PoS(BHs, GR and Strings)034 http://pos.sissa.it/ † ∗ [email protected] Speaker. A footnote may follow. We find eleven dimensional non-BPSby black using M-brane U-duality. solutions Under from the Kerrrotating transformation solution intersecting four in dimensional M-brane M-theory Kerr solutions. metricmust change need To to a easy supersymmetric non-BPS black application brane solution, for whiche.g., AdS/CFT limits M2-M2-M2 special correspondence configuration branes of we in M-branes, five dimensionalfour black dimensional one. brane In and this M5-M5-M5 case brane we mayblack with easily hole pp-wave apply solutions. in the AdS/CFT correspondence for non-BPS ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c ° Black Holes in General Relativity andAugust String 24-30 Theory 2008 Veli LoÅ ˛ainj,Croatia Makoto Tanabe Department of Physics, Waseda University, Tokyo, Japan E-mail: Kerr black hole and rotating blackintersecting string M-branes by PoS(BHs, GR and Strings)034 - 2 D ⊥ 2 D ⊥ ]. The paper 2 Makoto Tanabe 1 F M5-brane soltuion, which is ⊥ M5 ⊥ . There are some technical difficulties ) 4 ˜ D d ( ∗ = 4 2 . dD 4 D6 branes, which is impossible to extension to the ˜ D ⊥ -brane solution in M-theory, or intersecting ] using the same method we use below, but these solutions 2 3 -brane solutions from general axisymmetric four-dimensional M 4 ] is the most successful theory to describe strong coupling limit D ⊥ 1 2 ⊥ M 4 ], black hole solutions with flat extra dimensions in M-theory are only ] using the D0 4 D 5 ⊥ 2 ⊥ NS5 brane solutions in type IIA superstring theory. 5 M ⊥ NS D4 ⊥ -brane solutions from the Kerr metric. 2 M ⊥ 2 , which is the self-dual field strength 4 M D Adding the new charge for the solution, we apply the boost for the ordinary metric along the Before charging up the vacuum solution, we introduce the Kerr metric, which is a stationary AdS/CFT correspondence [ Non BPS black ring solutions [ We will construct the intersecting M-brane solution consistent to the intersecting rule, and In our previous work [ ⊥ 2 ], because we must choose typical coordinates for the compact space for near horizon limit, 1 exist the specific configuration ofstate intersecting M-branes. of Kerr However the black earlier hole study [ for the micro extra dimension. Applying T-duality forfield the boost solution, we find D3-brane given by four-form supersymmetric black brane solution inwhich four have the dimension. regular event The horizon supersymmetric with black finite brane surface area. means which is the kk-wave direction, butThus this we solution show has the not any typical direction for compact spaces. for boundary conformal field theory. Thereand is boundary a quantum holographic field duality theory. Four between dimensionalinterests, bulk CFT quantum and is gravity studied its in all dual aspectsrecent theory because years, of is CFT Kerr/CFT based correspondence on are found five only dimensional for supergravity the black extremal hole limit solutions. case [ In solution of four-dimensional vacuum Einstein equation. The Kerr metric in spherical coordinate is 2. From Kerr metric to Intersecting D-branes related to intersecting 1. Introduction Kerr BH and Rotating BS by Interseccting M-branes shows the reason of the finiteture entropy is of zero. the AdS/CFT extremal correspondence Kerr are blackcase very hole, we useful even to the have show Hawking no that tempera- correspondence reason. information we However about will non-extremal give AdS/CFT the correspondence, M-brane configuration thus about non-extremal for Kerr non-extremal solution. Kerr/CFT vacuum solution with two Killing vector,string e.g., solutions, Kerr which metric. is We also rotatingM get four-dimensional the black general hole rotating with black extra one dimension, by the for the Hodge dual fordiagonal the component