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Topological black holes in dilaton gravity theory
Physics Letters B 612 (2005) 127–136 www.elsevier.com/locate/physletb
Topological black holes in dilaton gravity theory Chang Jun Gao a , Shuang Nan Zhang a,b a Department of Physics and Center for Astrophysics, Tsinghua University, Beijing 100084, China b 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. Cvetiˇc
Abstract The static solutions of electrically and magnetically charged dilaton black holes with the topology of R 2 ⊗ S n−2 , R 2 ⊗ 1 S ⊗ S n−3 and R 2 ⊗ R 1 ⊗ S n−3 are constructed from the dilaton gravity theory with a cosmological constant. The spacetime structure 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 coupling constant α = 0 or the dilaton φ = const, the potential is reduced to the well-known Einstein’s cosmological constant. Using this “cosmological constant” term of the dilaton gravity, we have constructed the dilaton black hole solutions which are asymptotically (anti-)de Sitter in four and higher dimensions. Exact solutions of charged dilaton black holes have been previously constructed by many authors [2]. However, these solutions are asymptotically neither flat nor (anti-)de Sitter. Even if in the presence of one Liouville-type potential which was regarded as the generalization of the cosmological constant, the obtained class of charged black hole solutions [3] are still asymptotically neither flat nor (anti-)de Sitter. In this Letter, we extend our former work, the static spherical dilaton black hole solutions, to the torus-like and hyperboloidal-like black holes with a cosmological constant. It is generally believed that a black hole in the four dimensional spacetime always has a spherical topology. That is, the event horizon of a black hole has the topology of S 2 . This was proven by Friedman, Schleich and Witt [4] provided that the spacetime is globally hyperbolic, 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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asymptotically flat and the null energy condition is preserved. However, when the asymptotic flatness and the energy condition are given up, there are no fundamental reasons to forbid the existence of static or stationary black holes with nontrivial topologies. In fact, the solutions of black holes with nontrivial topologies have been found by many authors [5]. These investigations are mainly based on the Einstein(–Maxwell) theory with a negative cosmological constant. Cai, Ji and Soh have obtained some exact topological black hole solutions in the Einstein– Maxwell-dilaton theory with a Liouville-type dilaton potential [6]. But their solutions are asymptotically neither flat nor anti-de Sitter. Thus the purpose of the present Letter is to find the solution of topological dilaton black holes which is asymptotically both flat and (anti-)de Sitter. The Letter proceeds as follows: first, we will construct the solutions of the electrically charged dilaton black holes in four dimensions. Then we extend them to higher dimensions. In the next, we will construct the solutions of the electrically and magnetically charged dilaton black holes. Finally, a discussion on the horizons of the black holes 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 action √ S = d 4 x −g R − 2∂µ φ∂ µ φ − e−2αφ F 2 2 2 −2φ/α 1 2 2 2φα 2 φα−φ/α α 3α e , − λ − 1 e + 3 − α + 8α e (1) 3 (1 + α 2 )2 where R is the scalar curvature, F 2 = Fµν F µν is the usual Maxwell contribution, and V (φ) is the dilaton potential which is reduced to the cosmological constant in the Einstein–Maxwell theory when the coupling constant α = 0, 2 2 1 2 α 3α − 1 e−2φ/α + 3 − α 2 e2φα + 8α 2 eφα−φ/α . V (φ) = λ 2 2 3 (1 + α ) Varying the action with respect to the metric, Maxwell, and dilaton fields, respectively, yields 1 1 Rµν = 2∂µ φ∂ν φ + gµν V + 2e−2αφ Fµβ Fνβ − gµν F 2 , 2 4 √ −2αφ µν = 0, F ∂µ −ge ∂V α 1 − e−2αφ F 2 . ∂µ ∂ µ φ = 4 ∂φ 2
(2)
(3) (4) (5)
The most general form of the metric for the static spacetime can be written as ds 2 = −U (r) dt 2 +
1 2 dr 2 + f (r)2 dΩk,2 , U (r)
2 is the line element of a two-dimensional hypersurface with constant curvature where dΩk,2 2 2 2 dθ + sin θ dϕ , for k = 1, 2 2 2 2 dΩk,2 = dθ + θ dϕ , for k = 0, 2 2 2 dθ + sinh θ dϕ , for k = −1.
