arX

iv:1

408.

1998

v2 [

hep-

th]

19

Aug

201

4

(2 + 1)-dimensional charged black holes with scalarhair in Einstein-Power-Maxwell Theory

Wei Xu1 ∗†and De-Cheng Zou2 ‡

1School of Physics, Huazhong University of Science and Technology,

Wuhan 430074, China2Department of Physics and Astronomy, Shanghai Jiao Tong University,

Shanghai 200240, China

Abstract

We obtain an exact static solution to Einstein-Power-Maxwell (EPM) theoryin (2+1) dimensional AdS spacetime, in which the scalar field couples to gravityin a non-minimal way. After considering the scalar potential, a stable systemleads to a constraint on the power parameter k of Maxwell field. The solutioncontains a curvature singularity at the origin and is non-conformally flat. Thehorizon structures are identified, which indicates the physically acceptable lowerbound of mass in according to the existence of black hole solutions. Especiallyfor the cases with k > 1, the lower bound is negative, thus there exist scalarblack holes with negative mass. The null geodesics in this spacetime are alsodiscussed in detail. They are divided into five models, which are made up ofthe cases with the following geodesic motions: no-allowed motion, the circularmotion, the elliptic motion and the unbounded spiral motion.

Keywords: black hole with scalar hair, Einstein-Power-Maxwell (EPM) theory,(2 + 1)-dimensions, geodesics

PACS: 04.20.Jb, 04.40.Nr, 04.70.-s

1 Introduction

The no-hair theorems assert that the asymptotically flat spacetime can not admit anyhairy black hole solutions [1–3], since the scalar field was divergent on the horizon andstability analysis showed that they were unstable [4]. Nevertheless, when a negative

∗Correspondence author†email: xuweifuture@gmail.com

‡email: zoudecheng@sjtu.edu.cn

1

cosmological constant is considered, the no-hair theorems can usually be circumvented,and then there exists a broad literature of black hole solutions with a (non)minimalscalar field in the four or higher dimensional Einstein’s gravity, including static [5–14],and rotating [15–17] extensions with a complex, massive scalar field and in higher orderderivative gravity [18–21].

Inspiring by the pioneering work of Banados, Teitelboim and Zanelli (BTZ) [22],(2+ 1)-dimensional spacetimes admitting black hole solutions have attracted much at-tention. Besides sourced by a mass and a negative cosmological constant, pure BTZblack holes can be also to added new sources such as electric/magnetic fields fromMaxwell’s theory [23–25], Maxwell-dilaton [26], rotation [27], perfect fluid [28, 29] andothers [30, 31]. It is interesting to note that the nonlinear electrodynamics (NED) isuseful to overcome to eliminate the electromagnetic singularities due to point chargesthat occur in the linear Maxwell theory. Besides, the nonlinear electrodynamics hasbeen considered in general relativity, which are a good laboratories in order to con-struct black hole solutions [32–34]. Black hole solutions with nonlinear electrodynamicssources have interesting asymptotic behaviors and exhibit interesting thermodynamicsproperties [35–38]. For example, they satisfy the zeroth and first laws of black-holemechanics [39]. After considering the cosmological constant as a dynamical pressure,the Smarr relation work as well and there are rich phase structure which have the firstorder phase transitions and the reentrant phase transitions [40, 41].

On the other hand, with a negative cosmological constant in the action, the (2 +1)-dimensional black holes with the minimal [42–44] or non-minimal [45, 46] scalarfields have been constructed in the Einstein’s gravity. Furthermore, the charged [47],rotating [48–51], charged rotating [52] and Einstein-Born-Infeld [53] black holes withnon-minimally scalar hair and rotating black hole dressed with minimal scalar fieldhair [54] in the (2 + 1) dimensional Einstein’s gravity, and black hole dressed by a(non)minimally scalar field in New Massive Gravity [55] have been discussed. Beyondthe linear Maxwell electromagnetism in theory with scalar fields, in this paper, we studythe charged black hole solution with non-minimally coupled scalar field in the (2 +1)-dimensional Einstein-Power-Maxwell (EPM) theory. Actually, asymptotically AdSblack holes with nonlinear electrodynamics sources endowed with a extra scalar fieldhave been related to superconductors by means of the gravity/gauge duality [56–60].Especially, the larger power parameter k of the Power Maxwell field makes it harderfor the scalar hair to be condensated [56]. This makes it more interesting to study theblack hole solutions in this paper. In addition, we will present the null geodesics indetail, in order to have a further understanding of the properties of this solution.

