Abelian-Higgs hair on stationary axisymmetric black hole in Einstein-Maxwell-axion-dilaton gravity

# Abelian-Higgs hair on stationary axisymmetric black hole in Einstein-Maxwell-axion-dilaton gravity

## Abstract

We studied, both analytically and numerically, an Abelian Higgs vortex in the spacetime of stationary axisymmetric black hole being the solution of the low-energy limit of the heterotic string theory, acted as hair on the black hole background. Taking into account the gravitational backreaction of the vortex we show that its influence is far more subtle than causing only a conical defect. As a consequence of its existence, we found that the ergosphere was shifted as well as event horizon angular velocity was affected by its presence. Numerical simulations reveal the strong dependence of the vortex behaviour on the black hole charge and mass of the Higgs boson. For large masses of the Higgs boson and large value of the charge, black hole is always pierced by an Abelian Higgs vortex, while small values of the charge and Higgs boson mass lead to the expulsion of Higgs field.

###### pacs:
04.70.Bw, 98.80.Cq

## I Introduction

The no hair conjecture of black holes, or its mathematical formulation, the uniqueness theorem of black holes states that the static electrovacuum black hole spacetime is described by Reissner-Nordström solution whereas the circular one is diffeomorphic to Kerr-Newmann spacetime book (). The attempts of building a consistent quantum gravity theory and understanding the behaviour of matter field in a spacetime of higher-dimensional black holes rog12 () triggered the resurgence of works treating the mathematical aspects of higher-dimensional black objects. In higher dimensional spacetime the uniqueness theorem for static black holes is well justified nd (), while the stationary axisymmetric case is far more complicated (for the recent efforts of proving uniqueness theorem see, e.g., nrot ()). Consequently, these researches comprise also the low-energy limit of the string theory, like dilaton gravity, Einstein-Maxwell-axion-dilaton gravity and supergravities theories sugra ().

In addition, black holes and their properties consist the key ingredients of the AdS/CFT attitude adscft () to superconductivity acquire great attention. Questions about possible matter configurations in AdS spacetime arise naturally during aforementioned researches. In Ref.shi12 () it was shown that strictly stationary AdS spacetime could not allow for the existence of nontrivial configurations of complex scalar fields or form fields. The generalization of the aforementioned problem, i.e., strictly stationarity of spacetimes with complex scalar fields in Einstein-Maxwell-axion-dilaton gravity with negative cosmological constant was given in bak13 ().

From the above point of view the other theories of gravity are also under intensive researches. Namely, the strictly stationary static vacuum spacetimes in Einstein-Gauss-Bonnet theory were discussed in shi13 (), while in Ref. shi13a () it was revealed that a static asymptotically flat black hole solution is unique to be Schwarzschild spacetime in Chern-Simons-gravity. Applying the conformal positive energy theorem the uniqueness proof of static asymptotically flat electrically charged black hole in dynamical Chern-Simons-gravity was performed rog13 ().

The discovery of Bartnik and McKinnon ba88 () of a nontrivial particle like structure in Einstein-Yang-Mills systems opens new realms of nontrivial solutions to Einstein-non-Abelian-gauge systems. It turned out that black holes can be colored ba88 () - kun90 (), support a long-range Yang-Mills hair. However, these solutions are unstable str90 () - biz91 (), but nevertheless they exist.

The other kind of problems is the extension of the aforementioned no hair conjecture to the case when some field configurations have non-trivial topology. We may ask if topological defects vil () can constitute hair on black hole spacetimes. In Ref.ary86 () the metric describing a Schwarzschild black hole threaded by a cosmic string was provided. Then, the extension of the problem to the case of Abelian Higgs vortex on Euclidean Einstein dow92 () and Euclidean dilaton black hole systems mod98 () were elaborated. Further, numerical and analytical studies revealed that Abelian Higgs vortex could act as a long hair for a Schwarzschild ach95 () and Reissner-Nordström cha () black holes. The extremal Reissner-Nordström black hole displays an analog of Meissner effect, but the flux expulsion does not occur in all cases. The case of charged dilaton black hole Abelian Higgs vortex system was treated in mod99 (), where among all it was shown that all extremal dilaton black holes always expelled vortex flux. On the other hand, the problem of superconducting cosmic vortex and possible fermion condensations around Euclidean Reissner-Nordström gre92 () and dilaton black holes nak11 () were elaborated, while black string with superconducting cosmic string was studied in Ref.nak12 ().

