Topological black holes in dilaton gravity theory

b

etime

-uplingcon-ck holeed dila-llyas the

ke ande fourtopologylic,

Physics Letters B 612 (2005) 127–136

www.elsevier.com/locate/physlet

Topological black holes in dilaton gravity theory

Chang Jun Gaoa, Shuang Nan Zhanga,b

a Department of Physics and Center for Astrophysics, Tsinghua University, Beijing 100084, Chinab Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, China

Received 23 December 2004; received in revised form 9 March 2005; accepted 11 March 2005

Available online 21 March 2005

Editor: M. Cvetic

Abstract

The static solutions of electrically and magnetically charged dilaton black holes with the topology ofR2 ⊗ Sn−2, R2 ⊗S1 ⊗ Sn−3 andR2 ⊗ R1 ⊗ Sn−3 are constructed from the dilaton gravity theory with a cosmological constant. The spacstructure of the resulted solutions is studied. 2005 Elsevier B.V. All rights reserved.

PACS: 04.20.Ha; 04.50.+h; 04.70.Bw

1. Introduction

Recently we found the “cosmological constant term” in the dilaton gravity theory[1]. It is found that the cosmological constant proposed in the Einstein theory is coupled to Liouville-type dilaton potential. When the coconstantα = 0 or the dilatonφ = const, the potential is reduced to the well-known Einstein’s cosmologicalstant. Using this “cosmological constant” term of the dilaton gravity, we have constructed the dilaton blasolutions which are asymptotically (anti-)de Sitter in four and higher dimensions. Exact solutions of chargton black holes have been previously constructed by many authors[2]. However, these solutions are asymptoticaneither flat nor (anti-)de Sitter. Even if in the presence of one Liouville-type potential which was regardedgeneralization of the cosmological constant, the obtained class of charged black hole solutions[3] are still asymp-totically neither flat nor (anti-)de Sitter.

In this Letter, we extend our former work, the static spherical dilaton black hole solutions, to the torus-lihyperboloidal-like black holes with a cosmological constant. It is generally believed that a black hole in thdimensional spacetime always has a spherical topology. That is, the event horizon of a black hole has theof S2. This was proven by Friedman, Schleich and Witt[4] provided that the spacetime is globally hyperbo

E-mail addresses: [email protected](C.J. Gao),[email protected](S.N. Zhang).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2005.03.026

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mailto:[email protected]

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128 C.J. Gao, S.N. Zhang / Physics Letters B 612 (2005) 127–136

and theary blackfoundgativeinstein–

erblack

blackolutionse black

ction

l

of

. For

asymptotically flat and the null energy condition is preserved. However, when the asymptotic flatnessenergy condition are given up, there are no fundamental reasons to forbid the existence of static or stationholes with nontrivial topologies. In fact, the solutions of black holes with nontrivial topologies have beenby many authors[5]. These investigations are mainly based on the Einstein(–Maxwell) theory with a necosmological constant. Cai, Ji and Soh have obtained some exact topological black hole solutions in the EMaxwell-dilaton theory with a Liouville-type dilaton potential[6]. But their solutions are asymptotically neithflat nor anti-de Sitter. Thus the purpose of the present Letter is to find the solution of topological dilatonholes which is asymptotically both flat and (anti-)de Sitter.

The Letter proceeds as follows: first, we will construct the solutions of the electrically charged dilatonholes in four dimensions. Then we extend them to higher dimensions. In the next, we will construct the sof the electrically and magnetically charged dilaton black holes. Finally, a discussion on the horizons of thholes is given.

2. Topological electrically charged dilaton black holes

We consider the four-dimensional theory in which gravity is coupled to dilaton and Maxwell field with an a

S =∫

d4x√−g

R − 2∂µφ∂µφ − e−2αφF 2

(1)− 2

3λ

1

(1+ α2)2

[α2(3α2 − 1

)e−2φ/α + (

3− α2)e2φα + 8α2eφα−φ/α]

,

whereR is the scalar curvature,F 2 = FµνFµν is the usual Maxwell contribution, andV (φ) is the dilaton potentia

which is reduced to the cosmological constant in the Einstein–Maxwell theory when the coupling constantα = 0,

(2)V (φ) = 2

3λ

1

(1+ α2)2

[α2(3α2 − 1

)e−2φ/α + (

3− α2)e2φα + 8α2eφα−φ/α].

