these two points merge.

4.2 Black holes for γ>0

In this case, the solutions have to be constructed numerically. We have done so by using an adaptive grid collocation solver [35]. In order to solve the equations numerically, we have to fix appropriate boundary conditions. For we choose the following conditions :

 σ(r→∞)→1  ,  (rV)|r→∞→Q . (31)

On the black hole horizon, we impose . The requirement of regularity of the scalar field on the horizon then implies

 f2(ϕ′(rh))2+2f1ϕ′(rh)+f0=0 , (32)

where , and are expressions that contain , , and as follows:

 f2=4γrh(Q2(2γ2+r4h)+4Λr4h(r4h−2γ2)−4r6h) , (33)
 f1=(γ2Q4+4γ2Q2r2h+2Q2r6h−8r8h+8Λr4h(γ2Q2−6γ2r2h+r6h)+16Λ2γ2r8h) , (34)
 f0=γrh(−Q4+24Q2r2h−48r4h+8Λr4h(4r2h−Q2)−16Λ2r8h) . (35)

This specific condition clearly demonstrates that solutions with regular scalar hair on the horizon do not exist for arbitrary values of , and . We will discuss the restrictions on the parameters in 4.2.1.

Next to the regularity of the scalar field on the horizon, we also require the black hole temperature to be positive. For black holes with scalar-tensor hair, the Hawking temperature reads :

 TH=14π(1−Q24r2h−Λr2h)σ(rh)rh+γϕ′(r)|r=rh , (36)

where we have used the equations of motion. Note that independently of the choice of (finite) and as long as is finite and , the Hawking temperature becomes zero at , i.e. at the extremality condition for the RNAdS black holes. To put it differently, the Hawking temperature remains zero at the extremality condition for the RNAdS independent of . Note also that is not necessarily positive (see 4.2.2), while is positive. In general, the values of and , respectively, are only given numerically, i.e. it is not possible – as in the case of the RNAdS – to give analytical expressions for .

4.2.1 Restrictions on parameters

Keeping the dependence on for the moment, the solutions to equation (32) read :

 ϕ′(rh)±=−f1±√f21−f2f0f2=−f1±2K√Δf2 (37)

with

 K≡|Q2−4r2h+4Λr4h| (38)

and

 Δ≡16Λ2γ4r8h+8Λr4hγ2(Q2γ2+4r6h−12γ2r2h)+Q4γ4+8Q2γ2r2h(r4h+3γ2)+4r8h(r4h−12γ2) . (39)

Hence, in general, we would expect two branches of the solutions (one related to , the other to ) to exist in this model, unless . Note that is exactly the extremality condition for the corresponding RNAdS solutions (see (28)).

Moreover, these expressions clearly show that there exist parameters ranges for which the regularity condition (32) is not fulfilled. These are :

• for limited by the choice of parameters that gives for which ,

• for when  .

To demonstrate the restriction of parameters, we have studied the value of for fixed values of and in dependence of . In Fig. 2 we give for , and three different values of the charge including the uncharged case . In all plots, the vertical line parallel to the -axis corresponds to the parameters for which . This clearly shifts to larger values of with the increase of the charge . Moreover, as is clearly visible for and , respectively, small horizon black holes are separated by an interval of non-existence in from the large horizon black holes. This changes only at large values of the charge , here given for which in the limit corresponds to the maximal possible charge of the corresponding RNAdS solution.

As an example, we show the domain of existence of solutions with regular scalar hair for and three different values of in Fig. 3. For , the condition does not have solutions, but for and the solutions of are and , respectively. These are the corresponding horizontal lines. The two lines that join in a spike correspond to .

4.2.2 Numerical results

Uncharged black holes

Here, we briefly review the uncharged case . This has been studied previously [33], however, we present new results here that we believe to be important in order to understand the charged case. implies . As was shown in [33], the scalar-tensor black holes exist up to a maximal value of the scalar-tensor coupling (see Fig. 3 for ). This branch connects to the Schwarzschild-AdS (SAdS) solution in the limit (with value ) and can be constructed by continuous deformation of the latter. We will refer to this as the main branch in the following. In Fig. 4 (left) we show the behaviour of some parameters characterising the solution with and , i.e. the values of the metric function at the horizon, , and the value of the derivative of the scalar field function at the horizon, , as function of . In Fig. 4 (right) we show the profile of for and . As is obvious from this figure, .

Apart from this first branch of solutions we have been able to construct a second branch of solutions that starts at and extends back in reaching a singular limit for . This singular solution has and , see Fig. 4 (left). For a fixed value of , the solutions on the second branch have larger values of and smaller values of than the corresponding solution on the first branch.