stationary after spacetime, the since boost.get the Such the stationary a explicit black component description hole makes of the problem remains dual for the field integration non- in order to avoiding the difficulties of integrationdimension. we Thus apply we the final try boost to after find the the lifting sequence up to to get the the eleven M5 related to the D4 brane solution in typeTheory. IIA These configuration string in string theory theory is with[ impossible to compactification analysis on for AdS/CFT one-dimensional correspondence torus of M- PoS(BHs, GR and Strings)034 . . , 2 i a = + + , , r 2 dz ) ) ≥ ω ] 1 5 2 i z Ω → = m 6 i ( r dz + ∑ T direction 5 i 6 = dt , and + ∑ z i ( → 3 1 f , . To add first S and the gauge − 3 Makoto Tanabe 2 2 base 0 h − , 3 1 h → ds ]. = + = 2 6 ) 2 i + h 2 = 6 . 1 2 µν z φ ) dz ( h ds R α ξ 2 3 is boost for T Ω . / 4 = 2 ) ∑ 3 ω i 1 + f ω φ − 1 → z and t α d ( − 2 ξ ) dt t β s h α . θ 6 ( 1 1 ξ z = f 2 B − , − α 2 ( + and near horizon limit 2 ) ϕ ξ a mr 2 i − B Σ f c 2 3 a sin 2 z − 1 − + + dz f ( ∆ e is S-dual. To add second and third → = − α 2 1 − T = 2 h ) = S mr 1 ∑ + i m 2 ds φ 3 2 2 z 1 ˜ → A ) ( = − 1 2 ) − = , T f h 2 θ 2 f φ z [ j r , d t ( are corresponding to z 2 → i are the orthogonal three-dimensional metric β / φ z T = + ) ) 1 1 ˜ j d C 6 6 − 2 ξ ∆ z , . For simplicity we denote θ , → ( ξ ∆ dx 5 0 , 2 ) i 3 + dr S ( T − 1 θ , direction, and = ( dx ) sin 2 i − 2 = → 2 base 4 → i j ∆ a z , f t γ Σ ) ) ˜ . The dilaton field is 3 ds cos ( 1 A 5 ( z 2 mr + z θ 2 α = , , / ( ( 2 c a 2 ) 1 t 2 ) T mr cos α T 2 is the specific angular momentum witch bounded for Σ β ξ − + s 2 , Ω 2 base α α 1 a 2 2 1 ma + → s c → r − α Σ 3 + ds 2 ) 1 h s S ) 1 2 − α = 3 2 and z dt = s Ω h ( ( is T-dual for 2 ) = ( 1 intersecting brane solution as → 3 2 j . The metric has the two Killing vector f s m Σ Ω = ) c α ) = i , 2 φ 2 t i 1 z j j B − , is determined by = c ( z D ( z z i i 1 ( t θ z z + T c = , ˜ β ⊥ r T B r C , , 2 + 2 2 t α → ds D β dt ) ( 1 − ⊥ f z -brane solution in type IIB supergravity. We denote 3 2 ( 2 3 / h 1 1 2 α D D h − B 1 ξ h → − = ) . = 4 ξ z in below. The base metric Ω 2 ( 3 For the first example of a black hole solution, we continue to apply the U-duality for the Adding the extra six flat dimension, the metric change to c T ds 2 2 base c 1 The mass of the black hole is where the metric functions are defined by where written in Kerr BH and Rotating BS by Interseccting M-branes In the Kerr/CFT correspondence, we take the extremal limit where the event horizon where the pair of indices fields are c charging up the four-dimensional Kerrto solutions. the rotating Next black wedimension, string, try and these to which solutions apply can must another possess be the example described Gregory-Laflamme related instability by [ the Kerr metric with another one extra ds written by the variables This metric is also a solution of ten dimensional vacuum Einstein equation charge (D3-brane), we apply a sequence in below; then we find a with boost parameter charges, we take next sequence in below, then we find a PoS(BHs, GR and Strings)034 . 1 − i g (2.2) (2.1) 2 i c + f 2 i , on torus, we s , 2 a ) Makoto Tanabe − ) ˆ 7 ] ] ] A z φ t φ φ φ , = d β i Ω Ω Ω φ α t t t ˆ h ˆ c ··· ˜ ˜ A ˜ D D D 1 , ) ) ) 1 − α + 2 3 2 3 2 2 z s . c c c ( , 1 with 2 1 2 2 2 1 dt or like that combinations. ) t − α θ c c c 1 to the Poisson equation for 6 ) ˆ direction and boost for the ¯ 2 1 3 h 2 A z 3 i − i , ( h h ( φ h 7 ˆ h h ˆ 1 1 1 ) + z + C T , 2 ξ cos ) − − − 7 ) ∧ α t ξ ξ ξ θ 4 ( a ) = → ˆ dz A . The three-form fields related to 2 base 2 ˆ 2 Σ i D ( ( − − ) − 4 t 2 t ¯ h ˆ s 4 ] 3 8 ds 1 1 1 C ξ β z sin 2 i 1 3 2 d ( + 3 / − = − 2 2 dz T 4 + ( ∧ + ( + ( and s ϒ a 5 c − ) j 1 ma φ φ φ 6 ( 3 1 z = Ξ ∑ ( i ˜ ˜ − i i ˜ 7 + → ¯ h D D D ξ z ˆ ) h 1 2 2 ˜ ) [ C [ 1 [ + C 1 ) ¯ ) 2 h − 3 2 1 θ 3 − 3 ] z − 1 ¯ φ h φ c c c -brane with pp-wave solutions in M-theory; 2 ( h ≡ ) Ξ ¯ ˆ h 2 1 3 A d 5 2 3 s ) s s T + ¯ h 2 base h + ω 4 α = cos M 2 i − 1 2 ( a 2 4 = = = t 2 ˆ h h ds → + ˆ Ξ C ) ) ) dz ⊥ a 2 A 1 f s 1 2 ) 3 3 h ( φ + ( φ ( φ Σ 1 , 5 5 dt 4 ω − direction, then we find non-BPS four charge solution. = r ˆ ˆ ˆ z t 2 ∑ ( i ( − θ D D D ( 2 ) 2 M f 4 β 7 1 2 ξ + ( r s z φ T , , , α − 2 ϒ 1 t t 4 ⊥ t − c d ¯ h + + ˜ ˜ 1 ˜ ( cos − Σ D D D 5 → ω 2 3 = − α 2 3 3 + 3 ω Ξ c ) s i s c c = M 4 2 i 2 2 + 3 1 2 2 2 g c c 2 z ma = s c − α c -brane solutions and we lift up the 2 1 ( dz 1 3 1 dt 1 ¯ 2 h c 4 1 = ds ( s c c c T − 2 3 2 M5 brane solution, we must apply the sequence in following as f 2 1 3 3 = ( D ∑ ¯ i s . Because of non conformally flat base space, supersymmetrie breaks ϒ ω h h h 1 i 3 ⊥ 1 2 → 1 1 1 = s ¯ − h ⊥ c − 1 − 2 − ) − 7 2 ξ ¯ h s z 1 4 s ξ M5 ξ j ξ 1 z [ 1 z − s i D ( ⊥ − − − c 3 [ z / 1 T − 3 C 1 ⊥ s / = = = 1 Ξ ) ) = ) 4 ≡ = 1 2 3 Ξ ( ( t ( ) t t , then we find the φ D t = is determined by ˆ ˆ 7 ˆ ˆ ˆ α + A are determined by D D D A ( 2 7 z i ˆ g C ϒ ds and ¯ ω direction, thus in the ordinary base space configuration we took maximally charge for three. We note that Kerr metric with flat dimensions has no Chern-Simons term, but there are exist in In order to get M5 Finally we apply the boost for the φ -brane are given by 5 where eleven dimension after the charging up sequence, e.g., then we find the 2.1 Kerr solutions in String/M-theory Kerr BH and Rotating BS by Interseccting M-branes find the four dimensional charged solution as for same direction The Chern-Simons terms gives the non-trivialand effect this effect of change the the topology Laplace for equationthe for BPS non-harmonic the black function harmonic ring function solutions, However if we took same specialwith value conformally for flat physical base parameters, space. we To find Compactify supersymmetric extra solution seven dimensions where the component of M5-brane fields and the metric components are given by where the conformal factor can be written by The function M PoS(BHs, GR and Strings)034 a α + 4 c c 1 and 3 ( (2.3) (2.4) = c 0 m 2 2 c c = 1 = is related M2 brane c α = α ⊥ ρ 1 M . = c c 2 J Makoto Tanabe a M2 and − ⊥ 0 2 M5 brane solution m ⊥ = ). Since , √ 4 4 φ c M5 + 3 )] ⊥ c = m 2 + 3 a c r φ = 1 , c + 2 r m and 2 base π 2 + arctan 8 ds r , φ . The surface area of the outer event + α = 2 a r r = h + a = 2 ¯ ¯ ¯ ¯ + r ) 1 − + − 2 A mr φ ], which metric is given in = 2 ¯ 4 and the angular momentum 2 π + 4 ω 7 ) r r f 2 ( Ω ) / , which is as the same as ordinary Kerr metric. 4 i 2 α − + s a t c c i 3 ˆ A s φφ a = 5 4 + γ 1 s 2 1 2 = m 4 i dt − 1 + − ( ∑ ϒ 4 4 f ( c c m 1 ( 2 [ 2 − α the metric becomes flat and the ADM mass + Σ 2 3 t h √ c ) β 2 √ − ∞ c = 1 = φ = c d Q ) is given by → 2 κ + 0 θ r d ds ( mr = ∫ π κ 8 = = . Sen’s solution is included in our solution with the parameter A 2 α c are defined by the substitution for are exist. + + ) f t 2 , and this case the action is changing as ˆ ( A 2 α a 1 s ]. The four-dimensional black hole solution is given by M5 A 4 − = and 4 = , the conserved charge = c ) ) + 1 2 t α ( = a / β h 2 i 3 A s The four-dimensional solution is represented by the charged dilaton black hole. The Maxwell In this paper we have presented the charging up Kerr solution using the U-duality method. The extremal solution ( In the asymptotic region Sen gave the rotating charged black hole solution [ c 1 = 4 i with traveling wave, and the five-dimensional black string solution is given by M2 where charge and the dilatons are coupled to each other, and the angular momentum are represented by solution. Both of thethe solution Chern-Simons exist term the in non-vanishing the Chern-Simonsvanishing. ordinary term The Kerr in Chern-Simons metric M-theory, terms with although