Varying the action with respect to the metric, Maxwell, and dilaton fields, respectively, yields

(3)Rµν = 2∂µφ∂νφ + 1

2gµνV + 2e−2αφ

(FµβFβ

ν − 1

4gµνF

2)

,

(4)∂µ

(√−ge−2αφFµν) = 0,

(5)∂µ∂µφ = 1

4

∂V

∂φ− α

2e−2αφF 2.

The most general form of the metric for the static spacetime can be written as

(6)ds2 = −U(r) dt2 + 1

U(r)dr2 + f (r)2 dΩ2

k,2,

wheredΩ2k,2 is the line element of a two-dimensional hypersurface with constant curvature

(7)dΩ2k,2 =

dθ2 + sin2 θ dϕ2, for k = 1,

dθ2 + θ2 dϕ2, for k = 0,

dθ2 + sinh2 θ dϕ2, for k = −1.

Fork = 1, the spacetime of Eq.(6) has the topology ofR2⊗S2. The horizons of the black hole have the topologya two-dimensional sphere. Fork = 0, the spacetime of Eq.(6) has the topology ofR2 ⊗T 2 by identifiedφ = 0 withφ = 2π andθ = 0 with θ = π . The horizons of the black hole have the topology of a two-dimensional torus

C.J. Gao, S.N. Zhang / Physics Letters B 612 (2005) 127–136 129

gy

on field

n black

on

ighereld

-

k = −1, the spacetime of Eq.(6) has the topology ofR2 ⊗ H 2. The horizons of the black hole have the topoloof a two-dimensional hyperboloid.

The Maxwell equation Eq.(4) can be integrated to give

(8)F01 = Qe2αφ

f 2,

whereQ is the electric charge. With the metric Eq.(6) and the Maxwell field Eq.(8), the equations of motionEqs.(3)–(5)reduce to three independent equations

(9)1

f 2

d

dr

(f 2U

dφ

dr

)= 1

4

dV

dφ+ αe2αφ Q2

f 4,

(10)1

f

d2f

dr2= −

(dφ

dr

)2

,

(11)1

f 2

d

dr

(2Uf

df

dr

)= 2k

f 2− V − 2e2αφ Q2

f 4.

The solutions of dilaton black holes in the de Sitter universe[1] remind us that the dilaton fieldφ in the presenceof a cosmological constant is the same as that without the cosmological constant. So substituting the dilat

(12)e2αφ =(

1− r−r

) 2α2

1+α2

,

into Eqs.(9)–(11), we obtain the solution of topological dilaton black holes with the cosmological constant

f = r

(1− r−

r

) α2

1+α2

, U =(

k − r+r

)(1− r−

r

) 1−α2

1+α2 − 1

3λr2

(1− r−

r

) 2α2

1+α2

,

(13)Q2 = r+r−1+ α2

, 2M = r+ + 1− α2

1+ α2r−.

HereM is the mass of the black hole. Note that the solutions are almost identical to the spherically dilatohole solutions; the only difference is that the number “1” is replaced byk in the functionU(r). In other words,given thatk = 1, the solutions of Eqs.(12) and (13)will restore to the well-known spherically charged dilatblack hole solutions. It is apparent that the solutions are asymptotically both flat and (anti-)de Sitter.

3. Higher-dimensional topological dilaton black holes with cosmological constant

In this section, we will extend the topological dilaton black hole solutions from four dimensions to hdimensions. So let us consider then-dimensional theory in which gravity is coupled to dilaton and Maxwell fiwith an action

(14)S =∫

dnx√−g

[R − 4

n − 2∂µφ∂µφ − V (φ) − e− 4αφ

n−2 F 2],

whereR is the scalar curvature,F 2 = FµνFµν is the usual Maxwell contribution,α is an arbitrary constant gov

erning the strength of the coupling between the dilaton and the Maxwell field, andV (φ) is a potential of dilatonφwhich is with respect to the cosmological constant which is given by


h

he

of

V (φ) = λ

3(n − 3+ α2)2

[−α2(n − 2)(n2 − nα2 − 6n + α2 + 9

)e− 4(n−3)(φ−φ0)

(n−2)α

(15)+ (n − 2)(n − 3)2(n − 1− α2)e 4α(φ−φ0)

n−2 + 4α2(n − 3)(n − 2)2e−2(φ−φ0)(n−3−α2)

(n−2)α].

Hereλ is the cosmological constant andφ0 is the asymptotic value of dilaton. We note that Eq.(15) restores toEq.(2) if we rescale the dilatonφ and setn = 4 in Eq.(15).