(6)
(7)
For k = 1, the spacetime of Eq. (6) has the topology of R 2 ⊗ S 2 . The horizons of the black hole have the topology of a two-dimensional sphere. For k = 0, the spacetime of Eq. (6) has the topology of R 2 ⊗ T 2 by identified φ = 0 with φ = 2π and θ = 0 with θ = π . The horizons of the black hole have the topology of a two-dimensional torus. For
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k = −1, the spacetime of Eq. (6) has the topology of R 2 ⊗ H 2 . The horizons of the black hole have the topology of a two-dimensional hyperboloid. The Maxwell equation Eq. (4) can be integrated to give F01 =
Qe2αφ , f2
(8)
where Q is the electric charge. With the metric Eq. (6) and the Maxwell field Eq. (8), the equations of motion Eqs. (3)–(5) reduce to three independent equations 2 1 d 1 dV 2 dφ 2αφ Q (9) U , f = + αe dr 4 dφ f 2 dr f4 2 1 d 2f dφ (10) = − , f dr 2 dr 2 df 2k 1 d 2αφ Q 2Uf = (11) − V − 2e . dr f 2 dr f2 f4 The solutions of dilaton black holes in the de Sitter universe [1] remind us that the dilaton field φ in the presence of a cosmological constant is the same as that without the cosmological constant. So substituting the dilaton field
r− e2αφ = 1 − r
2α 2 1+α 2
,
(12)
into Eqs. (9)–(11), we obtain the solution of topological dilaton black holes with the cosmological constant
r− f =r 1− r Q2 =
r+ r− , 1 + α2
α2 1+α 2
,
r+ U = k− r
2M = r+ +
r− 1− r
1−α2 1+α 2
2α2 r− 1+α2 1 2 − λr 1 − , 3 r
1 − α2 r− . 1 + α2
(13)
Here M is the mass of the black hole. Note that the solutions are almost identical to the spherically dilaton black hole solutions; the only difference is that the number “1” is replaced by k in the function U (r). In other words, given that k = 1, the solutions of Eqs. (12) and (13) will restore to the well-known spherically charged dilaton black 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 higher dimensions. So let us consider the n-dimensional theory in which gravity is coupled to dilaton and Maxwell field with an action 4αφ 4 − n−2 n √ µ 2 ∂µ φ∂ φ − V (φ) − e S = d x −g R − (14) F , n−2 where R is the scalar curvature, F 2 = Fµν F µν is the usual Maxwell contribution, α is an arbitrary constant governing the strength of the coupling between the dilaton and the Maxwell field, and V (φ) is a potential of dilaton φ which is with respect to the cosmological constant which is given by
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V (φ) =
4(n−3)(φ−φ0 ) 2 λ −α (n − 2) n2 − nα 2 − 6n + α 2 + 9 e− (n−2)α 2 2 3(n − 3 + α ) −2(φ−φ0 )(n−3−α 2 ) 4α(φ−φ0 ) (n−2)α . + (n − 2)(n − 3)2 n − 1 − α 2 e n−2 + 4α 2 (n − 3)(n − 2)2 e
(15)
Here λ is the cosmological constant and φ0 is the asymptotic value of dilaton. We note that Eq. (15) restores to Eq. (2) if we rescale the dilaton φ and set n = 4 in Eq. (15). Varying the action with respect to the metric, Maxwell, and dilaton fields, respectively, yields 4αφ 4 1 1 − n−2 α 2 ∂µ φ∂ν φ + gµν V − 2e gµν F , Fµα Fν − Rµν = − (16) n−2 4 2n − 4 √ 4αφ ∂µ −ge− n−2 F µν = 0, (17) 4αφ n − 2 ∂V α ∂µ ∂ µ φ = (18) − e− n−2 F 2 . 8 ∂φ 2 Without the loss of generality, we set the required metric as follows 2 ds 2 = −U (r) dt 2 + W (r) dr 2 + f (r)2 dΩk,n−2 , 2 where r denotes the radial variable and dΩk,n−2 is the line element of a (n − 2)-dimensional constant curvature which is defined by 2 2 2 2 2 2 2 2 2 dθ1 + sin θ1 dθ2 + sin θ1 sin θ2 dθ3 + · · · + sin θ1 · · · sin θn−3 dϕ , 2 dΩk,n−2 = dθ12 + θ12 dθ22 + θ12 sin2 θ2 dθ32 + · · · + θ12 · · · sin2 θn−3 dϕ 2 , 2 dθ1 + sinh2 θ1 dθ22 + sinh2 θ1 sin2 θ2 dθ32 + · · · + sinh2 θ1 · · · sin2 θn−3 dϕ 2 ,
(19) hypersurface with k = +1, k = 0,
(20)
k = −1.
For k = 1, the spacetime of Eq. (33) has the topology of R 2 ⊗ S n−2 . The horizons of the black hole have the topology of a (n − 2)-dimensional sphere. For k = 0, the spacetime of Eq. (19) has the topology of R 2 ⊗ S 1 ⊗ S n−3 by identified ϕ = 0 with ϕ = 2π and θ1 = 0 with θ1 = π . The horizons of the black hole have the topology of a (n − 2)-dimensional torus. For k = −1, the spacetime of Eq. (19) has the topology of R 2 ⊗ R 1 ⊗ S n−3 also by identified ϕ = 0 with ϕ = 2π . The horizons of the black hole have the topology of a (n − 2)-dimensional hyperboloid. Inspecting the solution of the higher-dimensional (k = +1) dilaton black holes with cosmological constant n−3 1−γ (n−3) n−3 γ n−3 r− r− 1 r+ 1− − λr 2 1 − , U (r) = 1 − r r 3 r n−3 n−3 1−γ (n−3) n−3 γ −1 r+ r− r− 1 2 W (r) = 1 − − λr 1 − 1− r r 3 r n−3 −γ (n−4) r− × 1− , r n−3 γ r− 2 2 f (r) = r 1 − (21) , r and the solution of the four-dimensional topological dilaton black holes
1−α2 2α2 r+ r− 1+α2 r− 1+α2 1 2 U (r) = k − 1− − λr 1 − , r r 3 r 1−α2 2α2 r− 1+α2 −1 r+ r− 1+α2 1 2 W (r) = k − 1− − λr 1 − , r r 3 r
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2α2 r− 1+α2 f (r) = r 1 − , r 2
2
131
(22)
we find that the higher-dimensional version is given by n−3 n−3 1−γ (n−3) n−3 γ r+ r− r− 1 2 U (r) = k − − λr 1 − , 1− r r 3 r n−3 n−3 1−γ (n−3) n−3 γ −1 r+ r− r− 1 W (r) = k − 1− − λr 2 1 − r r 3 r n−3 −γ (n−4) r− × 1− , r n−3 γ r− 2 2 f (r) = r 1 − , r
(23)
where γ and the change of the black hole are given by γ=
2α 2 , (n − 3)(n − 3 + α 2 )
Q2 =
(n − 2)(n − 3)2 − 4αφ0 n−3 n−3 e n−2 r+ r− . 2(n − 3 + α 2 )
(24)
