On Hidden Symmetries of d > 4 NHEK-N-AdS Geometry

Commun. Theor. Phys. 63 205) 3 35 Vol. 63 No. January 205 On Hidden ymmetries of d > 4 NHEK-N-Ad Geometry U Jie ) and YUE Rui-Hong ) Faculty of cience Ningbo University Ningbo 352 China Received eptember 8 204; revised manuscript received October 29 204) Abstract As well nown all higher dimensional Kerr-NUT-Ads metrics with arbitrary rotation and NUT parameters in an asymptotically Ad spacetime have a new hidden symmetry. In this paper we show that in the near horizon the isometry group is enhanced to include the dilatation and special conformal transformation and find the conformal transformation contains the cosmological constant. It is demonstrated that for near horizon extremal Kerr-NUT-Ads NHEK-N-Ad) only one ran-2 Killing tensor decomposes into a quadratic combination of the Killing vectors in terms of conformal group while the others are functionally independent. PAC numbers: 04.50.-h 04.70.Bw Key words: near horizon blac holes Killing tensors hidden symmetries uniform reduced form Introduction It is well-nown that the Killing tensors generate the hidden symmetries of a spacetime which provide an important geometric characters of the spacetime. Carter first found that the new integral of motion is quadratic in momenta and governed by a ran-2 Killing tensor on the Kerr blac hole. ] ince then the similar properties are realized on other various ind of blac holes and theories in higher dimension. 2 3] The Kerr-NUT-Ad blac hole is an important blac hole solution with a rotating and a Nut parameters of Einstein equation which admits two linearly independent Killing vectors and possesses a hidden symmetry generated by a ran-2 Killing tensor. 4] Chen Lu and Pope generalized such it into the Kerr-Nut- Ads blac hole solution with arbitrary rotation and Nut parameters. 5] Kubizn and Frolov demonstrated that all of such solutions admit a universal Killing-Yano tensor. 6] Recently Kerr/CFT correspondence had attracted a lot of attention of theoretic physician. In order to discuss the near horizon geometry of extremal rotation Kerr blac hole Bardeen and Horowitz showed that the near horizon geometry was not asymptotic flat and the time translation symmetry became into conformal group. 7] Rasmussen studied the 4-dimensional extremal Kerr-NUT- Ad solution and obtained the hidden symmetry to be governed by a modified Killing Yano potential. The Killing tensor is reducible. 8] Chernyavsy studied the near-horizon geometry of an extremal Kerr blac hole and showed that a Killing tensor decomposed into a linear combination of the quadratic Casimir operator formed out of the Killing vectors corresponding to the conformal group. 9] This motivates us to consider the near horizon geometry of Kerr-Nut-Ads blac hole with arbitrary rotation and Nut parameters in high dimension. In this paper we first review the metric of Kerr-Nut- Ads blac hole which satisfies high-dimensional Einstein equation. Then by introducing a set of new coordinates and considering for the geometry near horizon we find that only one ran-2 Killing tensor of the NHEK-N-Ad geometry in arbitrary dimension is reducible. The similar results for 4-dimensional Kerr-Nut-Ads and n-dimensional Kerr are obtained in Refs. 8 9]. 2 Near-Horizon Killing Tensors in d 2n Let us consider the general Kerr-NUT-Ad blac hole solution in d 2n dimensional spacetime. The metric is given by 5] ds 2 U U α dr2 dyα 2 W d t α U and all functions and quantities are defined by U r 2 yα U α r 2 yα) 2 β γ i d φ ) i α W g 2 r 2 )d t U α g 2 yα 2 y 2 β y 2 α) α n ) W g 2 yα γ i a 2 i yα i n ) γ i a 2 i r2 )d φ i ) a 2 i y2 α upported by the National Natural cience Foundation of China under Grant Nos. 275099 and 435006 E-mail: yueruihong@nbu.edu.cn c 205 Chinese Physical ociety and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

