MG12-Paris - July 16 ’09 「 Properties of the Black Di-rings 」 Takashi Mishima (CST Nihon Univ.) Hideo Iguchi ( 〃)
I.Introduction （e.g.） asymptotically flat cases black rings Black Saturn black di-ring …. Black lense black bi-ring Generations of stationary 5-dim. spacetime solutions with BHs have succeeded to clarify interesiting variety of the topology and shape of five dimensional Black Holes never seen in four dimensions. 2
further progress Two ways Finding new proper higher dimensional BH solutions Some detailed analysis of previously obtained solutions （more exciting ！） （not so exciting but important ） 3
Here we consider black di-rings :5 dim. concentrically superimposed double S^1-rotating BRs the simillar method to the Backrund transformation. （ Kramer-Neugebauer’s Method，… ） Inverse Scattering Method（ISM） ( Belinsky-Zakharov technique ) I&M: hep-th/0701043 Phys. Rev. D75, 064018 (2007) Evslin & Krishnan: hep-th/0706.1231 CQG26:125018(2009) ( di-ring I ) ( di-ring II )
Solution-generation of di-ring I can be considered from the Pomeransky- type ISM. Differences between di-ring I and di-ring II are shown from the viewpoint of Pomeransky-type ISM. Some attempt to fix isometric equivalence of di-ring I and di-ring II with the aid of numerical calculations and the mathematical facts similar to four dimensional uniqueness theorem by Hollands & Yazadjiev. ＜Purpose of this talk＞ ？ diring II (E&K) diring I (I&M) … • Hard task ! ( Both the representations of di-rings are too complicated ! ) 5
Solitonic Methods and Rod structures ＜The spacetime considered here＞ • Assumptions c1 （5 dimensions） c2 （the solutions of vacuum Einstein equations） c3 （three commuting Killing vectors including time-translational invariance） c4 （Komar angular momentums for -rotation are zero） c5 （asymptotical flatness） ＜ metric （Weyl anzats : ） ＞ 6
＜ Basic Equations and Generation methods ＞ （Ernst system : ） （BZ system） （diring II） （diring I） Inverse Scattering Method（ISM） ( Belinsky & Zakharov + Pomeranski ) Backrund Transformation （ Neugebauer，… ） ( New solution ) ( Seed ) Adding solitons 7
∞ ∞ ∞ ＜Viewpoint of rod structure (interval structure) ＞ （Emparan & Reall , Harmark, Hollands & Yazadjiev … ） We see solitonic solution-generations from the viewpoint of ‘Rod diagram’ rod diagram : Convenient representation of the boundary structure of ‘Factor space/Orbit space’ （ e.g.5-dim.Black Ring spacetime） Φaxis Ψaxis 1 2 3 horizon 2 3 1 (direciton) 8
＜ Solitonic solution generations viewed from rod diagram＞ ( resultant rod diagram ) Adding soliton Transformation of boundary structure of Transformation of rod diagram （ e.g. generation of di-ring I） ( rod structure of the seed ) horizon Adding two solitons at these positions • A finite rod corresponding to ψ- rotational axis is lifted (transformed ) to horizon. 9
< Summary of Pomeransky’s Procedure based on ISM ( PISM ) > （i） Removing solitons with trivial BZ-parameters （ii） Scaling the metric obtained in the process (i) （iii） Recovering the same solitons as above with trivial BZ-parameters （iv） Scaling back of process (ii) （v） Adjusting parameters to remedy ‘flaws’ and add just physical effects (BZ-parameters used ) • The processes (i) and (iii) assure Weyl ansaz form. • Two parameters remain after adjusting. (the positions where solitons adding ) 10
＜Generation of di-ring I by using PISM (1) ＞ The di-ring I is regenerated using the PISM. Based on the fact that the two-block 2-soliton ISM （Tomizawa, Morisawa & Yasui, Tomizawa & Nozawa） is equivalent to ours. （Tomizawa, Iguchi & Mishima） ( rod structure of the seed ) Digging ‘holes’ horizon a1 a2 a4 a3 a5 a6 a7 11
Removing two anti-solitons ＜Generation of di-ring I by using PISM (2) ＞ ( intermediate state (static) ) （i）Removing : ＋ （ii）Scaling : horizon a1 a2 a4 a3 a5 a6 a7 • The seed of the original generation appears as an intermediate state.
a3 a4 a5 a6 a7 a1 a2 ＜Generation of the di-ring I by using PISM (3) ＞ ( Resultant rod diagram ) （iii）Recovering : （iv）Scaling back : horizon Elimination of flaws at a1 and a4 by arbitrariness of BZ -parameters （iii）Adjusting : • Only the soliton’s positions remains in the metrics to be free parameters. 13
a1 a2 a3 a4 a5 a6 a7 ＜difference of generations between di-ring I and di-ring II ＞ ( Seed of diring II :E&K ) （i）Removing : （ii）Scaling : （iii）Recovering (iv) Rescaling (v) Adjusting Differences: (i) positions of the holes (ii) axes mainly related to soliton • No coincidence when the parameters a1, a2, a3, a4, a5, a6and a7 are the same! (soliton positions are connected in complicated way! ) 14
IIIRelation between Di-ring I and Di-ring II Now we will try to fix the equivalence indirectly. Key mathematical facts The works by Hollands & Yazadjiev Here we use their discussions about the uniqueness of a higher dimensional BH to determine the equivalence of two given solutions which have different forms apparently . (statement) If all the corresponding rod lengths and the Komar angular momentums are the same, They are isometric. For Multi-BH systems (remarks) • For the single rotating two-BH system, to determine the solution two Komar angular momentums corresponding to -rotation are essential so that ADM mass and ADM angular momentum corresponding to -rotation may be used in place of the Komar angular momentums up to discrete degeneracy. • Existence of conical singularities seems to be harmless for this statement.
＜Behavior of physical quantities of di-ring I and di-ring II ＞ ( di-ring I ) 1. Moduli-parameters （a） Rod lengths for final states a1 a2 a3 a4 a5 a6 a7 s t （b） Soliton’s positions / hole’s lengths ( di-ring II ) p , q s , t or II I a1’ a4’ a2 a3 a5 a6 a7 p q • Other physical quantities can be represented with the above parameters.
2. Physical quantities ( diring I ) The quantities of di-ring I have been already given by us and complemented by Yazadijev. BZ parameters :
( di-ring II ) difference from E&K’s expression for ADM mass BZ parameters :
Coincidence except upper-right part ! t =16 (boundary) This region cannot be excluded even if the t goes to infinity. • It seems that the set of diring II includes diring I. (ADM j vs. ADM m) (I) (II) • Conical singularity is allowed.
＜Coincidence of regular solutions between di-ring I and di-ring II ＞ Branch 1 (di-ring I ) (di-ring II ) Branch 2 (di-ring I ) (di-ring II )
IV. Summary • Generation of di-ring I is regenerated by PISM. • It seems that the set of di-ring I solutions is included by the set of di-ring II when conical singularities are allowed. • The set of regular solution may be equivalent. • The set of regular solution set has two branches which correspond to conter-rotation case and anti-counter rotation. • More systematic analysis of the solution sets is needed.