We note that the solution Eqs. (23) is almost identical to the spherically dilaton black hole solution Eqs. (21); the only difference is that the number “1” is replaced by k in the function U (r). In the next, we will show our solution satisfies 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 find the non-vanishing component of the tensor for Maxwell field is n−3 γ (3−n) 4αφ r− , F01 = Qe n−2 r 2−n 1 − (25) r where Q is the electric charge of the black hole, and the dilaton is n−3 (n−2)√γ √2+3γ −nγ /2 r− 2φ 2φ0 1− e =e . r
(26)
Now it is very easy to verify that Eqs. (23)–(26) satisfy the Einstein equations (16). Thus Eq. (23) is just the metric of higher-dimensional topological dilaton black holes with cosmological constant. The physical mass of the black hole is obtained as follows n−3 (n−3)2 +α2 (n−3)(n−3+α 2 ) r− r+ (n − 2)(n − 3) n−3 r M = (n − 3) 1 − + . 2 r+ 2(n − 3 + α 2 ) −
(27)
The dilaton charge D of the black hole is given by 1 D= d n−2 Σ µ ∇µ φ 4π n−3 α2 −(n−3)2 (n−3)(n−3+α 2 ) r− α(n − 2)(n − 3) n−3 1 − r =− dΩk,n−2 − r+ 8π(n − 3 + α 2 )
π n−3 n−3 α2 −(n−3)2 θ1 dθ1 , k = +1, 0 sin (n−3)(n−3+α 2 ) r− α(n − 2)(n − 3) n−3 2π n−3 r =− dΩ1,n−3 1− dθ1 , k = 0, 0 θ1 r+ 8π(n − 3 + α 2 ) − ∞ sinhn−3 θ dθ , k = −1, 1 1 0
(28)
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where dΩk,n−2 and dΩ1,n−2 denote the area of (n − 2)-dimensional unit orthogonal hypersurface (sphere, torus, 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 orthogonal space. For the k = −1 case, the dilaton charge is infinite. However, it has the same surface density of charge as the k = +1 and k = 0 case. It also reveals that the dilaton charge D is determined by r+ , r− and k. On the other hand, Eqs. (24) and (27) reveal that r+ and r− are determined by the mass M and the charge Q. Thus the dilaton charge is determined by not only M, Q but also k. In other words, the quantity of the dilaton charge of a black hole is closely related to the topology of the spacetime. This is different from the mass and charge of the black hole.
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 constant, we consider the following action √ S = d 4 x −g R − 2∂µ φ∂ µ φ − V (φ) − e−2φ F 2 − e−2φ H 2 , (29) where R is the scalar curvature, F 2 = Fµν F µν and H 2 = Hµν H µν are the Maxwell electric and magnetic contributions, respectively, and V (φ) is given by λ 2φ e + e−2φ + 4 , (30) 3 which is the case of α = 1 in Eq. (2). This is due to the fact that the doubly charged black hole solutions only exist for the coupling constant α = 1 [2]. Varying the action with respect to the metric, Maxwell fields, and dilaton field, respectively, yields 1 1 1 Rµν = 2∂µ φ∂ν φ + gµν V + 2e−2φ Fµα Fνα − gµν F 2 + 2e−2φ Hµα Hνα − gµν H 2 , (31) 2 4 4 √ ∂µ −ge−2φ F µν = 0, (32) √ −2φ µν = 0, H ∂µ −ge (33) 1 1 ∂V 1 − e−2φ F 2 − e−2φ H 2 . ∂µ ∂ µ φ = (34) 4 ∂φ 2 2 V (φ) =
The most general form of the metric for the static spacetime can be written as ds 2 = −U (r) dt 2 +
1 2 dr 2 + f (r)2 dΩk,2 , U (r)
(35)
2 is defined as Eq. (7). Then the Maxwell equations Eqs. (32), (33) can be integrated to give where dΩk,2
F=
Qe2φ dt ∧ dr, f2
H = P e2φ dθ ∧ dϕ,
(36)
where Q and P are the electric and magnetic charges, respectively. With the metric Eq. (35) and the Maxwell fields Eq. (36), the equations of motion (31) and (34) reduce to four equations 1 dV 1 1 d 2 dφ f = + 4 Q2 e2φ − P 2 e−2φ , (37) U 2 dr 4 dφ f dr f 2 1 d 2f dφ (38) =− , f dr 2 dr
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df 2k 1 d 2 2Uf = 2 − V − 4 Q2 e2φ + P 2 e−2φ , dr f 2 dr f f 2 1 d 2 dU f = −V + 4 Q2 e2φ + P 2 e−2φ . 2 dr f dr f
(39) (40)
Eq. (40) can be followed from Eqs. (37)–(39) by virtue of the Bianchi identity if φ = 0. So only three of them are independent. Similar to the case of topological electrically charged dilaton black holes, we obtain the solutions of topological electrically and magnetically charged black holes with the cosmological constant 2Σ , f = r 2 − Σ 2, e2φ = 1 + r −Σ r− Σ 2 −1 1 2 r+ 1− 1− 2 U = k− − λ r − Σ2 , r r 3 r 2Q2 = kΣ 2 + r+ r− − Σ(r+ + kr− ),
2P 2 = kΣ 2 + r+ r− + Σ(r+ + kr− ),
2M = r+ + r− ,
(41)
where M and Σ are the mass and the dilaton charge (only for k = 1) of the black hole. The solution restores to the Gibbons–Maeda solution provided that k = 1 and λ = 0. The dilaton charge of the black hole is given by Σ dΩk,2 . D = 4π 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 consider the α = 1, n = 4 and electrically charged case. Then the metric of the black hole reads −1 2M 1 2M 1 2 2 ds = − k − (42) − λr(r − 2D) dt + k − − λr(r − 2D) dr 2 + r(r − 2D) dΩk2 , r 3 r 3 Q2 dΩk,2 is the absolute value of the dilaton charge. To study the event horizons of the black holes, where D ≡ 8πM we should rewrite the metric in the Schwarzschild coordinate system. So we set r = D + x 2 + D2, (43) then Eq. (42) becomes 2M 1 2 2 ds = − k − − λr dt 2 √ D + r 2 + D2 3 −1 2M D 2 −1 1 k− + 1+ 2 − λr 2 dr 2 + r 2 dΩk2 . √ r D + r 2 + D2 3
(44)
Here we have rewritten the variable r in placed of x. The metric becomes singular for k−
2M 1 − λr 2 = 0, √ 2 2 3 D+ r +D
(45)
which is the equation of horizons and at r = 0 (the curvature singularity). In order to simplify the discussions on the event horizons of the black holes, we make variable transformation in Eq. (45) ˜ y˜ + 2D). r = y( (46)
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Then the equation of the event horizons becomes k−
1 2M − λy( ˜ y˜ + 2D) = 0. y˜ + 2D 3
(47)
Set M > 0, λ˜ = 4λD 2 , D then the equation of the event horizons is simplified to be y˜ = 2Dy,
k−
β=
(48)
1 β ˜ − λy(y + 1) = 0. y+1 3
(49)