Varying the action with respect to the metric, Maxwell, and dilaton fields, respectively, yields

(16)Rµν = − 4

n − 2

(∂µφ∂νφ + 1

4gµνV

)− 2e− 4αφ

n−2

(FµαFα

ν − 1

2n − 4gµνF

2)

,

(17)∂µ

(√−ge− 4αφn−2 Fµν

) = 0,

(18)∂µ∂µφ = n − 2

8

∂V

∂φ− α

2e− 4αφ

n−2 F 2.

Without the loss of generality, we set the required metric as follows

(19)ds2 = −U(r) dt2 + W(r)dr2 + f (r)2 dΩ2k,n−2,

wherer denotes the radial variable anddΩ2k,n−2 is the line element of a(n − 2)-dimensional hypersurface wit

constant curvature which is defined by

(20)dΩ2k,n−2 =

dθ21 + sin2 θ1dθ2

2 + sin2 θ1 sin2 θ2 dθ23 + · · · + sin2 θ1· · ·sin2 θn−3 dϕ2, k = +1,

dθ21 + θ2

1 dθ22 + θ2

1 sin2 θ2dθ23 + · · · + θ2

1 · · ·sin2 θn−3 dϕ2, k = 0,

dθ21 + sinh2 θ1dθ2

2 + sinh2 θ1 sin2 θ2 dθ23 + · · · + sinh2 θ1· · ·sin2 θn−3 dϕ2, k = −1.

For k = 1, the spacetime of Eq.(33) has the topology ofR2 ⊗ Sn−2. The horizons of the black hole have ttopology of a(n−2)-dimensional sphere. Fork = 0, the spacetime of Eq.(19)has the topology ofR2 ⊗S1 ⊗Sn−3

by identifiedϕ = 0 with ϕ = 2π andθ1 = 0 with θ1 = π . The horizons of the black hole have the topologya (n − 2)-dimensional torus. Fork = −1, the spacetime of Eq.(19) has the topology ofR2 ⊗ R1 ⊗ Sn−3 alsoby identifiedϕ = 0 with ϕ = 2π . The horizons of the black hole have the topology of a(n − 2)-dimensionalhyperboloid.

Inspecting the solution of the higher-dimensional(k = +1) dilaton black holes with cosmological constant

U(r) =[1−

(r+r

)n−3][1−

(r−r

)n−3]1−γ (n−3)

− 1

3λr2

[1−

(r−r

)n−3]γ

,

W(r) =[

1−(

r+r

)n−3][1−

(r−r

)n−3]1−γ (n−3)

− 1

3λr2

[1−

(r−r

)n−3]γ −1

×[1−

(r−r

)n−3]−γ (n−4)

,

(21)f (r)2 = r2[1−

(r−r

)n−3]γ

,

and the solution of the four-dimensional topological dilaton black holes

U(r) =(

k − r+r

)(1− r−

r

) 1−α2

1+α2 − 1

3λr2

(1− r−

r

) 2α2

1+α2

,

W(r) =[(

k − r+)(

1− r−) 1−α2

1+α2 − 1λr2

(1− r−

) 2α2

1+α2]−1

,
r r 3 r


n

e black

(22)f (r)2 = r2(

1− r−r

) 2α2

1+α2

,

we find that the higher-dimensional version is given by

U(r) =[k −

(r+r

)n−3][1−

(r−r

)n−3]1−γ (n−3)

− 1

3λr2

[1−

(r−r

)n−3]γ

,

W(r) =[

k −(

r+r

)n−3][1−

(r−r

)n−3]1−γ (n−3)

− 1

3λr2

[1−

(r−r

)n−3]γ −1

×[1−

(r−r

)n−3]−γ (n−4)

,

(23)f (r)2 = r2[1−

(r−r

)n−3]γ

,

whereγ and the change of the black hole are given by

(24)γ = 2α2

(n − 3)(n − 3+ α2), Q2 = (n − 2)(n − 3)2

2(n − 3+ α2)e− 4αφ0

n−2 rn−3+ rn−3− .

We note that the solution Eqs.(23) is almost identical to the spherically dilaton black hole solution Eqs.(21); theonly difference is that the number “1” is replaced byk in the functionU(r). In the next, we will show our solutiosatisfies the field equations of the Einstein–Maxwell-dilaton theory.

