Dilatonic dyon-like black hole solutions in the model with two Abelian gauge fields

Dilatonic black hole dyon-like solutions in the gravitational $4d$ model with a scalar field, two 2-forms, two dilatonic coupling constants $\lambda_i \neq 0$, $i =1,2$, obeying $\lambda_1 \neq - \lambda_2$ and the sign parameter $\varepsilon = \pm 1$ for scalar field kinetic term are considered. Here $\varepsilon = - 1$ corresponds to a ghost scalar field. These solutions are defined up to solutions of two master equations for two moduli functions, when $\lambda^2_i \neq 1/2$ for $\varepsilon = - 1$. Some physical parameters of the solutions are obtained: gravitational mass, scalar charge, Hawking temperature, black hole area entropy and parametrized post-Newtonian (PPN) parameters $\beta$ and $\gamma$. The PPN parameters do not depend on the couplings $\lambda_i$ and $\varepsilon$. A set of bounds on the gravitational mass and scalar charge are found by using a certain conjecture on the parameters of solutions, when $1 +2 \lambda_i^2 \varepsilon>0$, $i =1,2$.


Introduction
In this paper we extend our previous work [1] devoted to dilatonic dyon black hole solutions. We note that at present there exists a certain interest in spherically symmetric solutions, e.g. black hole and black brane ones, related to Lie algebras and Toda chains, see [2]- [27] and the references therein. These solutions appear in gravitational models with scalar fields and antisymmetric forms.
Here we consider a subclass of dilatonic black hole solutions with electric and magnetic charges Q 1 and Q 2 , respectively, in the 4d model with metric g, scalar field ϕ, two 2-forms F (1) and F (2) , corresponding to two dilatonic coupling constants λ 1 and λ 2 , respectively. All fields are defined on an oriented manifold M. Here we consider the dyon-like configuration for fields of 2-forms: where τ = vol[S 2 ] is volume form on 2-dimensional sphere and * = * [g] is the Hodge operator corresponding to the oriented manifold M with the metric g. We call this noncomposite configuration a dyon-like one in order to distinguish it from the true dyon configuration which is essentially composite and may be chosen in our case either as: (i) F (1) = Q 1 e 2λ 1 ϕ * τ +Q 2 τ , F (2) = 0, or (ii) F (1) = 0, F (2) = Q 1 e 2λ 2 ϕ * τ + Q 2 τ . From a physical point of view the ansatz (1.1) means that we deal here with a charged black hole, which has two color charges: Q 1 and Q 2 . The charge Q 1 is the electric one corresponding to the form F (1) , while the charge Q 2 is the magnetic one corresponding to the form F (2) . For coinciding dilatonic couplings λ 1 = λ 2 = λ we get a trivial noncomposite generalization of dilatonic dyon black hole solutions in the model with one 2-form which was considered in ref. [1], see also [4,9,10,13,22,27] and references therein. The dilatonic scalar field may be either an ordinary one or a phantom (or ghost) one. The phantom field appears in the action with a kinetic term of the "wrong sign", which implies the violation of the null energy condition p ≥ −ρ. According to ref. [28], at the quantum level, such fields could form a "ghost condensate", which may be responsible for modified gravity laws in the infra-red limit. The observational data do not exclude this possibility [29].
Here we seek relations for the physical parameters of dyonic-like black holes, e.g. bounds on the gravitational mass M and the scalar charge Q ϕ .
As in our previous work [1] this problem is solved here up to a conjecture, which states a one-to-one (smooth) correspondence between the pair (Q 2 1 , Q 2 2 ), where Q 1 is the electric charge and Q 2 is the magnetic charge, and the pair of positive parameters (P 1 , P 2 ), which appear in decomposition of moduli functions at large distances. This conjecture is believed to be valid for all λ i = 0 in the case of an ordinary scalar field and for 0 < λ 2 i < 1/2 for the case of a phantom scalar field (in both cases the inequality λ 1 = −λ 2 is assumed).
We consider a family of dyonic-like black hole solutions to the field equations corresponding to the action (2.1) which are defined on the manifold M = (2µ, +∞) × S 2 × R, (2.2) and have the following form (2.6) Here Q 1 and Q 2 are (colored) charges -electric and magnetic, respectively, µ > 0 is the extremality parameter, dΩ 2 2 = dθ 2 + sin 2 θdφ 2 is the canonical metric on the unit sphere S 2 (0 < θ < π, 0 < φ < 2π), τ = sin θdθ ∧ dφ is the standard volume form on S 2 , i = 1, 2, and The functions H s > 0 obey the equations with the following boundary conditions imposed: for R → 2µ, and H s → 1 (2.11) for R → +∞, s = 1, 2. In (2.9) we denote where A 12 is defined in (2.8) and These solutions may be obtained just by using general formulas for nonextremal (intersecting) black brane solutions from [18,19,20] (for a review see [21]). The composite analogs of the solutions with one 2-form and λ 1 = λ 2 were presented in ref. [1].
The first boundary condition (2.10) guarantees (up to a possible additional requirement on the analyticity of H s (R) in the vicinity of R = 2µ) the existence of a (regular) horizon at R = 2µ for the metric (2.3). The second condition (2.11) ensures asymptotical (for R → +∞) flatness of the metric.
Remark 1. It should be noted that the main motivation for considering this and more general 4D models governed by the Lagrangian density L: where ϕ = (ϕ a ) is a set of l scalar fields, F (i) = dA (i) are 2 forms and λ i = (λ ia ) are dilatonic coupling vectors, i = 1, . . . , m, is coming from dimensional reduction of supergravity models; in this case the matrix (h ab ) is positive definite. For example, one may consider a part of bosonic sector of dimensionally reduced 11d supergravity [15] with l dilatonic scalar fields and m 2-forms (either originating from 11d metric or coming from 4-form) activated; Chern-Simons terms vanish in this case. Certain uplifts (to higher dimensions) of 4d black hole solutions corresponding to (2.14) may lead us to black brane solutions in dimensions D > 4, e.g. to dyonic ones; see [15,16,19,23,24] and the references therein. The dimensional reduction from the 12-dimensional model from ref. [30] with phantom scalar field and two forms of rank 4 and 5 will lead us to the Lagrangian density (2.14) with the matrix (h ab ) of pseudo-Euclidean signature. Equations (2.9) may be rewritten in the following form: A sl y l ), (2.15) s = 1, 2. Here and in the following we use the following notations: y s = ln H s , z = 2µ/R, q s = Q s /(2µ) and K s = h −1 s for s = 1, 2, respectively. We are seeking solutions to equations (2.15) for z ∈ (0, 1) obeying where y s 0 = ln H s0 are finite (real) numbers, s = 1, 2. Here z = 0 (or, more precisely z = +0) corresponds to infinity (R = +∞), while z = 1 (or, more rigorously, z = 1 − 0 ) corresponds to the horizon (R = 2µ).