For the torus-like black holes k = 0 and hyperboloidal-like black holes k = −1, it is found that neither black hole event horizon nor cosmic event horizon exist when λ˜ ≡ 3H 2 > 0. The total spacetime is dynamic. This is different from the solution of the Einstein–Maxwell theory where a cosmic event horizon exists. On the other hand, when λ˜ ≡ −3H 2 < 0, one black hole event horizon would appear for the dilaton black hole. Similar to the toruslike and hyperboloidal-like black holes in the Einstein–Maxwell theory, the horizon has the topology of T 2 or H 2 . Despite this similarity, there is one significant deference. When the dilaton is present, there is no analog of the inner Cauchy horizon in the Einstein–Maxwell theory [2] when λ˜ ≡ −3H 2 < 0. This reveals that the presence of dilaton has important consequences on the structure of black hole horizons and the black hole thermodynamics. For the spherical black holes, i.e., k = +1, we find that neither black√hole event horizon nor cosmic event horizon are presented when λ˜ ≡ −3H 2 < 0 provided that √ β 1, i.e., Q 2M. The total spacetime is static and the singularity is naked. However, when β > 1 (Q < 2M), one black hole event horizon would appear. This is also different from the solution of the Einstein–Maxwell theory where two horizons, the outer event horizon and the inner Cauchy, may exist. For the spherical black holes, the case of λ˜ ≡ 3H 2 > 0 is quite complicated, as shown below. (1) (2) (3) (4)
√ When H 1 (λ 3/(4M 2 )) and β 1 (Q √2M), there exists no horizons. The total spacetime is dynamic. When H 1 (λ 3/(4M 2 )) and β < 1 (Q < 2M), there √ exists only one cosmic event horizon. When 0 < H < 1 (0 < λ < 3/(4M 2 )) and β 1 (Q 2M), there exists only one cosmic event horizon. When 0 < H < 1 (0 < λ < 3/(4M 2 )) and 1 < β < F (H ), where F (H ) is defined by
1 9H + 2H 3 + 2H 2 3 + H 2 + 6 3 + H 2 , (50) 27H there exists a black hole event horizon and a cosmic event horizon. (5) When 0 < H < 1 (0 < λ < 3/(4M 2 )) and β = F (H ), the black hole event horizon and the cosmic event 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/(4M 2 )) and β > F (H ), there is no horizon in this spacetime and the total spacetime is dynamic. F (H ) =
Compared to the charged black holes in de Sitter universe, the dilaton version has some remarkable properties. In the first place, there may be three horizons in the Reissner–Nordström–de Sitter spacetime, i.e., black hole event horizon, black hole Cauchy horizon and cosmic event horizon. However, the charged dilaton black holes in de Sitter universe has at most two horizons. Here the inner Cauchy horizon disappears. This is due to the fact that the inner horizon is unstable, as pointed √ by Garfinkle, etc. [2]. Secondly, the transition between cosmic event horizon and the singularity occurs at Q = 2M rather than Q = M as in the Reissner–Nordström–de Sitter case. This is because the dilaton in dilaton gravity theory contributes an extra attractive force, so for a given M, one needs a larger Q to balance the forces between two black holes.
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For the case of α = 1, Eq. (12) gives the dilaton field 1 2D φ = ln 1 − . 2 r
135
(51)
It diverges at the singularity r = 2D. In the Schwarzschild coordinate system Eq. (44), the dilaton field becomes 1 2D φ = ln 1 − (52) . √ 2 D + x 2 + D2 It also diverges at the singularity x = 0 and behaves regular at the horizons. For the general case of α = 1, n > 4, we find that the spacetime structure is almost identical to the α = 1, n = 4 case. Here the surface r = r− is a curvature singularity except for the α = 0 case when it is a nonsingular inner horizon. Thus the solutions Eq. (23) describe black holes only when r > 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 presence of a cosmological constant. The solutions are asymptotically both flat and (anti-)de Sitter. It is found that, in general, the black holes with different topologies have different spacetime structures. It is also found that, unlike the mass and charge, the dilaton charge of the black holes is closely related to the topology of the spacetime. Thus dilaton charge is a topological charge. We had better note that the existence of such topological black holes does not violate the topology theorem since each surface of the spacetime at constant t and r are torus or hyperboloid which is essentially different from the asymptotically flat spacetime. In other words, the assumption of the theorem is not satisfied. Besides, one cannot rule out the possibility, at least at present, that some of the energy conditions are violated either by the existence of the bizarre matter, dark energy, or by taking into account of the quantum effects of matter fields. The discussion on the horizons of the black holes reveals that there are at most two horizons present in the spacetime of electrically charged dilaton black holes. Besides, for the T 2 and H 2 solutions, it is found that the cosmological constant in Eq. (1) must be negative in order to describe the black holes. These are quite different from 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 improved the Letter significantly. This study is supported in part by the Special Funds for Major State Basic Research Projects, the National Natural Science Foundation of China and the Directional Research Project of the Chinese Academy of Sciences.