As shown in Fig. 3 the value decreases strongly when the horizon radius is decreased. We find for , while for and our numerical analysis gives and , respectively. To state it differently : when the chosen horizon radius tends to the value of the corresponding AdS radius of a space-time without the scalar field , which with our choice of throughout the paper equates to , the interval in for which scalar-tensor black holes exist shrinks. Note also that the presence of the scalar field allows to construct black holes with .

Charged black Holes

The main difference to the uncharged case can be seen in Fig. 3, where we give the domain of existence of solutions in the --plane for two different values of non-vanishing . As can be clearly seen, the line, which restricts the existence of the solutions, does not extend to small horizon values anymore. In order to explain the pattern, also with view to 4.2.1, let us discuss cases with fixed value of . As mentioned above, the extremality condition of the RNAdS black holes equal that for . In the following, we will refer to the horizon value that fulfils as to (see (29)). The pattern can then be described as follows :

• for the RNAdS solution gets progressively deformed when increasing from zero, forming a branch of black holes with scalar-tensor hair limited by the maximal value of which corresponds to . An example in Fig. 3 would be , . The corresponding data for this case is shown in Fig. 4 (left). Again, we find two branches of solutions and a back-bending behaviour for , very similar to the case with .

• for the branch of scalar-tensor black holes can be constructed by deformation of the RNAdS solutions as above, however, before reaching the curve, the branch reaches the line at some . A typical case would be and in Fig. 3, for which . In order to understand what happens when increasing and with that the value of , we show as well as in Fig. 4 (left) for . The second branch is absent in this case and no backbending exists. From Fig. 4 (right) it also becomes clear what happens when increasing . When increasing , we find that becomes positive at some intermediate value of the charge, here and increases with the further increase of , see the profile for . On the other hand, is always positive. The profiles further suggest that when increasing the charge the derivative of close to the horizon increases strongly.

• for we have not been able to construct solutions, i.e. scalar-tensor black holes (seem to) exist only for horizon value larger than the corresponding horizon value of the extremal RNAdS solution.

With the domain of existence of solutions at hand, we have then studied the heat capacity of these solutions. Our results are shown in Fig. 5 for and two different values of . The curves for are indistinguishable from the ones shown, that is why we do not present them here. The first obvious difference to the case of the uncharged RNAdS case, which corresponds to a Schwarzschild-AdS (SAdS) solution, as well as the RNAdS case (see curves for and ) is that scalar-tensor black holes exist only on a finite interval of the horizon radius . In particular, as outlined above, small horizon black holes do not exist. In particular, we find that for large enough, remains positive on the full branch of solutions. This is indicated by the , curve. On the other hand, for small we observe that a transition from to with (or equivalently ) at exists. This is indicated by the curve for and . While for the SAdS case, which is for our choice of , we find that for , the value of . This means that the critical value of the horizon radius at which a transition from thermodynamically unstable () to thermodynamically stable () black holes appears shifts to larger values of .

5 Conclusions and Outlook

Whenever alternative theories of gravity are studied, black holes that possess non-trivial matter fields, e.g. scalar fields, on the horizon appear. Hence, for these black holes there is no equivalent of the No-hair theorems that exist for electro-vacuum black holes in 4 dimensional asymptotically flat space-time within General Relativity (GR). In this paper, we have studied a scalar-tensor extension of GR that contains a higher order curvature correction, the Gauss-Bonnet term, non-minimally coupled to a real scalar field. The resulting coupled, non-linear field equations of this model can only be solved numerically. The family of solutions obtained is a generalisation of the RNAdS black holes in the sense that the solutions are not only characterised by their charge, cosmological constant and horizon radius (or equivalently their mass parameter ), but by an additional parameter, the scalar-tensor coupling constant . We find that the domain of existence of these solutions shows a very complicated pattern and that solutions exist only in a finite domain of the -- parameter space for fixed value of the cosmological constant. We find parameters choice for which zero, one or two solutions exist. Critical values of the parameters beyond which no scalar-tensor black holes exist can be characterised by analytic expression and are related to the regularity of the scalar field at the black hole horizon. In particular, we find that small black holes, i.e. solutions with small horizon values, do not exist in our model.

While we have fixed the cosmological constant within this work, another approach would be to vary the cosmological constant. Then, the existence of phases of black holes in dependence of the pressure could be studied and compared to phase transitions in chemical systems [24]. This is currently under investigation.

Acknowledgements BH would like to thank FAPESP for financial support under grant number 2016/12605-2 and CNPq for financial support under Bolsa de Produtividade Grant 304100/2015-3.

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