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「 Properties of the Black Di-rings 」

Explore the properties and generation methods of black di-rings in 5-dimensional spacetime, using advanced mathematical techniques like Inverse Scattering Method and Rod Structures. Discover the differences between di-ring I and di-ring II through a detailed analysis and numerical calculations. Gain insights into the intriguing topology and shape of these unique black hole solutions. This talk aims to shed light on the complex nature of di-rings and deepen our understanding of higher-dimensional black holes.

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「 Properties of the Black Di-rings 」

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  1. MG12-Paris - July 16 ’09 「 Properties of the Black Di-rings 」 Takashi Mishima (CST Nihon Univ.) Hideo Iguchi ( 〃)

  2. 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

  3. 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

  4. 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 )

  5. 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

  6. 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

  7. ＜　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

  8. ∞ ∞ ∞ ＜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

  9. ＜ 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

  10. < 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

  11. ＜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

  12. 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.

  13. 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

  14. 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

  15. 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.

  16. ＜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.

  17. 2. Physical quantities ( diring I ) The quantities of di-ring I have been already given by us and complemented by Yazadijev. BZ parameters :

  18. ( di-ring II ) difference from E&K’s expression for ADM mass BZ parameters :

  19. 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.

  20. ＜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 )

  21. 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.

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