Substituted Eqs.(23)–(24)into the Maxwell equations Eq.(17) and the dilaton field equation Eq.(18), we findthe non-vanishing component of the tensor for Maxwell field is

(25)F01 = Qe4αφn−2 r2−n

[1−

(r−r

)n−3]γ (3−n)

,

whereQ is the electric charge of the black hole, and the dilaton is

(26)e2φ = e2φ0

[1−

(r−r

)n−3](n−2)√

γ√

2+3γ−nγ /2

.

Now it is very easy to verify that Eqs.(23)–(26)satisfy the Einstein equations(16). Thus Eq.(23) is just the metricof higher-dimensional topological dilaton black holes with cosmological constant. The physical mass of thhole is obtained as follows

(27)M = r+2

(n − 3)

[1−

(r−r+

)n−3] (n−3)2+α2

(n−3)(n−3+α2) + (n − 2)(n − 3)

2(n − 3+ α2)rn−3− .

The dilaton chargeD of the black hole is given by

D = 1

4π

∫dn−2Σµ∇µφ

= −α(n − 2)(n − 3)

8π(n − 3+ α2)rn−3−

[1−

(r−r+

)n−3] α2−(n−3)2

(n−3)(n−3+α2)∫

dΩk,n−2

(28)= −α(n − 2)(n − 3)

8π(n − 3+ α2)rn−3−

[1−

(r−r+

)n−3] α2−(n−3)2

(n−3)(n−3+α2)∫

dΩ1,n−3

∫ π

0 sinn−3 θ1 dθ1, k = +1,∫ 2π

0 θn−31 dθ1, k = 0,∫ ∞

0 sinhn−3 θ1dθ1, k = −1,


re,

l space.as the

,ee isle.

nstant,

ri-

exist

where∫

dΩk,n−2 and∫

dΩ1,n−2 denote the area of(n − 2)-dimensional unit orthogonal hypersurface (sphetorus, or identified hyperboloid) and(n − 3)-dimensional unit sphere, respectively.

Eq. (28) tells us that the value of the dilaton charge is proportional to the surface area of the orthogonaFor thek = −1 case, the dilaton charge is infinite. However, it has the same surface density of chargek = +1 andk = 0 case. It also reveals that the dilaton chargeD is determined byr+, r− andk. On the other handEqs.(24) and (27)reveal thatr+ andr− are determined by the massM and the chargeQ. Thus the dilaton chargis determined by not onlyM,Q but alsok. In other words, the quantity of the dilaton charge of a black holclosely related to the topology of the spacetime. This is different from the mass and charge of the black ho

4. Topological electrically and magnetically charged dilaton black hole

In order to find the topological solutions for the doubly charged dilaton black holes with cosmological cowe consider the following action

(29)S =∫

d4x√−g

[R − 2∂µφ∂µφ − V (φ) − e−2φF 2 − e−2φH 2],

whereR is the scalar curvature,F 2 = FµνFµν andH 2 = HµνH

µν are the Maxwell electric and magnetic contbutions, respectively, andV (φ) is given by

(30)V (φ) = λ

3

(e2φ + e−2φ + 4

),

which is the case ofα = 1 in Eq.(2). This is due to the fact that the doubly charged black hole solutions onlyfor the coupling constantα = 1 [2].

Varying the action with respect to the metric, Maxwell fields, and dilaton field, respectively, yields

(31)Rµν = 2∂µφ∂νφ + 1

2gµνV + 2e−2φ

(FµαFα

ν − 1

4gµνF

2)

+ 2e−2φ

(HµαHα

ν − 1

4gµνH

2)

,

(32)∂µ

(√−ge−2φFµν) = 0,

(33)∂µ

(√−ge−2φHµν) = 0,

(34)∂µ∂µφ = 1

4

∂V

∂φ− 1

2e−2φF 2 − 1

2e−2φH 2.