Some integrable cases
Explicit analytical solutions to eqs. (2.9), (2.10), (2.11) do not exist. One may try to seek the solutions in the form where P (k) s are constants, k = 1, 2, . . . , and s = 1, 2, but only in few integrable cases the chain of equations for P (k) s is dropped. For ε = +1, there exist at least four integrable configurations related to the Lie algebras Let us consider the case ε = 1 and We obtain For λ 1 = λ 2 we get a dilatonic coupling corresponding to string induced model. The matrix (3.2) is the Cartan matrix for the Lie algebra (2)). In this case For positive roots of (3.5) we are led to a well-defined solution for R > 2µ with asymptotically flat metric and horizon at R = 2µ. We note that in the case λ 1 = λ 2 the (A 1 + A 1 )-dyon solution has a composite analog which was considered earlier in [7,9]; see also [14] for certain generalizations.

A 2 -case
Now we put ε = 1 and We get This value of dilatonic coupling constant appears after reduction to four dimensions of the 5d Kaluza-Klein model. We get h s = 1/2 and (3.7) is the Cartan matrix for the Lie algebra A 2 = sl(3). In this case we obtain [19] In the composite case [1] the Kaluza-Klein uplift to D = 5 gives us the well-known Gibbons-Wiltshire solution [5], which follows from the general spherically symmetric dyon solution (related to A 2 Toda chain) from ref. [4].
The exact black hole (dyonic-like) solutions for Lie algebras B 2 = C 2 and G 2 will be analyzed in detail in separate publications. They do not exist for the case λ 1 = λ 2 . We note that for the B 2 = C 2 case (k = 2) the polynomials H i , i = 1, 2, were calculated in [31].