This paper is organized as follows. In Sec. 2, we present the charged black holesolution in the EPM gravity with non-minimally coupled scalar field, and then discussthe properties of the scalar potential. Moreover, the basic geometric properties andhorizon structure of the metric are also outlined. In Sec. 3 the geodesics motions aregiven for the photon. The Sec. 4 is devoted to the closing remarks.

2

2 Charged black hole in the EPM theory with non-

minimally coupled scalar field

2.1 Charged hairy black hole solution

The (2 + 1)-dimensional action in the Einstein-Power-Maxwell (EPM) theory withnon-minimally coupled scalar field is written as

I =

∫

d3x√−g

[

1

2π

(

R−∇µφ∇µφ− 1

8Rφ2 − 2V (φ)

)

− L(F)

]

, (1)

in which R is the Ricci scalar, φ is the scalar field, V (φ) is the self-coupling potentialof scalar field, and L(F) = Fk is the power Maxwell Lagrangian with the Maxwellinvariant F = FµνF

µν . Here power parameter k is a rational number whose range isk > 1

2, restricted by the weak energy conditions (WEC) and strong energy conditions

(SEC) [25, 61]. For k = 1, the theory reduces to Einstein-Maxwell theory with non-minimally coupled scalar field [47]. We does not consider this case in this paper.Besides, the theory with k = 3

4, in which the Maxwell source is conformally invariant,

is presented in [62] a few days ago.

Considering the variation of the action, we can obtain the field equations

Gµν − π T

µ

[A]ν − Tµ

[φ]ν + V (φ)δµν = 0, (2)

1√−g∂ν(

√−gF µνFk−1) = 0, (3)

∇µ∇µφ− 1

8Rφ− ∂φV (φ) = 0, (4)

where the energy-momentum tensor of the power Maxwell field and scalar field aregiven by

Tµ

[A]ν =Fk

2

(

4k(FνσFµσ)

F − δµν

)

, (5)

Tµ

[φ]ν = ∂µφ∂νφ− 1

2δµν∇ρφ∇ρφ+

1

8

(

δµν�−∇µ∇ν +Gµν

)

φ2. (6)

The metric ansatz is chosen as

ds2 = −f(r)dt2 +1

f(r)dr2 + r2dθ2 (7)

with the coordinate range −∞ < t < +∞, r > 0 and 0 < θ < 2π. In this setting, fork 6= 1, the vector potential of the power Maxwell field is given by the Maxwell equation

Aµdxµ =

(2k − 1)

2(k − 1)Qr

2(k−1)(2k−1)dt, (8)

3

where the parameter Q is related to the electric charge Qe of black hole in the EPMtheory [25]

Qe =

√2

4πk(2k − 1)

2k−12k Q2k−1.

In the following paper, we will still use the parameter Q for simplicity other than Qe.

Then, we can obtain the following solution from the other field equations with thesimplified form of scalar field shown in [47, 48]

φ(r) = ±√

8B

r +B, (9)

the metric function

f(r) =r2

ℓ2+

αB2(2B + 3r)

48r+

(−2)k(2k − 1)2π(

qB1

(2k−1)

)2k

2(k − 1)

(

r + 2kB(4k−1)

)

r1

(2k−1)

(10)

and self-coupling potential of scalar field

V (φ) = − 1

ℓ2+

1

512

(

1

ℓ2+

α

48

)

φ6 +(−2)k(2k − 1)π q2k

1024(k − 1)(4k − 1)

(

φ2

8− φ2

)2k

(2k−1)

×(

64kφ2 − 16k(2k − 1)φ4 + (2k − 1)2φ6

)

, (11)

where the constant q = Q

B1

2k−1makes the action of the theory invariant. Λ = V (0) =

− 1ℓ2

is the constant term emerging naturally in the potential which plays the role ofcosmological constant. The negative cosmological constant is necessary for obtainingblack hole solutions because of the No-Go theorem in three dimensions [63]. Theparameter α is related to the mass of black hole. Notice that the Maxwell field andself-coupling potential are singular when k = 1. Actually, For k = 1, the Maxwellvector potential reaches to the usual ln(r) form in three dimensions.