The case of stationary axisymmetric black hole vortex system turns out to be more complicated task. The first attempts to attack this problem were presented in Refs.ghe02 (), but the correct treatment of Kerr black hole Abelian Higgs vortex was presented in gre13 ().

In our paper we shall provide some continuity with our previous works concerning the dilaton static black hole topological defect systems mod98 (); mod99 () as well as the works concerning the problem of the existence of cosmic vortex in the spacetime of stationary axisymmetric black hole of Kerr type gre13 (). Namely, we shall take into account charged stationary axisymmetric solution of the low-energy limit of the heterotic string theory, Kerr-Sen black hole. The uniqueness of Kerr-Sen black hole was proved in Ref.rog10 () thus the next problem to consider will be the question of possible hair on the black hole in question. In the light of the arguments presented in gre13 () we shall look for the evidence that an Abelian Higgs vortex can act as a long hair for the Kerr-Sen black hole.

The paper is organized as follows. In Sec.II we briefly review an Abelian Higgs vortex configuration on stationary axisymmetric black hole solution in Einstein-Maxwell-axion-dilaton gravity. Then, we perform analytical considerations connected with the gravitational backreaction problem in order to achieve the line element describing Kerr-Sen black hole pierced by a vortex. We discuss its properties, especially the phenomenon of shifting the position of black hole ergosphere due to the presence of the vortex. In Sec.III we present a numerical analysis of the equations of motion for an Abelian Higgs vortex in the spacetime of the extremal and nonextremal Kerr-Sen black hole. Sec.IV will be devoted to the conclusions of our investigations.

## Ii Einstein-Maxwell-axion-dilaton gravity

In this section we shall study an Abelian Higgs vortex in the presence of Kerr-Sen black hole being the stationary axisymmetric solution of the low-energy limit of heterotic string theory, the so-called Einstein-Maxwell-axion-dilaton gravity. In our analysis we assume the complete separation of degrees of freedom for each of the objects in question. The resulting action for the considered system will be the sum of the action devoted to Einstein-Maxwell-axion-dilaton gravity

 S1=116πG ∫d4√−g[R−2 (∇ϕ)2−12 e4ϕ(∇a)2−e−2 ϕFμνFμν−a Fμν ∗Fμν], (1)

and the action for an Abelian Higgs model minimally coupled to gravity theory, which will be subject to spontaneous symmetry breaking. Its action implies the following:

 (2)

where we have denoted by a complex scalar field. The covariant derivative is given by and field strength tensor bounded with is of the form . Whereas the -gauge field strength tensor is given by . while the dual tensor to the -gauge field has the form .
As usual, we can define real fields by the following relations:

 Φ(xμ) = η X(xμ) eiχ(xμ), (3) Bα(xμ) = 1e [Pα(xμ)−∇αχ(xμ)]. (4)

The above real fields represent the physical degrees of freedom of the broken symmetric phase. Namely, is the scalar Higgs field, the massive vector boson, while is a gauge degree of freedom and it is not a local observable. However it can have a globally nontrivial phase factor which indicate the presence of the vortex in question. The equations of motion for and fields are provided by the relations

 ∇α∇αX − Pβ Pβ X−~λ η22 (X2−1) X=0, (5) ∇μ~Bμν − 2e2η2 X2 Pν=0. (6)

On the other hand, the field equations for the considered system yield

 Gμν = Tμν(F, a, ϕ)+8πG ~Tμν(vortex), (7) ∇α∇αϕ − 12 e4ϕ ∇μa ∇μa+12 e−2ϕ FαβFαβ=0, (8) ∇β(e4ϕ ∇βa) − Fμν ∗Fμν=0, (9) ∇μ(e−2 ϕFμνFμν + a ∗Fμν)=0, (10)