Special solution with two dependent charges
There exists also a special solution with P > 0 obeying Here b s = 0 is defined in (2.22). This solution is a special case of more general "block orthogonal" black brane solutions [32,33,34].
The calculations give us the following relations: where s = 1, 2 ands = 2, 1, respectively. Our solution is well defined if λ 1 λ 2 > 0, i.e. the two coupling constants have the same sign.
For positive roots of (3.18) we get for R > 2µ a well-defined solution with asymptotically flat metric and horizon at R = 2µ. It should be noted that this special solution is valid for both signs ε = ±1. We have where H = 1 + P R with P from (3.19) and By changing the radial variable, r = R + P , we get The metric in these variables is coinciding with the well-known Reissner-Nordström metric governed by two parameters: GM > 0 and Q 2 < 2(GM ) 2 . We have two horizons in this case. Electric and magnetic charges are not independent but obey eqs. (3.23).

The limiting A 1 -cases
In the following we will use two limiting solutions: an electric one with Q 1 = Q = 0 and Q 2 = 0, and a magnetic one with Q 1 = 0 and Q 2 = Q = 0, In both cases P s = −µ + µ 2 + K s Q 2 . These solutions correspond to the Lie algebra A 1 . In various notations the solution (3.26) appeared earlier in [2,6,7], and it was extended to the multidimensional case in [6,7,11,12]. The special case with λ 2 1 = 1/2, ε = 1, was considered earlier in [3,8].

Physical parameters
Here we consider certain physical parameters corresponding to the solutions under consideration.

Gravitational mass and scalar charge
For ADM gravitational mass we get from (2.3) where the parameters P s = P s appear in eq. (3.1) and G is the gravitational constant.
The scalar charge just follows from (2.4): For the special solution (3.15) with P > 0 we get For fixed charges Q s and the extremality parameter µ the mass M and scalar charge Q ϕ are not independent but obey a certain constraint. Indeed, for fixed parameters P s = P for z → +0, which after substitution into (2.18) gives (for z = 0) the following identity: By using eqs. (4.1) and (4.2) this identity may be rewritten in the following form: It is remarkable that this formula does not contain λ. We note that in the extremal case µ = +0 this relation for ε = 1 was obtained earlier in [13].

The Hawking temperature and entropy
The Hawking temperature corresponding to the solution is found to be where H s0 are defined in (2.10). Here and in the following we put c = = κ = 1. For special solutions (3.15) with P > 0 we get (4.8) In this case the Hawking temperature T H does not depend upon λ s and ε, when µ and P (or Q 2 ) are fixed. The Bekenstein-Hawking (area) entropy S = A/(4G), corresponding to the horizon at R = 2µ, where A is the horizon area, reads It follows from (4.7) and (4.9) that the product does not depend upon λ s , ε and the charges Q s . This product does not use an explicit form of the moduli functions H s (R).

PPN parameters
Introducing a new radial variable ρ by the relation R = ρ(1 + (µ/2ρ)) 2 (ρ > µ/2), we obtain the 3-dimensionally conformally flat form of the metric (2.3) where ρ 2 = |x| 2 = δ ij x i x j (i, j = 1, 2, 3) and The parametrized post-Newtonian (PPN) parameters β and γ are defined by the following standard relations: i, j = 1, 2, 3, where V = GM/ρ is Newton's potential, G is the gravitational constant and M is the gravitational mass (for our case see (4.1)). The calculations of PPN (or Eddington) parameters for the metric (4.11) give the same result as in [22]: These parameters do not depend upon λ s and ε. They may be calculated just without knowledge of the explicit relations for the moduli functions H s (R).
These parameters (at least formally) obey the observational restrictions for the solar system [35], when Q s /(2GM ) are small enough.