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Topological black holes in dilaton gravity theory Chang Jun Gao a , Shuang Nan Zhang a,b a Department of Physics and Center for Astrophysics, Tsinghua University, Beijing 100084, China b 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. Cvetiˇc
Abstract The static solutions of electrically and magnetically charged dilaton black holes with the topology of R 2 ⊗ S n−2 , R 2 ⊗ 1 S ⊗ S n−3 and R 2 ⊗ R 1 ⊗ S n−3 are constructed from the dilaton gravity theory with a cosmological constant. The spacetime structure 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 coupling constant α = 0 or the dilaton φ = const, the potential is reduced to the well-known Einstein’s cosmological constant. Using this “cosmological constant” term of the dilaton gravity, we have constructed the dilaton black hole solutions which are asymptotically (anti-)de Sitter in four and higher dimensions. Exact solutions of charged dilaton black holes have been previously constructed by many authors [2]. However, these solutions are asymptotically neither flat nor (anti-)de Sitter. Even if in the presence of one Liouville-type potential which was regarded as the generalization of the cosmological constant, the obtained class of charged black hole solutions [3] are still asymptotically neither flat nor (anti-)de Sitter. In this Letter, we extend our former work, the static spherical dilaton black hole solutions, to the torus-like and hyperboloidal-like black holes with a cosmological constant. It is generally believed that a black hole in the four dimensional spacetime always has a spherical topology. That is, the event horizon of a black hole has the topology of S 2 . This was proven by Friedman, Schleich and Witt [4] provided that the spacetime is globally hyperbolic, 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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asymptotically flat and the null energy condition is preserved. However, when the asymptotic flatness and the energy condition are given up, there are no fundamental reasons to forbid the existence of static or stationary black holes with nontrivial topologies. In fact, the solutions of black holes with nontrivial topologies have been found by many authors [5]. These investigations are mainly based on the Einstein(–Maxwell) theory with a negative cosmological constant. Cai, Ji and Soh have obtained some exact topological black hole solutions in the Einstein– Maxwell-dilaton theory with a Liouville-type dilaton potential [6]. But their solutions are asymptotically neither flat nor anti-de Sitter. Thus the purpose of the present Letter is to find the solution of topological dilaton black holes which is asymptotically both flat and (anti-)de Sitter. The Letter proceeds as follows: first, we will construct the solutions of the electrically charged dilaton black holes in four dimensions. Then we extend them to higher dimensions. In the next, we will construct the solutions of the electrically and magnetically charged dilaton black holes. Finally, a discussion on the horizons of the black holes 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 action √ S = d 4 x −g R − 2∂µ φ∂ µ φ − e−2αφ F 2 2 2 −2φ/α 1 2 2 2φα 2 φα−φ/α α 3α e , − λ − 1 e + 3 − α + 8α e (1) 3 (1 + α 2 )2 where R is the scalar curvature, F 2 = Fµν F µν is the usual Maxwell contribution, and V (φ) is the dilaton potential which is reduced to the cosmological constant in the Einstein–Maxwell theory when the coupling constant α = 0, 2 2 1 2 α 3α − 1 e−2φ/α + 3 − α 2 e2φα + 8α 2 eφα−φ/α . V (φ) = λ 2 2 3 (1 + α ) Varying the action with respect to the metric, Maxwell, and dilaton fields, respectively, yields 1 1 Rµν = 2∂µ φ∂ν φ + gµν V + 2e−2αφ Fµβ Fνβ − gµν F 2 , 2 4 √ −2αφ µν = 0, F ∂µ −ge ∂V α 1 − e−2αφ F 2 . ∂µ ∂ µ φ = 4 ∂φ 2
(2)
(3) (4) (5)
The most general form of the metric for the static spacetime can be written as ds 2 = −U (r) dt 2 +
1 2 dr 2 + f (r)2 dΩk,2 , U (r)
2 is the line element of a two-dimensional hypersurface with constant curvature where dΩk,2 2 2 2 dθ + sin θ dϕ , for k = 1, 2 2 2 2 dΩk,2 = dθ + θ dϕ , for k = 0, 2 2 2 dθ + sinh θ dϕ , for k = −1.