The most general form of the metric for the static spacetime can be written as

(35)ds2 = −U(r) dt2 + 1

U(r)dr2 + f (r)2 dΩ2

k,2,

wheredΩ2k,2 is defined as Eq.(7). Then the Maxwell equations Eqs.(32), (33)can be integrated to give

(36)F = Qe2φ

f 2dt ∧ dr, H = Pe2φ dθ ∧ dϕ,

whereQ andP are the electric and magnetic charges, respectively. With the metric Eq.(35)and the Maxwell fieldsEq.(36), the equations of motion(31)and(34) reduce to four equations

(37)1

f 2

d

dr

(f 2U

dφ

dr

)= 1

4

dV

dφ+ 1

f 4

(Q2e2φ − P 2e−2φ

),

(38)1 d2f = −

(dφ

)2

,
f dr2 dr


etions of

esby

onsider

holes,

rmation

(39)1

f 2

d

dr

(2Uf

df

dr

)= 2k

f 2− V − 2

f 4

(Q2e2φ + P 2e−2φ

),

(40)1

f 2

d

dr

(f 2dU

dr

)= −V + 2

f 4

(Q2e2φ + P 2e−2φ

).

Eq.(40)can be followed from Eqs.(37)–(39)by virtue of the Bianchi identity ifφ′ = 0. So only three of them arindependent. Similar to the case of topological electrically charged dilaton black holes, we obtain the solutopological electrically and magnetically charged black holes with the cosmological constant

e2φ =(

1+ 2Σ

r − Σ

), f =

√r2 − Σ2,

U =(

k − r+r

)(1− r−

r

)(1− Σ2

r2

)−1

− 1

3λ(r2 − Σ2),

(41)2Q2 = kΣ2 + r+r− − Σ(r+ + kr−), 2P 2 = kΣ2 + r+r− + Σ(r+ + kr−), 2M = r+ + r−,

whereM andΣ are the mass and the dilaton charge (only fork = 1) of the black hole. The solution restorto the Gibbons–Maeda solution provided thatk = 1 andλ = 0. The dilaton charge of the black hole is givenD = Σ

4π

∫dΩk,2.

5. Event horizons of topological dilaton black holes

In this section, we are devoted to the study of the event horizons. For simplicity in mathematics, we ctheα = 1, n = 4 and electrically charged case. Then the metric of the black hole reads

(42)ds2 = −[k − 2M

r− 1

3λr(r − 2D)

]dt2 +

[k − 2M

r− 1

3λr(r − 2D)

]−1

dr2 + r(r − 2D)dΩ2k ,

whereD ≡ Q2

8πM

∫dΩk,2 is the absolute value of the dilaton charge. To study the event horizons of the black

we should rewrite the metric in the Schwarzschild coordinate system. So we set

(43)r = D +√

x2 + D2,

then Eq.(42)becomes

ds2 = −[k − 2M

D + √r2 + D2

− 1

3λr2

]dt2

(44)+(

1+ D2

r2

)−1[k − 2M

D + √r2 + D2

− 1

3λr2

]−1

dr2 + r2 dΩ2k .

Here we have rewritten the variabler in placed ofx. The metric becomes singular for

(45)k − 2M

D + √r2 + D2

− 1

3λr2 = 0,

which is the equation of horizons and atr = 0 (the curvature singularity).In order to simplify the discussions on the event horizons of the black holes, we make variable transfo

in Eq.(45)

(46)r = √y(y + 2D).


kisr hand,torus-

g of theofamics.ventd

s ison and

mic.

ent

tal

roperties.ole events in det that thezonis is

Then the equation of the event horizons becomes

(47)k − 2M

y + 2D− 1

3λy(y + 2D) = 0.

Set

(48)y = 2Dy, β = M

D> 0, λ = 4λD2,

then the equation of the event horizons is simplified to be

(49)k − β

y + 1− 1

3λy(y + 1) = 0.

For the torus-like black holesk = 0 and hyperboloidal-like black holesk = −1, it is found that neither blachole event horizon nor cosmic event horizon exist whenλ ≡ 3H 2 > 0. The total spacetime is dynamic. Thisdifferent from the solution of the Einstein–Maxwell theory where a cosmic event horizon exists. On the othewhenλ ≡ −3H 2 < 0, one black hole event horizon would appear for the dilaton black hole. Similar to thelike and hyperboloidal-like black holes in the Einstein–Maxwell theory, the horizon has the topology ofT 2 or H 2.Despite this similarity, there is one significant deference. When the dilaton is present, there is no analoinner Cauchy horizon in the Einstein–Maxwell theory[2] whenλ ≡ −3H 2 < 0. This reveals that the presencedilaton has important consequences on the structure of black hole horizons and the black hole thermodyn

For the spherical black holes, i.e.,k = +1, we find that neither black hole event horizon nor cosmic ehorizon are presented whenλ ≡ −3H 2 < 0 provided thatβ 1, i.e.,Q

√2M . The total spacetime is static an

the singularity is naked. However, whenβ > 1 (Q <√

2M), one black hole event horizon would appear. Thialso different from the solution of the Einstein–Maxwell theory where two horizons, the outer event horizthe inner Cauchy, may exist.