Consequently, the energy momentum tensors for the adequate matter fields imply

 Tαβ(ϕ) = 2 ∇αϕ∇βϕ−gαβ(∇ϕ)2, (11) Tαβ(a) = 12 e4ϕ ∇αa∇βa−14gαβ(∇a)2, (12) Tαβ(F) = 2 e−2ϕFαγFβγ−12e−2ϕgαβFμνFμν, (13)

while for the Abelian Higgs field one gets the following form of the energy momentum tensor:

 ~Tμν(vortex)=2 η2 ∇μX∇νX+2 η2X2 PμPν+1e2~Bμα~Bνα+gμνL(Φ, Bμ), (14)

where is the Lagrangian density connected with and fields which yields

 L(Φ, Bμ)=−η2 ∇μX∇μX−η2 X2 PμPμ−14e2~Bμν~Bμν−~λ η44 (X2−1)2. (15)

In order to proceed further we shall use coordinates which contemplate the axial symmetry of the Kerr-Sen black hole Abelian Higgs vortex system. Namely, we take into account Weyl form of the axisymmetric line element described by

 ds2=−e2λ dt2+α2 e−2λ (dφ+β dt)2+e2(ν−λ) (dx2+dy2), (16)

where the functions under consideration are of dependences. Further, we define and coordinates by the relations

 x=∫dr√Δ,y=θ. (17)

As far as the considered metric of stationary axisymmetric black hole solution of Einstein-Maxwell-axion-dilaton gravity is concerned it can be written in the form as gal95 ()

 ds2 = −(1−2 G M (r−rm)~ρ2) dt2+~ρ2 (dr2Δ+dθ2)+4 G M (r−rm) a sin2θ~ρ2 dt dφ + [r (r−rm)+a2+2 G M (r−rm) a2 sin2θ~ρ2] sin2θ dφ2,

where we have denoted by and the rest of the quantities are defined by

 ~ρ2=r (r−rm)+a2 cosθ,Δ=(r−rm)(r−2GM)+a2. (19)

One can remark that the above metric reduces to the static dilaton black hole solution gar92 () when . On the other hand, when one puts , it contracts to the standard Kerr metric. The other form of the rotating axion dilaton black hole was conceived in Ref.sen92 (), but it turns out that by the adequate coordinate transformations and change of the parameters it can be brought to the line element (II). is the standard Schwarzschild mass while is the Kerr rotation parameter related to the black hole angular momentum by .

Taking account of the corresponding Kerr-Sen line element, one gets the following correspondence between Kerr-Sen metric and Weyl line element:

 α0=√Δ sinθ,e2λ0=~ρ2 ΔΣ2,β0=−2 M G a (r−rm)Σ2,e2ν0=~ρ4 ΔΣ2, (20)

where .

By virtue of the above one can rewrite the equations of motion for the Abelian Higgs vortex Kerr-Sen black hole system in the form as follows:

 Rαβ=Tαβ−12 T gαβ, (21)

where the energy momentum tensor is provided by the expression

 Tαβ=ϵ Tαβ(vortex)+Tαβ(F, a, ϕ). (22)

In the above equation, denotes the rescaled energy momentum tensor (see, e.g., dow92 (); mod99 ()) of the considered Abelian Higgs vortex, while . Thus, the explicit forms of equations of motion in the underlying theory are given by

 −12[2 ∇2α − α3 e−4λ (∇β)2+4α∇2ν+4 eλ ∇(α ∇e−λ)] = √−g ((Txx−12 T δxx)+(Tyy−12 T δyy)), −12[−α3 e−4λ β2,y + 2α∇2(ν−λ)−2 (αy νy−αx νx)−2 ∇α ∇λ+4 α λ2,y+2 α,yy] = √−g (Tyy−12 T δyy), −12(−3 ∇α ∇β + 4 α ∇β ∇λ−α ∇2β)=√−g Tϕt, (25) α2 e−2ν [2 ∇2α − 2 ∇α ∇λ−2 α ∇2λ+α3 e−4λ (∇β)2]=Tφφ−12 T gφφ, (26) −∇2α = √−g ((Ttt−12 T δtt)+(Tφφ−12 T δφφ)), (27)

where . On the other hand, equations of motion for the matter fields imply

 4 e4ϕ α e−2λ+2ν (∇ϕ ∇a e2λ−2ν) + e4ϕ ∇(α ∇a)−4 ∂yAφ ∂xAt=0, (28) ∇(α ∇ϕ) − 12 e4ϕ (∇a)2+α e−2ϕ−2λ[−∇At (∇At−β ∇Aφ) − ∇Aφ (−∇At−∇Aφ (e4λ−α2 β2α2))]=0, ∇[e−2ϕ−2λ α (∇At − β Aφ)]+∂x(a Aφ,y)−∂y(a Aφ,x)=0. (30)