(6)
(7)
For k = 1, the spacetime of Eq. (6) has the topology of R 2 ⊗ S 2 . The horizons of the black hole have the topology of a two-dimensional sphere. For k = 0, the spacetime of Eq. (6) has the topology of R 2 ⊗ T 2 by identified φ = 0 with φ = 2π and θ = 0 with θ = π . The horizons of the black hole have the topology of a two-dimensional torus. For
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k = −1, the spacetime of Eq. (6) has the topology of R 2 ⊗ H 2 . The horizons of the black hole have the topology of a two-dimensional hyperboloid. The Maxwell equation Eq. (4) can be integrated to give F01 =
Qe2αφ , f2
(8)
where Q is the electric charge. With the metric Eq. (6) and the Maxwell field Eq. (8), the equations of motion Eqs. (3)–(5) reduce to three independent equations 2 1 d 1 dV 2 dφ 2αφ Q (9) U , f = + αe dr 4 dφ f 2 dr f4 2 1 d 2f dφ (10) = − , f dr 2 dr 2 df 2k 1 d 2αφ Q 2Uf = (11) − V − 2e . dr f 2 dr f2 f4 The solutions of dilaton black holes in the de Sitter universe [1] remind us that the dilaton field φ in the presence of a cosmological constant is the same as that without the cosmological constant. So substituting the dilaton field
r− e2αφ = 1 − r
2α 2 1+α 2
,
(12)
into Eqs. (9)–(11), we obtain the solution of topological dilaton black holes with the cosmological constant
r− f =r 1− r Q2 =
r+ r− , 1 + α2
α2 1+α 2
,
r+ U = k− r
2M = r+ +
r− 1− r
1−α2 1+α 2
2α2 r− 1+α2 1 2 − λr 1 − , 3 r
1 − α2 r− . 1 + α2
(13)
Here M is the mass of the black hole. Note that the solutions are almost identical to the spherically dilaton black hole solutions; the only difference is that the number “1” is replaced by k in the function U (r). In other words, given that k = 1, the solutions of Eqs. (12) and (13) will restore to the well-known spherically charged dilaton black 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 higher dimensions. So let us consider the n-dimensional theory in which gravity is coupled to dilaton and Maxwell field with an action 4αφ 4 − n−2 n √ µ 2 ∂µ φ∂ φ − V (φ) − e S = d x −g R − (14) F , n−2 where R is the scalar curvature, F 2 = Fµν F µν is the usual Maxwell contribution, α is an arbitrary constant governing the strength of the coupling between the dilaton and the Maxwell field, and V (φ) is a potential of dilaton φ which is with respect to the cosmological constant which is given by
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V (φ) =
4(n−3)(φ−φ0 ) 2 λ −α (n − 2) n2 − nα 2 − 6n + α 2 + 9 e− (n−2)α 2 2 3(n − 3 + α ) −2(φ−φ0 )(n−3−α 2 ) 4α(φ−φ0 ) (n−2)α . + (n − 2)(n − 3)2 n − 1 − α 2 e n−2 + 4α 2 (n − 3)(n − 2)2 e
(15)
Here λ is the cosmological constant and φ0 is the asymptotic value of dilaton. We note that Eq. (15) restores to Eq. (2) if we rescale the dilaton φ and set n = 4 in Eq. (15). Varying the action with respect to the metric, Maxwell, and dilaton fields, respectively, yields 4αφ 4 1 1 − n−2 α 2 ∂µ φ∂ν φ + gµν V − 2e gµν F , Fµα Fν − Rµν = − (16) n−2 4 2n − 4 √ 4αφ ∂µ −ge− n−2 F µν = 0, (17) 4αφ n − 2 ∂V α ∂µ ∂ µ φ = (18) − e− n−2 F 2 . 8 ∂φ 2 Without the loss of generality, we set the required metric as follows 2 ds 2 = −U (r) dt 2 + W (r) dr 2 + f (r)2 dΩk,n−2 , 2 where r denotes the radial variable and dΩk,n−2 is the line element of a (n − 2)-dimensional constant curvature which is defined by 2 2 2 2 2 2 2 2 2 dθ1 + sin θ1 dθ2 + sin θ1 sin θ2 dθ3 + · · · + sin θ1 · · · sin θn−3 dϕ , 2 dΩk,n−2 = dθ12 + θ12 dθ22 + θ12 sin2 θ2 dθ32 + · · · + θ12 · · · sin2 θn−3 dϕ 2 , 2 dθ1 + sinh2 θ1 dθ22 + sinh2 θ1 sin2 θ2 dθ32 + · · · + sinh2 θ1 · · · sin2 θn−3 dϕ 2 ,
(19) hypersurface with k = +1, k = 0,
(20)
k = −1.
For k = 1, the spacetime of Eq. (33) has the topology of R 2 ⊗ S n−2 . The horizons of the black hole have the topology of a (n − 2)-dimensional sphere. For k = 0, the spacetime of Eq. (19) has the topology of R 2 ⊗ S 1 ⊗ S n−3 by identified ϕ = 0 with ϕ = 2π and θ1 = 0 with θ1 = π . The horizons of the black hole have the topology of a (n − 2)-dimensional torus. For k = −1, the spacetime of Eq. (19) has the topology of R 2 ⊗ R 1 ⊗ S n−3 also by identified ϕ = 0 with ϕ = 2π . The horizons of the black hole have the topology of a (n − 2)-dimensional hyperboloid. Inspecting the solution of the higher-dimensional (k = +1) dilaton black holes with cosmological constant n−3 1−γ (n−3) n−3 γ n−3 r− r− 1 r+ 1− − λr 2 1 − , U (r) = 1 − r r 3 r n−3 n−3 1−γ (n−3) n−3 γ −1 r+ r− r− 1 2 W (r) = 1 − − λr 1 − 1− r r 3 r n−3 −γ (n−4) r− × 1− , r n−3 γ r− 2 2 f (r) = r 1 − (21) , r and the solution of the four-dimensional topological dilaton black holes
1−α2 2α2 r+ r− 1+α2 r− 1+α2 1 2 U (r) = k − 1− − λr 1 − , r r 3 r 1−α2 2α2 r− 1+α2 −1 r+ r− 1+α2 1 2 W (r) = k − 1− − λr 1 − , r r 3 r
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2α2 r− 1+α2 f (r) = r 1 − , r 2
2
131
(22)
we find that the higher-dimensional version is given by n−3 n−3 1−γ (n−3) n−3 γ r+ r− r− 1 2 U (r) = k − − λr 1 − , 1− r r 3 r n−3 n−3 1−γ (n−3) n−3 γ −1 r+ r− r− 1 W (r) = k − 1− − λr 2 1 − r r 3 r n−3 −γ (n−4) r− × 1− , r n−3 γ r− 2 2 f (r) = r 1 − , r
(23)
where γ and the change of the black hole are given by γ=
2α 2 , (n − 3)(n − 3 + α 2 )
Q2 =
(n − 2)(n − 3)2 − 4αφ0 n−3 n−3 e n−2 r+ r− . 2(n − 3 + α 2 )
(24)