For the spherical black holes, the case ofλ ≡ 3H 2 > 0 is quite complicated, as shown below.

(1) WhenH 1 (λ 3/(4M2)) andβ 1 (Q √

2M), there exists no horizons. The total spacetime is dyna(2) WhenH 1 (λ 3/(4M2)) andβ < 1 (Q <

√2M), there exists only one cosmic event horizon.

(3) When 0< H < 1 (0< λ < 3/(4M2)) andβ 1 (Q √

2M), there exists only one cosmic event horizon.(4) When 0< H < 1 (0< λ < 3/(4M2)) and 1< β < F(H), whereF(H) is defined by

(50)F(H) = 1

27H

(9H + 2H 3 + 2H 2

√3+ H 2 + 6

√3+ H 2

),

there exists a black hole event horizon and a cosmic event horizon.(5) When 0< H < 1 (0 < λ < 3/(4M2)) and β = F(H), the black hole event horizon and the cosmic ev

horizon would coalesce. That is a version of the Nariai solution which has been discussed by Bousso[7].(6) When 0< H < 1 (0 < λ < 3/(4M2)) and β > F(H), there is no horizon in this spacetime and the to

spacetime is dynamic.

Compared to the charged black holes in de Sitter universe, the dilaton version has some remarkable pIn the first place, there may be three horizons in the Reissner–Nordström–de Sitter spacetime, i.e., black hhorizon, black hole Cauchy horizon and cosmic event horizon. However, the charged dilaton black holeSitter universe has at most two horizons. Here the inner Cauchy horizon disappears. This is due to the facinner horizon is unstable, as pointed by Garfinkle, etc.[2]. Secondly, the transition between cosmic event horiand the singularity occurs atQ = √

2M rather thanQ = M as in the Reissner–Nordström–de Sitter case. Thbecause the dilaton in dilaton gravity theory contributes an extra attractive force, so for a givenM , one needs alargerQ to balance the forces between two black holes.


er

sence ofeneral,e mass

s dilatondoes nothorem isitions areeffects

s presenttferent

ved therojects,cademy

For the case ofα = 1, Eq.(12)gives the dilaton field

(51)φ = 1

2ln

(1− 2D

r

).

It diverges at the singularityr = 2D. In the Schwarzschild coordinate system Eq.(44), the dilaton field becomes

(52)φ = 1

2ln

(1− 2D

D + √x2 + D2

).

It also diverges at the singularityx = 0 and behaves regular at the horizons.For the general case ofα = 1, n > 4, we find that the spacetime structure is almost identical to theα = 1, n = 4

case. Here the surfacer = r− is a curvature singularity except for theα = 0 case when it is a nonsingular innhorizon. Thus the solutions Eq.(23)describe black holes only whenr > r− as discussed by Horn and Horowitz[2].

6. Conclusion and discussion

In conclusion, we have constructed electrically and magnetically charged black hole solutions in the prea cosmological constant. The solutions are asymptotically both flat and (anti-)de Sitter. It is found that, in gthe black holes with different topologies have different spacetime structures. It is also found that, unlike thand charge, the dilaton charge of the black holes is closely related to the topology of the spacetime. Thucharge is a topological charge. We had better note that the existence of such topological black holesviolate the topology theorem since each surface of the spacetime at constantt andr are torus or hyperboloid whicis essentially different from the asymptotically flat spacetime. In other words, the assumption of the thenot satisfied. Besides, one cannot rule out the possibility, at least at present, that some of the energy condviolated either by the existence of the bizarre matter, dark energy, or by taking into account of the quantumof matter fields. The discussion on the horizons of the black holes reveals that there are at most two horizonin the spacetime of electrically charged dilaton black holes. Besides, for theT 2 andH 2 solutions, it is found thathe cosmological constant in Eq.(1) must be negative in order to describe the black holes. These are quite diffrom the spherically symmetric black holes in the Einstein–Maxwell theory.

Acknowledgements

We thank the anonymous referee for the expert and insightful comments, which have certainly improLetter significantly. This study is supported in part by the Special Funds for Major State Basic Research Pthe National Natural Science Foundation of China and the Directional Research Project of the Chinese Aof Sciences.

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Topological black holes in dilaton gravity theory