In the considered case one should also take into account and components of the Abelian Higgs gauge field. In what follows we shall apply an iterative procedure of solving equations of motion, expanding the equations in terms of . This is well justified because of the fact that, e.g., for the grand unified theory string one has . Our starting background solution will be described by and gre13 () and will constitute the Nielsen-Olesen vortex solution nil73 (), provided by

 X≃X0(R),Pφ≃P0(R),Pt≃ζ Pφ. (31)

It can be easily found that , where we put equal to

 ρ=r (r−rm)+a2, (32)

and . Near the core of the cosmic string one gets that , which results in the relation . This implies in turn that to the zeroth order in , the components describing the energy momentum tensor of Abelian Higgs vortex yield

 T(0)yy ≃ X′20−X20 P20R2+~β P′20R2−14 (X20−1)2, (33) T(0)φφ ≃ −X′20+X20 P20R2+~β P′20R2−14 (X20−1)2, (34) T(0)tt = Txx≃−X′20−X20 P20R2−~β P′20R2−14 (X20−1)2, (35) T(0)xy ≃ √Δρ (r−rm) R (2 X′20+2 ~β P′20), (36) T(0)tφ ≃ 0, (37)

where is the Bogomol’nyi parameter while . A prime denotes differentiation with respect to -coordinate.

As in Refs.ach95 () we expand the adequate quantities in series, i.e., , etc. and solve the underlying equations of motion iteratively. In the case under consideration except Abelian Higgs vortex fields one has to do with dilaton, axion and Maxwell gauge field which appears at the level of -order. It caused that to -order the geometry of the problem in question is not only modified by the vortex fields but also one should take into account the backreaction of the rest of the matter fields appearing in EMAD-gravity. As we can see the energy momentum components are functions of -coordinate and this leads us to the conclusion that the modification of the Kerr-Sen stationary axisymmetric solution will also depend on this coordinate. Therefore, we assume that the first order perturbed solutions of the equations of motion in the theory in question will imply

 α1 = α0 ~α1(R),λ1=λ1(R),ν1=ν1(R),β1=β1(R), (38) a1 = a1(R),ϕ1=ϕ1(R),A(1)μ=f(R) A(0)μ,

where denotes Maxwell field for Kerr-Sen solution. Having in mind the form of the energy momentum tensor for cosmic vortex, near the string core, and the fact that is equal to zero, we conclude that to the leading order one attains

 d2dR2~α1+2R ddR~α1=2 X20 P20R2+12 (X20−1)2. (39)

It can be easily checked that will be given by

 ~α1(R)=∫∞R 1R2dR ∫R0 R′2(2 X20 P20R′2+12 (X20−1)2)dR′. (40)

On the other hand, the fact that is subdominant quantity and its derivatives are also subdominant, one reveals that -order Einstein-Maxwell-axion-dilaton gravity equations of motion can be readily write down in the forms

 R (R ~α′′1+2 ~α′1 − R λ′′1−λ′1)=R2 [2 X20 P20R2+~β (P′0)2R2+14 (X20−1)2] + R2 [Q2 f0ρ4+4 f0 Q2 a2ρ4], ~α′′1 + 2 ~α′1R+2 ν′′1−2 λ′′1−2 λ′1R=2 X20+12 (X20−1)2, (42) −~α′′1 − ~α′1R−ν′′1+λ′′1+ν′1R+λ′1R=2 X20+~β (P′0)2R2+34 (X20−1)2 + 2 f Q4ρ4+2 Q2 R f′ρ5, −√Δρ (r − rm2) (R2 ~α′′1+2 R ~α′1−4 λ′1)≃√Δρ (r−rm2) R [2 X20+2 ~β (P′0)2], (44) ϕ′′1 + ϕ′1R−2 f Q2ρ4+2 f Q2 a2ρ4 R2≃0, (45) a′′1 + a′1R−4 ϕ′1ρ4−rm a ~α1ρ4≃0, (46) f′′ + f′R≃−2 ρR r (ϕ′1+λ′1)+2 ~α1R2 r. (47)