We note that the solution Eqs. (23) is almost identical to the spherically dilaton black hole solution Eqs. (21); the only difference is that the number “1” is replaced by k in the function U (r). In the next, we will show our solution satisfies 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 find the non-vanishing component of the tensor for Maxwell field is n−3 γ (3−n) 4αφ r− , F01 = Qe n−2 r 2−n 1 − (25) r where Q is the electric charge of the black hole, and the dilaton is n−3 (n−2)√γ √2+3γ −nγ /2 r− 2φ 2φ0 1− e =e . r
(26)
Now it is very easy to verify that Eqs. (23)–(26) satisfy the Einstein equations (16). Thus Eq. (23) is just the metric of higher-dimensional topological dilaton black holes with cosmological constant. The physical mass of the black hole is obtained as follows n−3 (n−3)2 +α2 (n−3)(n−3+α 2 ) r− r+ (n − 2)(n − 3) n−3 r M = (n − 3) 1 − + . 2 r+ 2(n − 3 + α 2 ) −
(27)
The dilaton charge D of the black hole is given by 1 D= d n−2 Σ µ ∇µ φ 4π n−3 α2 −(n−3)2 (n−3)(n−3+α 2 ) r− α(n − 2)(n − 3) n−3 1 − r =− dΩk,n−2 − r+ 8π(n − 3 + α 2 )
π n−3 n−3 α2 −(n−3)2 θ1 dθ1 , k = +1, 0 sin (n−3)(n−3+α 2 ) r− α(n − 2)(n − 3) n−3 2π n−3 r =− dΩ1,n−3 1− dθ1 , k = 0, 0 θ1 r+ 8π(n − 3 + α 2 ) − ∞ sinhn−3 θ dθ , k = −1, 1 1 0
(28)
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where dΩk,n−2 and dΩ1,n−2 denote the area of (n − 2)-dimensional unit orthogonal hypersurface (sphere, torus, 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 orthogonal space. For the k = −1 case, the dilaton charge is infinite. However, it has the same surface density of charge as the k = +1 and k = 0 case. It also reveals that the dilaton charge D is determined by r+ , r− and k. On the other hand, Eqs. (24) and (27) reveal that r+ and r− are determined by the mass M and the charge Q. Thus the dilaton charge is determined by not only M, Q but also k. In other words, the quantity of the dilaton charge of a black hole is closely related to the topology of the spacetime. This is different from the mass and charge of the black hole.
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 constant, we consider the following action √ S = d 4 x −g R − 2∂µ φ∂ µ φ − V (φ) − e−2φ F 2 − e−2φ H 2 , (29) where R is the scalar curvature, F 2 = Fµν F µν and H 2 = Hµν H µν are the Maxwell electric and magnetic contributions, respectively, and V (φ) is given by λ 2φ e + e−2φ + 4 , (30) 3 which is the case of α = 1 in Eq. (2). This is due to the fact that the doubly charged black hole solutions only exist for the coupling constant α = 1 [2]. Varying the action with respect to the metric, Maxwell fields, and dilaton field, respectively, yields 1 1 1 Rµν = 2∂µ φ∂ν φ + gµν V + 2e−2φ Fµα Fνα − gµν F 2 + 2e−2φ Hµα Hνα − gµν H 2 , (31) 2 4 4 √ ∂µ −ge−2φ F µν = 0, (32) √ −2φ µν = 0, H ∂µ −ge (33) 1 1 ∂V 1 − e−2φ F 2 − e−2φ H 2 . ∂µ ∂ µ φ = (34) 4 ∂φ 2 2 V (φ) =
The most general form of the metric for the static spacetime can be written as ds 2 = −U (r) dt 2 +
1 2 dr 2 + f (r)2 dΩk,2 , U (r)
(35)
2 is defined as Eq. (7). Then the Maxwell equations Eqs. (32), (33) can be integrated to give where dΩk,2
F=
Qe2φ dt ∧ dr, f2
H = P e2φ dθ ∧ dϕ,
(36)
where Q and P are the electric and magnetic charges, respectively. With the metric Eq. (35) and the Maxwell fields Eq. (36), the equations of motion (31) and (34) reduce to four equations 1 dV 1 1 d 2 dφ f = + 4 Q2 e2φ − P 2 e−2φ , (37) U 2 dr 4 dφ f dr f 2 1 d 2f dφ (38) =− , f dr 2 dr
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df 2k 1 d 2 2Uf = 2 − V − 4 Q2 e2φ + P 2 e−2φ , dr f 2 dr f f 2 1 d 2 dU f = −V + 4 Q2 e2φ + P 2 e−2φ . 2 dr f dr f
(39) (40)
Eq. (40) can be followed from Eqs. (37)–(39) by virtue of the Bianchi identity if φ = 0. So only three of them are independent. Similar to the case of topological electrically charged dilaton black holes, we obtain the solutions of topological electrically and magnetically charged black holes with the cosmological constant 2Σ , f = r 2 − Σ 2, e2φ = 1 + r −Σ r− Σ 2 −1 1 2 r+ 1− 1− 2 U = k− − λ r − Σ2 , r r 3 r 2Q2 = kΣ 2 + r+ r− − Σ(r+ + kr− ),
2P 2 = kΣ 2 + r+ r− + Σ(r+ + kr− ),
2M = r+ + r− ,
(41)
where M and Σ are the mass and the dilaton charge (only for k = 1) of the black hole. The solution restores to the Gibbons–Maeda solution provided that k = 1 and λ = 0. The dilaton charge of the black hole is given by Σ dΩk,2 . D = 4π 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 consider the α = 1, n = 4 and electrically charged case. Then the metric of the black hole reads −1 2M 1 2M 1 2 2 ds = − k − (42) − λr(r − 2D) dt + k − − λr(r − 2D) dr 2 + r(r − 2D) dΩk2 , r 3 r 3 Q2 dΩk,2 is the absolute value of the dilaton charge. To study the event horizons of the black holes, where D ≡ 8πM we should rewrite the metric in the Schwarzschild coordinate system. So we set r = D + x 2 + D2, (43) then Eq. (42) becomes 2M 1 2 2 ds = − k − − λr dt 2 √ D + r 2 + D2 3 −1 2M D 2 −1 1 k− + 1+ 2 − λr 2 dr 2 + r 2 dΩk2 . √ r D + r 2 + D2 3
(44)
Here we have rewritten the variable r in placed of x. The metric becomes singular for k−
2M 1 − λr 2 = 0, √ 2 2 3 D+ r +D
(45)
which is the equation of horizons and at r = 0 (the curvature singularity). In order to simplify the discussions on the event horizons of the black holes, we make variable transformation in Eq. (45) ˜ y˜ + 2D). r = y( (46)
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Then the equation of the event horizons becomes k−
1 2M − λy( ˜ y˜ + 2D) = 0. y˜ + 2D 3
(47)
Set M > 0, λ˜ = 4λD 2 , D then the equation of the event horizons is simplified to be y˜ = 2Dy,
k−
β=
(48)
1 β ˜ − λy(y + 1) = 0. y+1 3
(49)