32 Communications in Theoretical Physics Vol. 63 g 2 r 2 ) a 2 r 2 ) 2Mr α g 2 yα) 2 a 2 yα 2L α y α α n ). 2) We have actually already included the new NUT parameters here. They appear just in the definitions of the functions α. By introducing new variables x α y α x n ir and letting M n im M α L α we will obtain a compact form 5] n { ds 2 U dx 2 W γ i U g 2 x 2 d t a 2 i d φ 2} i 3) x2 n U x 2 ν x 2 ) g 2 x 2 ) a 2 x 2 ) 2M x W n g 2 x 2 v) γ i with From Eq. 3) one can find the inverse of the metric to be Taing n a 2 i x 2 v). 2 s) n { U x U Ξ t g 2 x 2 } Ξ B a 2 x2 ) φ 4) t t a 2 x 2 B j n Ξ i φ i a i Ξ i φi the inverse metric Eq. 4) becomes into the slightly simpler form s n j { U x U t a 2 j a 2 ). 5) a 2 i a 2 ) Ξ i g 2 a 2 i g 2 x 2 ) a 2 x2 ) }. 6) φ For this Kerr-NUT-Ad blac hole solution there are n functionally independent Killing tensors constructed in Ref. 0] n { A A K x U t g 2 x 2 ) } a 2 7) x2 ) φ U A n ν <ν 2 <ν ν i x 2 ν x 2 ν 2 x 2 ν 0... n and K 0) is the inverse metric Eq. 6). Now let us discuss the condition about the near horizon limit of Eq. 7). Considering the extremal case in which the horizon radius r is defined through the double zero of ] r r 0 r r 0. To extract the near-horizon geometry one needs to redefine the coordinates by r λr r r φ i φ i φ i β 0tρ i λ t t β 0t λ β 0 r2 a 2 i ) V. 8) V r 2 r r Near the horizon the isometry group is enhanced. 9] We just mae a change on c c 4 r g 2 a 2 ) j r2 a 2 j ) r 2 a 2 )2... n.9) First we denote the inverse metric by I II s I II n n U x U t g 2 x 2 ) a 2 x2 ) φ. 0) Applying the definition of new coordinates Eq. 8) and taing the limit λ 0 then we obtain II n λ 2 n Ũ n n β 0 t 2r r 2 g 2 a 2 ) r 2 a 2 )2 ρ g 2 r 2 ) φ r 2 a 2. )

No. Communications in Theoretical Physics 33 We can also obtain the results I n V r2 I Ũ r II Ũ x Here n Ũ Ũ are the functions n U U with x n i r. The near horizon illing tensors can be written in the form K r 2 x 2 Ã ) Ũ A) n V r 2 V Ũ t Ũ g 2 a 2 ) l a2 l x2 ) r 2 a 2 )a2 x2 ) g 2 a 2 ) r2 x 2 ) r 2 a 2 )a2 x2 ) φ Ã ) Ũ x. 2) φ 2r r g 2 a 2 ) l r2 a 2 l ) r 2 V A ) n 3) r 2 a 2 )2 φ Ũ r here Ã) are the functions A ) with x n i r. Considering the following linear combination the nearhorizon Killing tensor can be reduced to the form 9] Q AB In terms of the two results r 2 ) Ã ) 0 r 2 ) A ) n r 2 ) K AB ). 4) l r 2 a 2 l ). 5) It is immediately proved that Q AB can be reduced as the Killing vectors Q AB 2 A ) B 3) A 3) B ) ) V A 2) B 2) n2 l4 c l 3l) A ) n2 l4 c l 3l) B ) 6) 4V ) t 2) V t t r ) r 3) V r 2 t2) t 2V rt r c i r φ i correspond to the time translation t t ɛ dilatation t t δ) r r δ) and conformal transformation t t V r 2 t 2 ) r r 2trV φ i φ i c i r. The others l) are related to the shifts of the azimuthal angular variables φ i. The constants c ) which appear in the right side of Eq. 6) are defined by Eq. 9). In fact one can change the factor r 2 ) in Eq. 4) to be an undetermined since the contribution of x to Eq. 4) must vanish and find no other solution excepting r 2 ). We thus conclude that only one of second ran Killing tensors near horizon decomposes into a quadratic combination of the illing vectors while the others are functionally independent. imilar results for d 4 have been obtained earlier in Refs. 8 2 4]. 3 Near-Horizon Killing Tensors in d 2n The general Kerr-NUT-Ad solutions in d 2n can be written in the form 5] ds 2 U U α dr2 dyα 2 W d t a 2 i γ i d α U φ i n a2 α W g 2 r 2 ) U α g 2 yα 2 d t r 2 y2 α U r 2 yα g 2 r 2 )W d t n γ i a 2 i a2 i r2 ) a 2 i d φ i y2 α n γ i a 2 i r 2 )d φ i 7) U α r 2 yα β y2 β yα α n ) W g 2 yα γ i a 2 i yα i n) g2 r 2 ) r 2 α g2 y 2 α) y 2 α n a 2 r 2 ) 2M n a 2 yα)2l 2 α α ).8) We will obtain a simpler metric ds 2 n { U dx 2 W U g 2 x 2 d t by defining functions n U x 2 ν x 2 ) g2 x 2 ) x 2 M n im M α L α x n ir x α y α. n a 2 i γ i a 2 i d φ } n i a2 n W d t x2 x2 n γ i d φ i 9) n a 2 x 2 ) 2M W n g 2 x 2 v) γ i n a 2 i x 2 v)