As in Refs.mod98 ()-mod99 () we shall work in the so-called thin string limit. It means that one assumes that the mass of the black hole in question is subject to the inequality . Thus we shall neglect terms of the order . First, let us consider equation (II) and relation for . They give us the following expression for :

 λ1≃∫∞RdRR ∫R0dR′ R′ [14 (X20−1)2−~β (P′0)2R′2]. (48)

Hence, from equation (47), dropping terms of order , one concludes that if we put the first integration constant to zero, then finally , where is a constant value. Turning our attention to the relations (42) and (II), we observe that . Then, from equation describing the dilaton field we find that

 ϕ1=∫∞RdR 2R ∫R0dR′ f0 Q4ρ4 (R′−a2R′). (49)

Referring our studies to the relations describing axion field, it entails that the following is satisfied:

 a1=∫∞RdR 1R ∫R0dR′ R′ρ4 (4 ϕ′1+rm a ~α1). (50)

The essential point, however, is that the first order correction of cannot be established from an asymptotical analysis, due to the fact that -quantity is a subdominant function in the problem in question. Just, taking the divergent part of the Ricci curvature tensor one has that to -oder we arrive at the relation

 e−4λ0 α20 β0,x β1,y≃2 M a R2ρ2 √Δ (−1+4 r (r−rm)ρ2+O(1ρ4)) β1,y. (51)

On this account it is customary to examine the right-hand side of equation in question, i.e., studying the adequate components of the energy momentum tensor, both for Abelian Higgs vortex and matter fields, one infers that we cannot find which has the mandatory functional dependence on the coordinates. It is remarkable fact that, the required form of -order correction required for a pure deficit angle leads to the divergence of -component of the Ricci curvature tensor at the event horizon of the considered black hole. In view of these arguments, we deduce that .

After rescaling coordinates , one achieves to the line element describing thin string in Kerr-Sen black hole spacetime. The resultant metric is provided by

 ds2 = −(^Δ−^a2 sin2θ^ρ2) d^t2−8 G2 ^M2 ^a2 (^r−^rm)2^Σ4 ϵ ^μ [^r (^r−^rm)+^a2] sin2θ d^t2 + ^Σ2^Δ d^r2+^ρ2 d^θ2+^Σ2 sin2θ^ρ2(1−2 ϵ ^μ) dφ2−4 G ^M ^a (^r−^rm)^ρ2 sin2θ (1−2 ϵ ^μ) d^t dφ,

where we have denoted by mass per unit length of the cosmic string dow92 (), while the other quantities are defined as follows:

 ^Δ = (^r−^rm) (^r−2 G ^M)+^a2, (53) ^ρ2 = ^r (^r−^rm)+^a2 cos2θ, (54) ^Σ2 = (55)

One can remark that the Abelian Higgs vortex on Kerr-Sen rotating black hole causes not only an angular deficit angle which is felt by -coordinate but also the deficit which influences -coordinate. This fact leads us to a quite new physical phenomenon. Namely, because of the fact that the deficit angle appears in and the condition on the radius of the ergosphere is , then the position of the ergosphere is shifted (we shall pay more attention to this problem in Sec.III). The same occurrence takes place in Kerr Abelian Higgs vortex system gre13 (). One should remark that recently the problem of ergoregions attracted much attention, see, e.g., gib13 () and references therein.
The angular velocity of the observer in Kerr-Sen black hole Abelian Higgs vortex system belongs to the interval , where , with . Thus, it is affected by the presence of the Abelian Higgs vortex. At the event horizon of the black hole threaded by a vortex, where , coincides with . The limiting angular velocity is of the form

 Ωh=ω2(rh, θ)=ω2Kerr−Sen [1+ϵ ^μ (2−8 G ^M (^r−^rm))], (56)

where we denoted

 ω2Kerr−Sen=^a2[^r (^r−^rm)+^a2]. (57)

The limiting angular velocity , sometimes called the angular velocity of the event horizon, is also modified by the presence of the Abelian Higgs vortex.
It is worth mentioning that, when we consider the area of event horizon of the the Kerr-Sen black hole penetrated by an Abelian Higgs vortex it satisfies up to -order

 AKerr−Sen−vortex=∫∫√gφφ gθθ dφ dθ≃AKerr−Sen √1−2 ϵ ^μ, (58)

which means that it is affected by the vortex presence.