For the torus-like black holes k = 0 and hyperboloidal-like black holes k = −1, it is found that neither black hole event horizon nor cosmic event horizon exist when λ˜ ≡ 3H 2 > 0. The total spacetime is dynamic. This is different from the solution of the Einstein–Maxwell theory where a cosmic event horizon exists. On the other hand, when λ˜ ≡ −3H 2 < 0, one black hole event horizon would appear for the dilaton black hole. Similar to the toruslike and hyperboloidal-like black holes in the Einstein–Maxwell theory, the horizon has the topology of T 2 or H 2 . Despite this similarity, there is one significant deference. When the dilaton is present, there is no analog of the inner Cauchy horizon in the Einstein–Maxwell theory [2] when λ˜ ≡ −3H 2 < 0. This reveals that the presence of dilaton has important consequences on the structure of black hole horizons and the black hole thermodynamics. For the spherical black holes, i.e., k = +1, we find that neither black√hole event horizon nor cosmic event horizon are presented when λ˜ ≡ −3H 2 < 0 provided that √ β 1, i.e., Q 2M. The total spacetime is static and the singularity is naked. However, when β > 1 (Q < 2M), one black hole event horizon would appear. This is also different from the solution of the Einstein–Maxwell theory where two horizons, the outer event horizon and the inner Cauchy, may exist. For the spherical black holes, the case of λ˜ ≡ 3H 2 > 0 is quite complicated, as shown below. (1) (2) (3) (4)
√ When H 1 (λ 3/(4M 2 )) and β 1 (Q √2M), there exists no horizons. The total spacetime is dynamic. When H 1 (λ 3/(4M 2 )) and β < 1 (Q < 2M), there √ exists only one cosmic event horizon. When 0 < H < 1 (0 < λ < 3/(4M 2 )) and β 1 (Q 2M), there exists only one cosmic event horizon. When 0 < H < 1 (0 < λ < 3/(4M 2 )) and 1 < β < F (H ), where F (H ) is defined by
1 9H + 2H 3 + 2H 2 3 + H 2 + 6 3 + H 2 , (50) 27H there exists a black hole event horizon and a cosmic event horizon. (5) When 0 < H < 1 (0 < λ < 3/(4M 2 )) and β = F (H ), the black hole event horizon and the cosmic event 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/(4M 2 )) and β > F (H ), there is no horizon in this spacetime and the total spacetime is dynamic. F (H ) =
Compared to the charged black holes in de Sitter universe, the dilaton version has some remarkable properties. In the first place, there may be three horizons in the Reissner–Nordström–de Sitter spacetime, i.e., black hole event horizon, black hole Cauchy horizon and cosmic event horizon. However, the charged dilaton black holes in de Sitter universe has at most two horizons. Here the inner Cauchy horizon disappears. This is due to the fact that the inner horizon is unstable, as pointed √ by Garfinkle, etc. [2]. Secondly, the transition between cosmic event horizon and the singularity occurs at Q = 2M rather than Q = M as in the Reissner–Nordström–de Sitter case. This is because the dilaton in dilaton gravity theory contributes an extra attractive force, so for a given M, one needs a larger Q to balance the forces between two black holes.
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For the case of α = 1, Eq. (12) gives the dilaton field 1 2D φ = ln 1 − . 2 r
135
(51)
It diverges at the singularity r = 2D. In the Schwarzschild coordinate system Eq. (44), the dilaton field becomes 1 2D φ = ln 1 − (52) . √ 2 D + x 2 + D2 It also diverges at the singularity x = 0 and behaves regular at the horizons. For the general case of α = 1, n > 4, we find that the spacetime structure is almost identical to the α = 1, n = 4 case. Here the surface r = r− is a curvature singularity except for the α = 0 case when it is a nonsingular inner horizon. Thus the solutions Eq. (23) describe black holes only when r > 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 presence of a cosmological constant. The solutions are asymptotically both flat and (anti-)de Sitter. It is found that, in general, the black holes with different topologies have different spacetime structures. It is also found that, unlike the mass and charge, the dilaton charge of the black holes is closely related to the topology of the spacetime. Thus dilaton charge is a topological charge. We had better note that the existence of such topological black holes does not violate the topology theorem since each surface of the spacetime at constant t and r are torus or hyperboloid which is essentially different from the asymptotically flat spacetime. In other words, the assumption of the theorem is not satisfied. Besides, one cannot rule out the possibility, at least at present, that some of the energy conditions are violated either by the existence of the bizarre matter, dark energy, or by taking into account of the quantum effects of matter fields. The discussion on the horizons of the black holes reveals that there are at most two horizons present in the spacetime of electrically charged dilaton black holes. Besides, for the T 2 and H 2 solutions, it is found that the cosmological constant in Eq. (1) must be negative in order to describe the black holes. These are quite different from 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 improved the Letter significantly. This study is supported in part by the Special Funds for Major State Basic Research Projects, the National Natural Science Foundation of China and the Directional Research Project of the Chinese Academy of Sciences.
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