「 Black Di-ring 解の諸性質」

’09 Dec. 26 / 於京都（高次元BH） 「Black Di-ring解の諸性質」三島隆　　(日大理工) （井口英雄氏(日大理工)との共同に基づく）

I.Introduction （e.g.） asymptotically flat cases black ring black saturn black di-ring …. Black lense orthogonal di-ring Solitonic generations of stationary 5-dim. spacetime solutions with BHs have been used to clarify some variety of the topology and shape of five dimensional Black Holes never seen in four dimensions. 2

Further progress : two ways Hunting of new proper higher dimensional BH solutions Some detailed analysis of previously obtained solutions （not so exciting but important ）（more exciting ！） Here we consider black dirings : 5 dim. concentrically superimposed double S^1-rotating BRs Two different presentations of diring ( diring I ) ( diring II ) the Backlund-like 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) 3

(1) We clarify the correspondence between the diring I and diring II from the viewpoint the rod structure and establish the equivalence with the aid of numerical calculations and the mathematical facts similar to four dimensional uniqueness theorem. ( Hollands & Yazadjiev ) ＜Plan of this talk＞？ diring II (E&K) diring I (I&M) … (2) We show some physical properties of the diring systems, especially thermo-dynamical diring systems (i.e. iso-thermal and iso-rotational systems) 4 4

II. Rod structure viewpoint for the solitonic generations ＜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 ( )　＞ Metric coefficients are the functions of and . 5

＜　Basic Equations and Generation methods　＞ （Ernst system : ）（BZ system）（diring II）（diring I） Inverse Scattering Method（ISM） ( Belinsky & Zakharov + Pomeranski ) Backlund Transformation （ Neugebauer，… ） ( New solution ) ( Seed ) Adding solitons 6

∞ ∞ ∞ ＜Viewpoint of rod structure : Emparan & Reall , Harmark＞ We see solitonic solution-generations from the viewpoint of ‘Rod diagram’ useful representation of the boundary structure of ‘ Orbit space ’ rod diagram : （ e.g.5-dim.Black Ring spacetime） Φaxis 1 2 Ψaxis 3 1 2 3 horizon 2 3 1 ( direction vectors ) 7

＜ Original solitonic solution generations viewed from rod diagram＞ ( resultant rod diagram ) Adding soliton Transformation of boundary structures of at least for the seeds of vacuum static spacetimes Transformation of rod diagram （ e.g. generation of di-ring I by Backlund transformation） ( 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. 8

< 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 using the arbitrariness of BZ-parameters • The processes (i) and (iii) assure Weyl form. • Two parameters remain after adjusting. (the positions where solitons are added ) ( Seed : static spacetime with solitons ) Removing trivial solitons ( New solution : non-trivial spacetime with solitons ) Adding non-trivial solitons 9

＜Generation of di-ring I by using PISM ＞ The di-ring I can be generated by using the PISM. Based on the fact that the original solitonic method adopted by us is equivalent to the two-block 2-soliton ISM （Tomizawa, Morisawa & Yasui, Tomizawa & Nozawa） is equivalent to ours. ( See Tomizawa, Iguchi & Mishima ’06 ) ( rod structure of the seed ) Digging ‘holes’ horizon a1 a2 a4 a3 a5 a6 a7 10 ( corresponding to a soliton )

( intermediate state (static) ) （i）Removing : ＋（ii）Scaling : horizon a1 a2 a4 a3 a5 a6 a7 Removing two anti-solitons with BZ-vectors (1,0) and (0,1) at a1 and a4 • The seed of the original construction appears as an intermediate state. 11

a3 a4 a5 a6 a7 a1 a2 ( Resultant rod diagram ) （iii）Recovering : （iv）Scaling back : horizon Recovering two anti-solitons with BZ-vectors (1, bI) and (cI, 1) at a1 and a4 elimination of flaws at a1 and a4 by arbitrariness of BZ -parameters （v）Adjusting : Regular new solution • Only the soliton’s positions remain to be free parameters in the metrics. 12

a1 a2 a3 a4 a5 a6 a7 ＜difference of procedures to generate diring I and diring II ＞ ( Generation of diring II :E&K ) （i）Removing : （ii）Scaling : （iii）Recovering (iv) Rescaling (v) Adjusting ( BZ-vectors ( 1, bII ) and ( 1, cII ) ) • No coincidence even when the parameters a1, a2, a3, a4, a5, a6and a7 are the same! • Essential difference is the positions of the holes that are dug in the seeds. The hole positions are connected in complicated way! even if two di-rings are physically equivalent. 13