## Iii Numerical results

In this section one performs a number of numerical calculations to study the behaviour of the Abelian Higgs vortex in Kerr-Sen spacetime. We commence our studies with introducing dimensionless quantities useful in numerical solutions of the underlying equations. Namely, one establishes the following rescaling:

 (~Q,~a,~r,1) = 1GM(Q,a,r,GM), (59) (~mX,~mV) = GM(mX,mV), (60) (~Pϕ,~Pt) = (G2M2Pϕ,GMPt), (61)

where we have denoted the Higgs boson mass by and the mass of the vector field in the broken phase by . In what follows, for the brevity of our notation, one drops the tilde from the rescaled quantities.

It may be recalled that, in analogous to Kerr-Newmann solution in Einstein-Maxwell gravity, Kerr-Sen solution is characterized by it mass, electric charge and rotation parameter related to the black hole angular momentum and mass. The maximal value of the rotation parameter can be inferred from the condition for the extremal black hole. Namely, one has that the outer and inner horizons coalesce , where . Hence, in our rescaling . Consequently, the maximal value of the black hole charge is fixed by the relation . It leads to the conclusion that .

### iii.1 Location of the ergosphere - numerical results

To begin with we pay more attention to the problem of the ergosphere shifting due to the presence of Abelian-Higgs vortex. On this account, in Fig.1a we plotted the distance between the event horizon and the ergosphere for Kerr-Sen Abelian Higgs vortex system, as a function of black hole charge and angular momentum parameter . We fix and perform this figure and the subsequent ones in the equatorial plane for which . It may be seen that the increase of the value of black hole charge causes the diminishing of the distance in question and in the case of the ergosphere collapses onto the Kerr-Sen event horizon.
The difference between the location of the ergosphere for pure Kerr-Sen black hole and Kerr-Sen black hole pierced by an Abelian Higgs vortex is presented in Fig.(b)b. It was done as a function of black hole charge. One can conclude that the sign of the difference indicates that after taking into account the presence of the vortex the ergosphere is shifted towards the black hole event horizon. The magnitude of this shift is of order equal to .

In Fig.(a)a we depicted the difference between the location of the ergosphere for Kerr-Sen black hole and the ergosphere of Kerr-Sen Abelian Higgs vortex system, as a function of and angular momentum parameter , for fixed value of the black hole charge equal to . It can be observed that for the fixed and the black hole charge , the shift increases in its magnitude as the considered black hole approaches extremality. In Fig.(b)b the difference between the location of the ergosphere for Kerr-Sen black hole and the ergosphere of Kerr-Sen Abelian Higgs vortex system, as a function of and is plotted, for the maximal value of the angular momentum parameter. From this figure we infer that the shift decreases as the black hole charge approaches the maximal one. The angular momentum parameter is fixed but the relation between the maximal allowed and implies that the bigger is the smaller one obtains. Summing it all up, one can draw a conclusion that the increase of the value of black hole charge (decrease of ) influences the decrease in the magnitude of the shift in the ergosphere position. On the other hand, Fig.(a)a and Fig.(b)b indicate that for fixed values of the black hole charge and angular momentum parameter , the distance between the black hole event horizon and the ergosphere is insensitive to the modification of the value of (in the considered range of this parameter).

### iii.2 Numerical solution of the equations of motion for Kerr-Sen Abelian Higgs vortex system

In this subsection we shall analyze numerically equations of motion for an Abelian Higgs vortex on the background of Kerr-Sen black hole. Namely, one considers the following relations:

 ∇μ∇μX − P2X−m2X2X(X2−1)=0, (62) ∇μ∇μPα − m2V PαX2−Rα  βPβ=0,α=ϕ, t. (63)

The relevant components of the Ricci curvature tensor are as follows: , , and , while their explicit forms imply