For the scalar field, the Eq.(9) shows that the potential V (φ) always keeps regularwhen φ2 < 8. In order to obtain a stable system, the potential should be bounded frombelow. Firstly, when φ2 → 8−, the scalar potential

V (φ) =α

48− (2k − 1)(−2)kπq2k

2(4k − 1)lim

φ2→8−

1(

1− φ2

8

)2k

(2k−1)

≃ −(−1)k × (+∞)

must be positive, hence

(−1)k = −1 (12)

must holds. This leads to k = 2a+12b+1

or k = 2c+ 1, where a, b, c are all positive integer.

Note for the case with conformally invariant Maxwell source, i.e. k = 34, presented

4

in [62] a few days ago, is clear out of this constraint Eq.(12), for the reason that theauthors begin with the general form of the scalar field other than the simplified oneEq.(9). Then, the horizon function Eq. (10) becomes

f(r) =r2

ℓ2+

αB2(2B + 3r)

48r−

2k(2k − 1)2πq2k(

B2k

(2k−1)

)

2(k − 1)

(

r + 2kB(4k−1)

)

r1

(2k−1)

. (13)

Meanwhile, one can rewrite the potential as

V (φ) = − 1

ℓ2+

1

512

(

1

ℓ2+

α

48

)

φ6 − 2k(2k − 1)π q2k

1024(k − 1)(4k − 1)

(

φ2

8− φ2

)2k

(2k−1)

×(

64kφ2 − 16k(2k − 1)φ4 + (2k − 1)2φ6

)

. (14)

On the other hand, when φ2 → 0, the leading terms in the power series expansionof V (φ) behave as

V (φ) = − 1

ℓ2+

1

512

(

1

ℓ2+

α

48

)

φ6 +(2k − 1)2(k−1)πq2k

26k

(2k−1)

φ4k

(2k−1)

×(

− k

8(k − 1)(4k − 1)φ2 +

k

32(2k − 1)φ4 +O(φ6)

)

,

where we have inserted Eq.(12). One can find some discussions:

• If k ∈ (1,+∞), hence 4k(2k−1)

< 4. When φ is small, the main contribution of

V (φ) is from − k(2k−1)8(k−1)(4k−1)

2(k−1)πq2k

26k

(2k−1)

φ4k

(2k−1)+2

term whose coefficient is negative.

Thus V (φ) will possess a maximum at φ = 0 and two minimum at some non-vanishing constant scalar field. This is more easily seen in the left plot of Fig.(1).

• If k ∈ (12, 1), then 4k

(2k−1)> 4. When φ is small, the main contribution of V (φ) is

from 1512

(

1ℓ2+ α

48

)

φ6 term whose coefficient determines the behavior of potential.If, in addition,

(

1ℓ2+ α

48

)

> 0, V (φ) will only possess a minimum at φ = 0. Thiscan be seen in the middle plot of Fig.(1). If

(

1ℓ2+ α

48

)

≤ 0, V (φ) will also possessa maximum at φ = 0 and two minimum. One can see it in the right plot ofFig.(1).

For all above cases, one can find that the system is always stable against smallperturbations in φ.

2.2 Geometric properties and horizon structure

For this charged hairy black hole, the mass can be calculated simply by adoptingthe Brown-York method [64]. The quasilocal mass m(r) at a r takes the following

5

Figure 1: Self-coupling potential V (φ) versus φ with q = 2 and ℓ = 1.

form [64–66]

m(r) = 2√

f(r)(√

f0(r)−√

f(r)). (15)

As a result, the mass can be obtained as

M ≡ limr→∞

m(r) = −αB2

16, (16)

using which, we can rewrite the horizon function as

f(r) =r2

ℓ2−

(

1 +2B

3r

)

M −X(2k − 1)

(

r + 2Bk(4k−1)

)

r1

(2k−1)

. (17)

where X = 2k(2k−1)πQ2k

2(k−1).

In order to have a further understanding of the solutions, we calculate some geo-metric quantities. Firstly, the Ricci scalar reads as

R = − 6

ℓ2+

2(k − 1)X

(2k − 1)

(

(4k − 3)− 2Bk

(4k − 1)r

)

r−2k

(2k−1) ,

which shows that the solution has an essential singularity at r = 0 whenever Q 6= 0.Higher order curvature invariants such as RµνR

µν and RµνρσRµνρσ both have much

complicated expressions. One can also find the further behavior as O(Mr), hence the

solution is also singular at r = 0 whenever M 6= 0. As we are interested in the blackholes, the solutions need to contain a event horizon to surround the singularity.