change additionalfunction the given flat by metric extra Poisson function dimensions equation. given must by be Laplece equation to the horizon Black hole solutions are presented byM-brane intersecting solution, M-brane, which and consistent we have to constructvious the the work supersymmetrical intersecting [ black hole solution given by our pre- 3. Concluding remark In the Kerr metric case,the the metric regularity has condition no conical for singularity the at rotating the axis ordinary are event horizon the same as before, thus and we can show thementum thermodynamics and with the the temperature, physical but parameter the as dilaton the fields does charge not and contribute. angular mo- Kerr BH and Rotating BS by Interseccting M-branes where The area surface is vanishingsurface in in the M-theoretical extremal Kerr limit metric of doesn’t the vanishing ordinary ( Kerr metric, however the area ∑ to the charge ofM-brane. M-brane (or D-brane), this non-vanishing area surface gives the microstate of are given in the asymptotic metric form. The surface gravity change as below and only PoS(BHs, GR and Strings)034 for ∞ → R Makoto Tanabe -brane solutions given by 6 D ⊥ -brane is the same configuration 0 2 D M ⊥ 2 ]. The black hole has the regular extremal M ? ⊥ 2 M 6 ], and tihs solution include the limit for 3 ] , however we gave the another solution with regular event 9 ]. 10 ]. 5 [arXiv:hep-th/9905111]. By the way of this paper we only consider from the vacuum solutions, but adding the extra Since there are kk-wave mode in the ten-dimensional metric, the D-brane configuration of this The five-dimensional solution is fully understood as the charged dilaton Kerr black hole with [7] A. Sen, Phys.Rev.Lett. 69 (1992) 1006,[8] [arXiv:hep-th/9204046]. , G. W. Gibbons and K.[9] Maeda, Nucl. Phys. Y. Brihaye B and 298 T. (1998) Delsate, 741. [ arXiv:gr-qc/0703146]. [4] K. Maeda and M Tanabe, Nucl.[5] Phys. B738 (2006) G. 184, T. horowitz [arXiv:hep-th/0510082]. and M. M.[6] Roberts, [arXiv:0708.1346]. R. Gregory and R. Laflamme, Phys. Rev. D 37 (1988) 305. [1] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y.[2] Oz, Phys. M. Rept. Guica, 323, T. 183 Hartman, (2000) W. Song, A.[3] Strominger, [arXiv:0809.4266]. H. Elvang, R Emparan and P. Figueras, JHEP 0502 (2005) 031, [arXiv:hep-th/0412130]. dimension we can extend to theEinstein and Einstein Maxwell manifold equation with in the lower and constantrotating higher gauge dimension. black field, In ring which this case, satisfy formalism, we including the hole the can dynamics. one apply the more interesting case, especially cosmology and black References metric is suitable to applysolutions the in the AdS/CFT context correspondence. ofwith AdS/CFT We the correspondence, will specific and show physical we the parameters will microfor in also state the subsequent show CFT, of the paper, we these regular witch compare solutions we the are micro writing state of now. Kerr In black the hole limit by the non-BPS two rotatingGregory-Laflamme charged instability black [ ring solution. The black string solution must have the [10] B. Kleihaus, J. Kunz and E. Radu the charges. Thus wedilaton, are and only this possible limit gives to static take black the brane solution limit [ to vanishing the charges with the trivial horizon coupled to dilaton. The representation for Kerr BH and Rotating BS by Interseccting M-branes flat extra one dimension with travelingstring wave solution along with this regular extra event direction. horizon is Thewith given charged-rotating a in black positive a cosmological higher constant dimensional [ Einstein-Maxwell theory limit to the BPS solutionare in represented by four the dimension micro with state non-vanishing of M- are or surface, D-branes. and the area surface Horowitz et. al., [ as the non-BPS black ring solution given by [