34 Communications in Theoretical Physics Vol. 63 We find the inverse of the metric Eq. 9) is given by Note that 2 s) n { U x x 4 U Ξ ) t g 2 x 2 } Ξ B a 2 x2 ) φ n a2 n x2 ν t t n Ξ t 20) a 2 B Ξ φ n a 2 x 2 B j n n n Ξ i φ i a i Ξ i φi a2 j a 2 ). a2 i a 2 ) Ξ i g 2 a 2 i the inverse metric Eq. 20) taes the slightly simpler form 2 s) n { n U x x 4 U t g 2 x 2 ) } a 2 x2 ) φ n a2 n x2 ν In this case there are n functionally independent second ran Killing tensors. 0] n { ) A A ) n K U x x 4 U t g 2 x 2 ) } a 2 A) n x2 ) φ a2 n x2 ν t n t n φ. 2) φ 22) A ) n ν <ν 2...<ν ν i x 2 ν x 2 ν 2 x 2 ν 0... n and K 0) is the metric Eq. 2). In order to describe the extremal case we consider that has a double zero at the horizon radius r r r 0 r r 0 then one redefines the coordinates r λr r r φ i φ i φ i β 0tρ i λ t t β 0t λ n β 0 r2 a 2 i ) V r 3 V 2 r r. 23) Near the horizon the isometry group is enhanced. 9] We just mae a change on c c 4 g 2 a 2 ) n j r2 a 2 j ) r r 2 a 2 )2... n. 24) First we thin the inverse metric can be written as the following three terms I II III 25) s I II n n III U x n U t n a2 n x2 ν t n g 2 x 2 ) a 2 x2 φ. φ Applying the definition of new coordinates Eq. 23) and taing the limit λ 0 then we obtain the results as follow ρ g 2 r 2 ) r 2 a 2 We can also obtain the results II I n x 4 Ũ II n λ 2 n x 4 nũn n β 0 t n g 2 a 2 ) r2 x 2 ) r 2 a 2 )a2 x2 ) r 4 III Ũ x à n) n Here n Ũ Ũ are the functions n U U with x n i r. b 2r r 2 g 2 a 2 ) r 2 a 2 )2. 26) φ In V r2 φ Ũ r b g2 a 2 ) n j a j φ r 2 a 2 ). 27)

No. Communications in Theoretical Physics 35 The near horizon illing tensors can be written in the form r 2 x 2 à ) n g 2 a 2 K ) n l a2 l x2 ) x 4 Ũ r 2 a 2 )a2 x2 ) φ A) n V r 2 r 2 V Ũ r n t here Ã) à ) are the functions A ) A ) with x n i r. Considering the following linear combination one of the near-horizon Killing tensors is reducible Q AB à ) Ũ x 2r g 2 a 2 ) n l r2 a 2 l ) r 2 V A ) n r 4 à ) n r 2 a 2 )2 φ Ũ r à n) r 2 ) K AB. 29) In terms of Eq. 5) and another result à ) r 2 ) x 2 l. 30) l It is immediately proved that Q AB can be reduced as the Killing vectors Q AB 2 A ) B 3) A 3) B ) ) V A 2) B 2) n3 l4 c l 3 A l) ) n3 l4 c l 3 B l) ) 4V with ) t r 2 n3 b l 3 l) A l4 ) n3 l4 b 2) V t t r ) r 3) V r 2 t2) t 2V rt n r 28) φ ) b l 3 l) B 3) c i r φ i c b are given in Eqs. 24) and 27) above and the others l) besides the ) A A 2) A 3) are related to the shifts of the azimuthal angular variables φ i. Based on the same argument given in last section we summarize that only one of second ran Killing tensors near horizon is reducible. Furthermore we can use a unified form to describe the reduced ran-2 Killing tensor of d > 4 NHEK-N-Ad geometry. References ] B. Carter Phys. Rev. D 74 968) 559; Commun. Math. Phys. 0 96880. 2] R.C. Myers and M.J. Perry Ann. Phys. 72 986) 304;.W. Hawing C.J. Hunter and M.M. Taylor-Robinson Phys. Rev. D 59 999) 064005. 3] G.W. Gibbons H. Lu D.N. Page and C.N. Pope Phys. Rev. Lett. 93 2004) 702; G.W. Gibbons H. Lu D.N. Page and C.N. Pope J. Geom. Phys. 53 2005) 49. 4] W. Chen H. Lu and C.N. Pope Nucl. Phys. B 762 2007) 38 hep-th/060002]. 5] W. Chen H. Lu and C.N. Pope Class. Quant. Grav. 23 2006) 5323 hep-th/060425]. 6] D. Kubizna and V.P. Frolov Class. Quant. Grav. 24 2007) F gr-qc/06044]. 7] J.M. Bardeen and G.T. Horowitz Phys. Rev. D 60 999) 04030 hep-th/9905099]. 8] J. Rasmussen J. Geom. Phys. 6 20) 922 ariv: 009.4388 gr-qc]. 9] D. Chernyavsy J. Geom. Phys. 83 204) 2 ariv: 309.299 hep-th]. 0] D. Kubizna ariv:0809.2452 gr-qc]. ] H. Lu J. Mei and C.N. Pope J. High Energy Phys. 0904 2009) 054 ariv:08.2225 hep-th]. 2] A. Galajinsy J. High Energy Phys. 0 200) 26 ariv:009.234 hep-th]]. 3] A. Galajinsy and K. Orehov Nucl. Phys. B 850 20) 339 ariv:03.047 hep-th]]. 4] A.M. Al Zahrani V.P. Frolov and A.A. hoom Int. J. Mod. Phys. D 20 20) 649 ariv:00.570 gr-qc]].