On the other hand, in order to indicate that the metric is non-conformally flat, wecan find some non-vanishing components of the Cotton tensor, e.g.

Cθθ r =MB

r2+

k

(

(1− k) + Bkr

)

X

(2k − 1)2r1

(2k−1)

,

6

which does not vanishes whenever B 6= 0, M 6= 0 or Q 6= 0.

When Q = 0, the solutions reduce to the uncharged scalar solutions, whose horizonstructure is given in [47]. In what follows, we will only focus on the charged case.However, the expression of f(r) with Q 6= 0 is complicated so that we can not solvedirectly the black hole horizons. Thus we focus on the extreme charged black holefirstly. As the Hawking temperature is

T =f ′(r+)

4π=

1

2π

r+

ℓ2+

BM

3r2+− X

(2k − 1)

(

(k − 1)− Bk(4k−1)r+

)

r1

(2k−1)

+

. (18)

Now we can evaluate the extreme charged black hole T = 0. Then the mass of extremalblack hole is

Mex = − 3(4k − 1)kr3ex

(k − 1)

(

3(4k − 1)rex + 4Bk

)

ℓ2, (19)

where the extreme black hole horizon rex(X) is the biggest root of the following equation

X =3(2k − 1)(4k − 1)r

(4k−1)(2k−1)ex

(k − 1)

(

3(4k − 1)rex + 4Bk

)

ℓ2. (20)

As it is still difficult to find analytical result for rex(X) and Mex, we can consideranother way to know more about them. From Eq.(19) and Eq.(20), we can rewrite thethe extreme black hole horizon as

rex(Mex, X) =

(

− Xk

(2k − 1)Mex

)

(2k−1)2(1−k)

, (21)

while the condition for extreme charged black hole, i.e. Mex, being the biggest root ofthe following equation

Mex = − 3(4k − 1)krex(Mex, X)3

(k − 1)

(

3rex(Mex, X)(4k − 1) + 4Bk

)

ℓ2. (22)

As rex is real and positive, Eq.(21) reveals the classification of the horizon structure ofthis solutions, which are divided into the following cases:

• for k ∈ (12, 1): one can find X < 0, thus Mex > 0:

1. When M < Mex, the solutions have no horizon, thus there is a bare singu-larity in the origin of the spacetime which is physically unaccepted;

7

2. When M = Mex, the solutions have a single horizon rex, which correspondsto the extreme black hole;

3. When M > Mex, the solutions have two horizons rE and rC , which corre-sponds to the non-extreme black hole.

• for k ∈ (1,+∞): one can find X > 0, thus Mex < 0:

1. When M < Mex, the solutions have no horizon which leads to a bare singu-larity in the origin of the spacetime, thus it is physical unaccepted;

2. When M = Mex, the solutions have a single horizon rex. This correspondsto the extreme black hole;

3. When Mex < M < 0, the solutions have two horizons rE and rC . This isthe non-extreme black hole;

4. When M ≥ 0, there is no another extreme black hole case for solutions withpositive mass, thus these solutions can only have either zero or one horizon.Besides, as f(0)f(+∞) < 0, f(r) must come across zero once at least, thusthe solutions are the black holes with a single horizon.

In Fig.(2), we plot the horizon function f(r) with Q = 0.5, B = 0.1, ℓ = 1 anddifferent values of parameters M, k, in order to have a more clear understanding of theabove horizon structure.

Figure 2: The horizon structure with Q = 0.5, B = 0.1, ℓ = 1. On both plots, the massdecreases from top to bottom. The dashed curve corresponds to the extreme blackholes. The mass for the extreme black hole with k = 7

9on the left is Mex ≃ 2.1242,

while it for k = 5 on the right is Mex ≃ −0.3030.

Totally, in order to avoid the appearance of a naked singularity, a horizon in thespacetime is needed at least. In this sense, we see that the physically acceptable region

8

for mass M is M ≥ Mex for any values of parameter k. Note for the cases k > 1,the lower bound Mex is negative, which shows that there exist scalar black holes withnegative mass in three dimensions. Actually, the negative mass is acceptable, as itis well-known that the mass for some kinds of black holes is bounded from belowby a negative value [7, 67–71], especially for the four dimensional charged scalar blackholes [7]. Obviously, the three dimensional charged scalar black holes with k ∈ (1,+∞)in EPM theory present in this paper share the same feature.