Bounds on mass and scalar charge
Here we outline the following hypothesis, which is supported by certain numerical calculations [1,36]. For h 1 = h 2 this conjecture was proposed in ref. [1].
The conjecture implies the following proposition. Proposition 1. For h s > 0, Q s = 0, λ s = 0 (s = 1, 2) and λ 1 + λ 2 = 0 we have the following bounds on the gravitational mass M and the scalar charge Q ϕ : for ε = +1 (0 < h s < 2) and Here we illustrate the bounds on M and Q ϕ graphically by four figures, which represent a set of physical parameters GM and Q ϕ for Q 2 1 +Q 2 2 = Q 2 = 2 and µ = 1.
The left panel of Fig. 1 corresponds to the case ε = +1, λ 1 = 1 2 and λ 2 = 1/2, while the right panel of this figure describes the case ε = +1, In proving Proposition 1 we use the following lemma.
Proposition 1 and the lemma imply the following proposition. Proposition 2. In the framework of the conditions of Proposition 1, the following bounds on the mass and scalar charge are valid for all µ > 0: for ε = +1 (0 < h s < 2), and In proving (5.13) and (5.15) the following (obvious) relation was used: In ref.
Remark 3. When one of h s , say h 1 , is negative, the conjecture is not valid. This may be verified just by analyzing the solutions with small enough charge Q 2 .
We note that here we were dealing with a special class of solutions with phantom scalar field (ε = −1). Even in the limiting case Q 2 = +0 and Q 1 = 0 there exist phantom black hole solutions which are not covered by our analysis [37] (see also [38].) Remark 4. The inequalities on the mass (5.1) and (5.3) in Proposition 1 can be refined when λ 1 λ 2 < 0. For both cases which are considered in Proposition 1, we get (see right panels of Figs. 1 and 2) f (µ, h min ; Q 2 ) < GM < f (µ, h max ; Q 2 ), (5.16) where Q 2 = Q 2 1 + Q 2 2 and f (µ, h; Q 2 ) is defined in (5.6). The bounds on mass (5.16) are a specific feature of the model with two different dilatonic couplings of opposite sign. For λ 1 λ 2 > 0, e.g. for λ 1 = λ 2 , one should use relations (5.1) and (5.3). We also note that in the proof of Proposition 1 the condition λ 1 = −λ 2 was used. For the case λ 1 = −λ 2 the arcs on the right panels of Figs. 1, 2 reduce to points and we get GM = f (µ, h 1 ; Q 2 ).

Conclusions
In this paper a family of non-extremal black hole dyon-like solutions in a 4d gravitational model with a scalar field and two Abelian vector fields is presented. The scalar field is either ordinary (ε = +1) or phantom (ε = −1). The model contains two dilatonic coupling constants λ s = 0, s = 1, 2, obeying The solutions are defined up to two moduli functions H 1 (R) and H 2 (R), which obey two differential equations of second order with boundary conditions imposed. For ε = +1 these equations are integrable for four cases, corresponding to the Lie algebras A 1 + A 1 , A 2 , B 2 = C 2 and G 2 . In the first case (A 1 + A 1 ) we have λ 1 λ 2 = 1/2, while in the second one (A 2 ) we get λ 1 = λ 2 = λ and λ 2 = 3/2. Two other solutions, corresponding to the Lie algebras B 2 = C 2 and G 2 , will be considered in separate publications.
Here we have also calculated some physical parameters of the solutions: gravitational mass M , scalar charge Q ϕ , Hawking temperature, black hole area entropy and post-Newtonian parameters β, γ. The PPN parameters γ = 1 and β do not depend upon λ s and ε, if the values of M and Q ϕ are fixed.
We have also obtained a formula, which relates M , Q ϕ , the dyon charges Q 1 , Q 2 , and the extremality parameter µ for all values of λ s = 0. Remarkably, this formula does not contain λ s and coincides with that of ref. [1]. As in the case λ 1 = λ 2 , the product of the Hawking temperature and the Bekenstein-Hawking entropy do not depend upon ε, λ s and the moduli functions H s (R).
Here we have obtained lower bounds on the gravitational mass and upper bounds on the scalar charge for 1 + 2λ 2 s ε > 0, which are based on the conjecture (from Sect. 5) on the parameters of solutions P 1 = P 1 (Q 2 1 , Q 2 2 ), P 2 = P 2 (Q 2 1 , Q 2 2 ). In [1] we have presented several results of numerical calculations which support our bounds for λ 1 = λ 2 . A rigorous proof of this conjecture may be the subject of a separate publication. For ε = +1 the lower bound on the gravitational mass is in agreement for λ 1 = λ 2 with that obtained earlier by Gibbons et al. [10] by using certain spinor techniques.
It was noted in Sect. 3.3 that for λ 1 = λ 2 there exist two integrable cases corresponding to the Lie algebras C 2 and G 2 , which will be analyzed in separate papers. They do not occur for An open question here is to find the conditions on the dilatonic coupling constants λ s which guarantee the existence of the second (hidden) horizon and the existence of the extremal black hole in the limit µ = +0. For ε = +1, λ 1 = λ 2 this problem was analyzed in refs. [13,27]. This question can be addressed to a separate publication.