Black Hole solutions in Einstein-Maxwell-Yang-Mills-Gauss-Bonnet Theory
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
arXiv:0801.2110v2 [gr-qc] 12 Mar 2008
Black Hole solutioห้องสมุดไป่ตู้s in Einstein-Maxwell-Yang-Mills-Gauss-Bonnet Theory
S. Habib Mazharimousavi∗ and M. Halilsoy† Department of Physics, Eastern Mediterranean University, G. Magusa, north Cyprus, Mersin-10, Turkey ∗ habib.mazhari@emu.edu.tr † mustafa.halilsoy@emu.edu.tr March 12, 2008
R − (N − 1) (N − 2) Λ − Fµν F µν − 3
a=1
1 (a) A(b) A(c) C 2σ (b)(c) µ ν
(N −1)(N −2) 2
(2)
Fµν
(a)
= ∂µ Aν − ∂ν Aµ parameter Lie
(a) Aµ
in which C(b)(c) stands for the structure constants of group G and σ is a coupling constant.
(N −1)(N −2)/2 (a) (a)µν Fµν F
(1)
M
Here, R is the Ricci Scalar while the YM and Maxwell fields are defined respectively by
(a) Fµν a) (a) = ∂µ A( ν − ∂ν Aµ +
(N −1)(N −2)/2
(4)
+
a=1
1 (a) (a)λ (a) 2Fµ Fνλ − Fλσ F (a)λσ gµν . 2
(a)
Variation with respect to the gauge potentials Aµ and Aµ yield the respective YM and Maxwell equations
Abstract We consider Maxwell and Yang-Mills (YM) fields together, interacting through gravity both in Einstein and Gauss-Bonnet (GB) theories. For this purpose we choose two different sets of Maxwell and metric ansatzes. The combined effect in the first ansatz modifies the features for dimensions N≥5, with Maxwell term dominating over YM, or vice versa, depending on the asymptotic conditions. For N=3 and N=4, where the GB term is absent, we recover the well-known Ba˜ nados-Teitelboim-Zanelli (BTZ) and Reissner-Nordstrom metrics, respectively. The second ansatz corresponds to the case of constant radius function for SN −2 part in the metric. This leads to the Bertotti-Robinson type solutions in the underlying theory.
2
Action, Field Equations and our Ansatzes
The action which describes Einstein-Maxwell-Yang-Mills gravity with a cosmological constant in N dimensions reads IG = 1 2 √ dxN −g ×
1
INTRODUCTION
We introduce Maxwell field alongside with Yang-Mills (YM) field in general relativity and present spherically symmetric black hole solutions in any higher dimensions. These two gauge fields, one Abelian the other non-Abelian, are coupled through gravity and to our knowledge they have not been considered together in a common geometry. Being linear, a multitude of Maxwellian fields can be superimposed at equal case, but for the YM field there is no such freedom. To obtain exact solutions the Maxwell field is chosen pure electric while the YM field is pure magnetic. Historically, starting with the Reissner-Nordstrom metric, its higher dimensional extensions known as the Einstein-Maxwell (EM) black holes are well-known by now. Einstein-Yang-Mills (EYM) black holes in higher dimensions have also attracted interest in more recent works [1,2,3]. In this paper we combine these two foregoing black holes (i.e. EM and EYM)
are the SO(N − 1)gauge group YM
2
potentials while Aµ represents the usual Maxwell potential. We note that the internal indices {a, b, c, ...} do not differ whether in covariant or contravariant form. Variation of the action with respect to the space-time metric gµν yields the field equations (N − 1) (N − 2) Λgµν = Tµν (3) 6 where the stress-energy tensor is the superposition of the Maxwell and YM parts, namely Gµν + Tµν = 1 λ 2Fµ Fνλ − Fλσ F λσ gµν 2
1
under the common title of EMYM black holes. Further, we add to it the GaussBonnet (GB) term overall which we phrase as the EMYMGB black holes. Such black holes will be characterized by mass (m), electric charge (q), YM charge (Q), GB parameter α and cosmological constant (Λ) . It is remarkable that exact solutions to such a highly non-linear theory can be found for two different Maxwell and metric ansatzes. Before adding the GB term we find the EMYM solutions in which the relative contributions from the Maxwell /YM charges can be compared. It is shown that for r → 0 Maxwell term dominates over the YM which is reminiscent of asymptotic freedom from the YM charge. On the other hand for r → ∞ the YM term becomes dominant. Such behaviors, we speculate, have relevance in connection with the mini (super-massive) black holes as well as asymptotically (anti)-de Sitter space times. The solutions for N>5 turn out to be distinct from the lower (i.e., N=3,4) dimensional cases. We show that in the latter case the role of a YM charge is similar to the electric charge as far as the space time metric is concerned. Namely, one is still the BTZ, while the other is Reissner-Nordstrom. It is found that for N=3 and N=5 the metric function contains a logaritmic term while in the other cases not. Our second ansatz leads beside the black hole solution to a Bertoti-Robinson (BR) type metric within EMYMGB theory. The paper is organized as follows: In Sec. (2) we introduce our action metric Maxwell YM ansatz and field equations. Sec. (3) follows with the solution of the field equations for different dimensionalities. In Sec. (4) we add the GaussBonnet (GB) form and present the most general solution. Sec. (5) deals with a new set of ansatz for the Maxwell field and metric function. The paper ends with concluding remarks in Sec. (6).