For the analytical results, we can give some detailed discussions about the case withB = 0, which corresponds to the black holes without scalar hair in EPM theory. Forthis simplified case, Eq.(19) and Eq.(20) reduce to

Mex = − kr2ex(k − 1)ℓ2

, X =(2k − 1)r

2k(2k−1)ex

(k − 1)ℓ2,

which lead to the condition and horizon for extreme charged black hole

rex = (2Q2)(2k−1)

2

(π

2ℓ2)

(2k−1)2k

, Mex = − k

(k − 1)ℓ2(2Q2)(2k−1)

(π

2ℓ2)

(2k−1)k

. (23)

Consider the parameter k: for k ∈ (12, 1), Mex > 0; while for k ∈ (1,+∞), we get

Mex < 0. The above discussion about the horizon structure still works. Hence we cansee that the scalar field does not affect the horizon structure, which is consistent withthe k = 1 case shown in [47, 54]. The effect of scalar field is only related to the valueof horizons and Mex. This can be seen from comparing with the horizon structureof black holes with B = 0.1 shown in Fig(2). Let us consider the black holes withB = 0, Q = 0.5, ℓ = 1, then Eq.(23) shows that the mass for extreme black hole withk = 7

9is Mex ≃ 2.1606, while it for k = 5 is Mex ≃ −0.2873.

3 Null geodesics

Let us consider the geodesic equations for uncharged test particles around the solutionwith scalar hair in EPM Theory. Since this spacetime has two Killing vectors ∂t and∂θ, there are two constants of motions, i.e.

E = f(r)dt

dλ, (24)

L = r2dθ

dλ, (25)

where λ is the affine parameter along the geodesics. The geodesic equation can bederived from the Lagrangian for a test particle

f(r)

(

dt

dλ

)2

− 1

f(r)

(

dr

dλ

)2

− r2(

dθ

dλ

)

= −m2,

9

where m = 0 corresponding to null geodesics and m = 1 corresponding to time-likegeodesics (without loss of generality). Inserting Eq.(24) and Eq.(25) into the aboveequation, one can obtain the effective equation

1

2

(

dr

dλ

)2

+ Veff(r) = 0, (26)

where Veff(r) is the effective potential and takes the form as

Veff(r) =1

2

[

f(r)

(

L2

r2+m2

)

−E2

]

. (27)

Then from Eq.(25) and Eq.(26), we get the orbit equation

(

dr

dθ

)2

= −2r4

L2Veff (r) (28)

As we consider the null geodesics, i.e. the geodesics for a photon. The effectivepotential (27) reduces to

Veff(r) =1

2

[

f(r)

(

L2

r2

)

− E2

]

. (29)

In the following subsections, we focus on the orbit equation (28) and effective potential(29) to classify all possible geodesic motions.

3.1 Radial geodesics where L = 0

Firstly we examine the radial geodesics where L = 0. The corresponding effectivepotential Eq.(29) further reduces to

Veff(r) = −E2

2.

Obviously, the behavior of these geodesics do not depend on the electric charge Qe andmass M of the black hole. As E is a constant, this resembles the geodesic motion of afree photon.

Combining Eq.(24) and Eq.(26), we can get

dt

dr= ± 1

f(r). (30)

Here we are only interested in the geodesics of the black holes, then one can rewritethe metric function as

f(r) = (r − rE)(r − rC)F (r) (31)

10

with rE being the Event horizon and rC being the Cauchy horizon of the black holes,respectively. However, it is difficult to find the form of F (r), but we still know thatF (r) can have either no real roots or negative roots. For non-extreme charged blackhole with two horizons, rE and rC are both positive. For charged black hole withsingle horizon, only rE is real and positive. For extreme charged black hole, rE = rCis positive.