III.Relation between Di-ring I and Di-ring II We try to fix the equivalence. 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 . (Their 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 S^1-rotating two-BH system, two Komar angular momentums corresponding to - • rotation are zero so that other two Komar angular momentums corresponding to - rotation • are essential to determine the solution. This means that two independent physical quantities can • be used to determine the solution at least locally. • Their proof seems not to depend on the existence of conical singularities on the axes • so that their statement remains effective to determine the diring solution with conical deficits. ( The Singularities in the inner region of the orbit space are dangerous. ) 14

＜Moduli-parameters and physical quantities of diring I and diring II ＞ ( di-ring I ) 1. Moduli-parameters （a） rod lengths for final states a1 a2 a3 a4 a5 a6 a7 s ‘hole-positions’ t （b） positions of solitons(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. 15

2. Physical quantities ( diring I ) ( diring II ) • Lengths of other intervals are represented with moduli-parameters. 16

＜Equivalence of diring I and diring II ＞ The corresponding rod lengths and other two physical quantities must be equal each other due to the fact by Hollands & Yazadjiev. ( 1 ) ( 2 ) The relation can be given explicitly. Using the above relation we can investigate the relation of the solution-sets of diring I and diring II, which allow conical deficits. 17

Correspondence between ( s, t ) and ( p, q ) From the relation , the schematic picture describing the correspondence is given as follows : ( space of ( s, t )for diring I ) ( space of ( p, q )for diring II ) • The correspondence is one-to-one and onto. 18

For further investigations, we can use the simpler expression of physical quantities of diring I and diring II ! Next step: we must try to clarify the physical properties of the diring systems. 19

IV.some physical properties of the di-ring systems In the rest we show some examples to show proper characteristics of the system. ＜ distribution of regular dirings in ( j2 , ah )-space ＞ ah : total horizon area (i) j: total angular mom Gray point : one diring system (ii) ( i ) : MP-BH ( ii ) : Single BR ( mass is fixed to be one. ) • Diring system can simulate single BR, but cannot simulate MP-BH. 20

＜ thermodynamic diring system ＞ Thermo-dynamical multi-BH system is defined by the iso-thermality and iso-rotationality of it’s horizons. The first problem is whether the thermo-dynamical black di-ring systems exist or not. Suggestion by Emparan et.al. OK ？ (BS) (BD) 21

(1) Existence of thermo-dynamical systems of dirings and comparison with other systems Black : MP-BH Green : Single BR (BR) Blue : Black Saturn (BS) ah Red : Black Diring (BD) j Thermo-dynamical di-ring systems exist ! • As a thermo-dynamical system, the diring system is the most sub-dominate . 22

(2) Difference between diring systems and other systems ( Behavior of total area against large j ) fat BS and fat BD thin BS ah thin BD j • As j increases, BS immediately approaches tBR, while BD’s approaching • is very slow. 23

(3) Behavior of horizon-area’s ratio between the inner BR(BH) and the outer BR Rh = (area of inner BR/BH )/(area of outer BR) Rh Red : Black Diring (BD) Blue : Black Saturn (BS) j The ratio of BS rapidly decreases. , The influence of the inner BH quickly vanishes so that the BS behaves like single BR. The ratio of BD takes a constant value one ! 24

V. Summary • Di-ring I solutions are regenerated by PISM. • The solutions set of diring I coincides with the set of diring II. • The thermo-dynamical systems of di-ring exsist. • As a thermo-dynamical system, the diring system is the most sub- • dominate among these four systems. • For the thermo-dynamical diring system, equi-partition of entropy (!) is realized. More systematic analysis of the diring system and the generalization to multi-BH systems and higher dimensional cases will be an interesting problems. 25

「 Black Di-ring 解の諸性質 」