a)µν F;( + µ
1 (a) A(b) F (c)µν C σ (b)(c) µ F;µν µ 1 (a) A(b) ∗ F (c)µν C σ (b)(c) µ ∗F;µν µ
Black Hole solutioห้องสมุดไป่ตู้s in Einstein-Maxwell-Yang-Mills-Gauss-Bonnet Theory
S. Habib Mazharimousavi∗ and M. Halilsoy† Department of Physics, Eastern Mediterranean University, G. Magusa, north Cyprus, Mersin-10, Turkey ∗ habib.mazhari@emu.edu.tr † mustafa.halilsoy@emu.edu.tr March 12, 2008
R − (N − 1) (N − 2) Λ − Fµν F µν − 3
a=1
1 (a) A(b) A(c) C 2σ (b)(c) µ ν
(N −1)(N −2) 2
(2)
Fµν
(a)
= ∂µ Aν − ∂ν Aµ parameter Lie
(a) Aµ
in which C(b)(c) stands for the structure constants of group G and σ is a coupling constant.
(N −1)(N −2)/2 (a) (a)µν Fµν F
(1)
M
Here, R is the Ricci Scalar while the YM and Maxwell fields are defined respectively by
(a) Fµν a) (a) = ∂µ A( ν − ∂ν Aµ +
(N −1)(N −2)/2
(4)
+
a=1
1 (a) (a)λ (a) 2Fµ Fνλ − Fλσ F (a)λσ gµν . 2
(a)
Variation with respect to the gauge potentials Aµ and Aµ yield the respective YM and Maxwell equations
Abstract We consider Maxwell and Yang-Mills (YM) fields together, interacting through gravity both in Einstein and Gauss-Bonnet (GB) theories. For this purpose we choose two different sets of Maxwell and metric ansatzes. The combined effect in the first ansatz modifies the features for dimensions N≥5, with Maxwell term dominating over YM, or vice versa, depending on the asymptotic conditions. For N=3 and N=4, where the GB term is absent, we recover the well-known Ba˜ nados-Teitelboim-Zanelli (BTZ) and Reissner-Nordstrom metrics, respectively. The second ansatz corresponds to the case of constant radius function for SN −2 part in the metric. This leads to the Bertotti-Robinson type solutions in the underlying theory.
2
Action, Field Equations and our Ansatzes
The action which describes Einstein-Maxwell-Yang-Mills gravity with a cosmological constant in N dimensions reads IG = 1 2 √ dxN −g ×
1
INTRODUCTION
We introduce Maxwell field alongside with Yang-Mills (YM) field in general relativity and present spherically symmetric black hole solutions in any higher dimensions. These two gauge fields, one Abelian the other non-Abelian, are coupled through gravity and to our knowledge they have not been considered together in a common geometry. Being linear, a multitude of Maxwellian fields can be superimposed at equal case, but for the YM field there is no such freedom. To obtain exact solutions the Maxwell field is chosen pure electric while the YM field is pure magnetic. Historically, starting with the Reissner-Nordstrom metric, its higher dimensional extensions known as the Einstein-Maxwell (EM) black holes are well-known by now. Einstein-Yang-Mills (EYM) black holes in higher dimensions have also attracted interest in more recent works [1,2,3]. In this paper we combine these two foregoing black holes (i.e. EM and EYM)
are the SO(N − 1)gauge group YM
2
potentials while Aµ represents the usual Maxwell potential. We note that the internal indices {a, b, c, ...} do not differ whether in covariant or contravariant form. Variation of the action with respect to the space-time metric gµν yields the field equations (N − 1) (N − 2) Λgµν = Tµν (3) 6 where the stress-energy tensor is the superposition of the Maxwell and YM parts, namely Gµν + Tµν = 1 λ 2Fµ Fνλ − Fλσ F λσ gµν 2
1
under the common title of EMYM black holes. Further, we add to it the GaussBonnet (GB) term overall which we phrase as the EMYMGB black holes. Such black holes will be characterized by mass (m), electric charge (q), YM charge (Q), GB parameter α and cosmological constant (Λ) . It is remarkable that exact solutions to such a highly non-linear theory can be found for two different Maxwell and metric ansatzes. Before adding the GB term we find the EMYM solutions in which the relative contributions from the Maxwell /YM charges can be compared. It is shown that for r → 0 Maxwell term dominates over the YM which is reminiscent of asymptotic freedom from the YM charge. On the other hand for r → ∞ the YM term becomes dominant. Such behaviors, we speculate, have relevance in connection with the mini (super-massive) black holes as well as asymptotically (anti)-de Sitter space times. The solutions for N>5 turn out to be distinct from the lower (i.e., N=3,4) dimensional cases. We show that in the latter case the role of a YM charge is similar to the electric charge as far as the space time metric is concerned. Namely, one is still the BTZ, while the other is Reissner-Nordstrom. It is found that for N=3 and N=5 the metric function contains a logaritmic term while in the other cases not. Our second ansatz leads beside the black hole solution to a Bertoti-Robinson (BR) type metric within EMYMGB theory. The paper is organized as follows: In Sec. (2) we introduce our action metric Maxwell YM ansatz and field equations. Sec. (3) follows with the solution of the field equations for different dimensionalities. In Sec. (4) we add the GaussBonnet (GB) form and present the most general solution. Sec. (5) deals with a new set of ansatz for the Maxwell field and metric function. The paper ends with concluding remarks in Sec. (6).
a)µν F;( + µ
1 (a) A(b) F (c)µν C σ (b)(c) µ F;µν µ 1 (a) A(b) ∗ F (c)µν C σ (b)(c) µ ∗F;µν µ