Rewriting the right side of Eq.(30), it is equal to

1

f(r)=

1

(rE − rC)

[

1

F (rE)

(

1

(r − rE)− G(r, rE)

F (r)

)

− 1

F (rC)

(

1

(r − rC)− G(r, rC)

F (r)

)]

for the non-extreme black hole, where G(r, ri) =F (r)−F (ri)

(r−ri)with i being (E,C). Note

G(ri, ri) = F ′(ri) is finite. After integrating Eq.(30), we find

t = ± 1

(rE − rC)

[

1

F (rE)

(

ln(r − rE)−∫

G(r, rE)

F (r)dr

)

− 1

F (rC)

(

ln(r − rC)−∫

G(r, rC)

F (r)dr

)]

.

Similarly, for the extreme black hole case and the black hole case with a single horizon,it leads to

t = ± 1

F (rE)

(

1

(rE − r)−

∫

G(r, rE)

F (r)(r − rE)dr

)

.

For all cases, the sign “ + ” denotes the out going null rays and the sign “− ” denotesthe ingoing null ray. Consider the ingoing null rays, when r → rE, the coordinate timet → +∞ for the black holes.

On the other hand, the geodesic equation Eq.(26) can be integrated to give

r(λ) = ±Eλ,

From which, we find when r → rE (in-going case), λ has a finite value rEE. Hence one

can see that a photon without angular momentum arrives the horizons in its own finiteproper time, while it is an infinite coordinate time.

3.2 Radial geodesics where L 6= 0

Then we consider the radial geodesics where L 6= 0. From Eq.(29), the effective poten-tial can be obtained as

Veff(r) =L2

2r2

[(

1

ℓ2− Ξ2

)

r2 −(

1 +2B

3r

)

M −X(2k − 1)

(

r + 2Bk(4k−1)

)

r1

(2k−1)

]

, (32)

11

where Ξ = EL. When the minimum value of Veff is zero, the geodesic reaches to the

circular motion. This leads to the critical condition

Ξcri =

√

1

ℓ2+

(k − 1)M

r2cri

(

1

k+

4B

3(4k − 1)rcri

)

(33)

and the radius of the circular motion

rcri(M,X) =

(

− Xk

(2k − 1)M

)2k−12(1−k)

(34)

As rcri is real and positive, Eq.(34) shows that rcri can only exist in two cases: (1)for k ∈ (1

2, 1), we get X < 0, thus M > 0 is needed; (2) for k ∈ (1,+∞), we find X > 0,

thus M < 0 is needed. For other cases, rcri is imaginary, hence the circular motioncan not exist. However, even for the above two cases, one can find (k − 1)M < 0,thus Eq.(33) leads to another constraint on M , in order to keep Ξcri real. For thisone, we can firstly focus on the critical case, i.e. Ξcri = 0 with the effective potentialV (r) = L2

2r2f(r). Then the vanishing minimum of this effective potential is equal to that

of horizon function f(r), which immediately leads to the critical condition M = Mex

and the radius of circular motion rcri = rex. However, when Ξcri > 0, it is difficult tofind the physical constraint of M , because rcri is not equal to rex in this case. But thediscussion about the effective potential in what follows will show that the constraintfor real Ξcri is M ≤ Mex for k ∈ (1

2,+∞). (One can also see Fig.3 and Fig.4.)

Putting the above discussions together, we can obtain

• for k ∈ (12, 1): as Mex > 0, one can get that only the geodesics in the spacetime

with 0 < M ≤ Mex can contain the circular motion with its radius r = rcri;

• for k ∈ (1,+∞): as Mex < 0, only the geodesics in the spacetime with M ≤Mex(< 0) can contain the circular motion with the radius r = rcri.

These will classify the discussion of the geodesics. One can find that the geodesicsare divided into several cases according to the above constraint and the horizon struc-ture. This can also be seen from the behavior of the effective potntial, which is shownin Fig.3 and Fig.4. Actually, there are five models in these spacetime:

• Model I: this corresponds to the cases with 12< k < 1, 0 < M < Mex and

k > 1,M < Mex. These corresponding metric do not have a horizon. For theeffective potential, one can look at the left plots of Fig.3 and Fig.4, respectively.When the constant of motions Ξ changes, all the geodesic motions of a photoncan be categorized by four subcases:

1. when 0 ≤ Ξ < Ξcri, the effective potential is always positive in the wholespacetime. From the orbit equation Eq.(28), one can only get a imaginaryorbit equation, thus there is no allowed motion. Actually, the orbits areallowed only when Ξ ≥ Ξcri as shown later.

12

Figure 3: The effective potential V (r) with k = 79, Q = 0.5, B = 0.1, ℓ = 1. For this

case, the mass of extreme black holes Mex ≃ 2.1242. From left plot to right, the massfor solutions are M = 1.5,M = 2.5 and M = −0.5, respectively. On all plots, theconstant of motions Ξ decreases from top to bottom. The dashed curve correspondsto the case with circular motion.

Figure 4: The effective potential V (r) with k = 5, Q = 0.5, B = 0.1, ℓ = 1. For thiscase, the mass of extreme black holes Mex ≃ −0.3030. From left plot to right, the massfor solutions are M = −0.5,M = −0.1 and M = 0.1, respectively. On all plots, theconstant of motions Ξ decreases from top to bottom. The dashed curve correspondsto the case for circular motion.

2. when Ξ = Ξcri, the photon can only stay at the circular motion with theradius rcri. For example, the critical quantities in the left plots of Fig.3and Fig.4 are Ξcri ≃ 0.8427, rcri ≃ 1.2398 and Ξcri ≃ 0.6929, rcri ≃ 0.8942,respectively.

3. when Ξcri < Ξ < 1ℓ, the photon has the elliptic motions. Namely, all orbits

are bounded between aphelion and perihelion. For this subcase, we can onlyget the numerical results; consider the cases with Ξ = 0.9 in the left plots ofFig.3 and Fig.4, the aphelion and perihelion are rap ≃ 2.0658, rph ≃ 0.9439and rap ≃ 3.8709, rph ≃ 0.5239,respectively.

4. when Ξ ≥ 1ℓ, the geodesic motions are unbounded spiral motions at a large

13

scale. . When Ξ > 1ℓ, the orbit equation Eq.(28) is not integrable, thus the

numerical analysis is a useful tool for those geodesic motions. Note thatthere also exists perihelion but we cannot obtain it analytically. However,when Ξ = 1

ℓ, Eq.(28) becomes integrable, which is shown in detail here. For

simplicity, we consider the case with k = 79, Q = 1

2, B = 0, then the explicit

form of the integrable orbits corresponding to Ξ = 1ℓis

e25

√Mθ = 12

√Mr

25 +

√

2(

72Mr45 − 25× 2

29π

)

, (35)

which can be simplified to

r =

[

1

24√M

(

e25

√Mθ + 50× 2

29π e−

25

√Mθ

)]52

. (36)

The perihelion of this analytical orbit in Eq.(35) is trivially obtained as

rph =

[

1

24√M

(

1 + 50× 229π

)]52

. (37)

• Model II: this corresponds to the cases with 12< k < +∞,M = Mex. These

solution are the extreme black holes. For this cases, the effective potential hasthe same shape with the left plots of Fig.3 and Fig.4. The only difference is Ξcri

will reach to zero. As a result, the circular motion for this extremal charged EPMscalar black hole is the stopped motion at the degenerated horizon rcri = rex,which means eventually that this circular motion is not allowed. Then whenthe constant of motions Ξ changes, all the geodesic motions of a photon can becategorized by three subcases:

1. when Ξ = 0, there is no allowed motion, as the circular motion is notphysical.

2. when 0 < Ξ < 1ℓ, the photon has the similar elliptic motions as that of

Model I.

3. when Ξ ≥ 1ℓ, the geodesic motions of the photon are also unbounded spiral

motions. We only consider the integrable case with Ξ = 1ℓhere. In additional

with k = 79, Q = 1

2, B = 0, the orbits can be trivially obtained from Eq.(36)

r =

[

1

24√Mex

(

e25

√Mexθ + 50× 2

29π e−

25

√Mexθ

)]52

with the perihelion obtained from Eq.(37)

rexph =

[

1

24√Mex

(

1 + 50× 229π

)]52

.

14

• Model III: this corresponds to the cases with 12< k < 1,M > Mex and k >

1,Mex < M < 0. These solutions have two horizons. For the effective potential,one can focus on the middle plots of Fig.3 and Fig.4, respectively. From thefigures, we can get that the subcase without allowed motion does not exist, i.e.all subcases have allowed motions. When the constant of motions Ξ changes, allthe geodesic motions of a photon can be categorized by two subcases:

1. when 0 ≤ Ξ < 1ℓ, the geodesic motions are the elliptic motions. Consider

the subcases with Ξ = 0.9 in the middle plots of Fig.3 and Fig.4, the aphe-lion and perihelion are rap ≃ 3.2832, rph ≃ 0.4446 and rap ≃ 4.3720, rph ≃0.0849,respectively.

2. when Ξ ≥ 1ℓ, the photon stays at the unbounded spiral motions. The corre-

sponding analytical orbits for Ξ = 1ℓare the same with Eq.(36) with same

perihelion in Eq.(37) as shown in Model I.

• Model IV: this corresponds to the cases with 12< k < 1,M ≤ 0, which have no

horizon. One can take the right plots of Fig.3 for an example. We find that themeta-critical subcase, i.e. the one with Ξ = 1

ℓ, always has the positive effective

potential. Hence this meta-critical subcase has no allowed motion, other thanthe unbounded spiral motions as other Models. When the constant of motions Ξchanges, all the geodesic motions of a photon can be divided into two subcases:

1. when 0 ≤ Ξ ≤ 1ℓ, the positive effective potential indicates that there is no

allowed motion.

2. when Ξ > 1ℓ, the photon has the unbounded spiral motions. We calculate

the numerical results for case with Ξ = 1.2 in the right plot of Fig.3: theperihelion is rph ≃ 1.7597.

• Model V: this corresponds to the cases with k > 1,M > 0. These correspondingmetric have a single horizon. An example is presented in the right plots of Fig.4.When the constant of motions Ξ changes, the geodesic motions of a photon areall the unbounded spiral motions. For subcases with 0 ≤ Ξ < 1

ℓand Ξ ≥ 1

ℓ,

there is still one difference: the latter one has no perihelion, thus the photon forthe latter one will fall into the black holes. Follow the similar discussion withthe radial geodesics of un-rotating photon in the previous subsection, one canfind that the photon with Ξ ≥ 1

ℓalso arrive the black hole horizon in an infinite

coordinate time.

Considering all the above cases together, we present totally the geodesics of solutionswith different mass M in Tab.(1) and Tab.(2) corresponding to k ∈ (1

2, 1) and k ∈

(1,+∞), respectively. Comparing with the null geodesics of the three dimensionalregular charged black hole with k = 1 [72, 73], that of the EPM solutions with scalarhair have extra Model V which only contains the unbounded spiral motions.

15

Solutions M ≤ 0 M ∈ (0,Mex) M = Mex M > Mex

Geodesics Model IV Model I Model II Model III

Table 1: The models of geodesic for solutions with k ∈ (12, 1).

Solutions M < Mex M = Mex M ∈ (Mex, 0) M ≥ 0Geodesics Model I Model II Model III Model V

Table 2: The models of geodesic for solutions with k ∈ (1,+∞).

4 Closing remarks

In this paper, we have presented an exact static solution to Einstein-Power-Maxwelltheory in (2 + 1) dimensional AdS spacetimes, in which the scalar field couples togravity in a non-minimal way. We focus on the simplified form of scalar field, thenafter considering the scalar potential, we have found that a stable system leads to aconstraint on the power parameter k of Maxwell field. The solution contains a curva-ture singularity at the origin and is non-conformally flat. The horizon structure areidentified, then the existence of black hole solutions leads to the physically acceptablemass bound M ≥ Mex, where Mex is the mass of the extreme black holes. Especiallyfor the cases k > 1, the lower bound Mex is negative, which shows that there existscalar black holes with negative mass in three dimensions.

The geodesic motions for a photon in this spacetime have been discussed in detail.They are divided into five models, which are made up of the cases with the followinggeodesic motions: no-allowed motion, the circular motion, the elliptic motion and theunbounded spiral motion. We also present some analytical results for the unboundedspiral motion. Differing from the null geodesics of the three dimensional regular chargedblack hole with k = 1 [72,73], that of the EPM black holes with scalar hair have extraModel V which only contains the unbounded spiral motions. To consider the geodesicsof this spacetime further, the time-like one for a photon, even the one for chargedparticle are left as a future task. On the other hand, it would be also interesting tostudy the cause structure and thermodynamics of this black hole solution, especiallyfor the one with negative mass.

Acknowledgements

We would like to thank Liu Zhao for useful conversations. Wei Xu is supported bythe Research Innovation Fund of Huazhong University of Science and Technology(2014TS125). This work was supported by the National Natural Science Foundationof China.

16

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