「 Properties of the Black Di-

rings 」

Takashi Mishima (CST Nihon Univ.)

Hideo Iguchi ( 　　 〃　 )

MG12-Paris - July 16 ’09

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Black Saturn black di-ring

black bi-ring….Black lense

black rings

（ e.g. ） asymptotically flat cases

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.

I. 　 Introduction

Finding new proper higher dimensional BH solutions

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Two ways

Some detailed analysis of previously obtained solutions（ more

exciting ！）

further progress

（ not so exciting but important ）

we consider black di-rings :5 dim. concentrically superimposed

double S^1-rotating BRs

Inverse Scattering Method （ ISM ） ( Belinsky-Zakharov technique )

Here

I&M: hep-th/0701043

Phys. Rev. D75, 064018 (2007) Evslin & Krishnan: hep-th/0706.1231

CQG26:125018(2009)

the simillar method to the Backrund transformation.（ Kramer-Neugebauer’s Method ，… ）

( di-ring I ) ( di-ring II )

？？

i. Solution-generation of di-ring I can be considered from the Pomeransky- type ISM.

ii. Differences between di-ring I and di-ring II are shown from the viewpoint of Pomeransky-type ISM. 　　　　　

iii. 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 ＞

( Both the representations of di-rings are too complicated ! )

diring I

(I&M)

Hard task !

diring II

(E&K)

…

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 ）

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II. Solitonic Methods and Rod structures

＜ The spacetime considered here ＞

　 Assumptions

＜　 metric （ Weyl anzats : ）　＞

（ Ernst system : ）

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＜　 Basic Equations and Generation methods 　＞

（ BZ system ）

（ diring I ）

Backrund Transformation （ Neugebauer ，… ）

Inverse Scattering Method （ ISM ） ( Belinsky & Zakharov + Pomeranski )

（ diring II ）

Adding solitons( Seed ) ( New solution )

＜ Viewpoint of rod structure (interval structure) ＞

We see solitonic solution-generations from the viewpoint of ‘Rod diagram’

（ Emparan & Reall , Harmark, Hollands & Yazadjiev … ）

rod diagram :

Convenient representation of the boundary structure of ‘Factor space/Orbit space’

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（ e.g. 5-dim. Black Ring spacetime ）

Φaxis

Ψaxis∞∞

∞

1 2 3

1 2 3

horizon

Φ -axis

Ψ - axis

(direciton)

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＜ Solitonic solution generations viewed from rod diagram ＞

A finite rod corresponding to ψ- rotational axis is lifted (transformed ) to horizon.

( rod structure of the seed )

－∞

　 Adding soliton 　　　　　　　 Transformation of boundary structure of 　

( resultant rod diagram )

（ e.g. generation of di-ring I ）

horizon

Φ -axis

Ψ - axis

Adding two solitons at these positions

Transformation of rod diagram

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< 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

　 The processes (i) and (iii) assure Weyl ansaz form.

　 Two parameters remain after adjusting.(the positions where solitons adding )

(BZ-parameters used )

1111

( rod structure of the seed )

－∞ a1 a2 a3 a4 a5 a6 a7

Based on the fact that the two-block 2-soliton ISM （ Tomizawa, Morisawa & Yasui, Tomizawa & Nozawa ） is equivalent to ours.

The di-ring I is regenerated using the PISM.

（ Tomizawa, Iguchi & Mishima ）

horizon

Φ -axis

Ψ - axis

Digging ‘holes’

＜ Generation of di-ring I by using PISM (1) ＞

Removing two anti-solitons

＜ Generation of di-ring I by using PISM (2) ＞ ( intermediate state (static) )

（ i ） Removing : （ ii ） Scaling : ＋

－∞ a1 a2 a3 a4 a5 a6 a7

horizon

Φ -axis

Ψ - axis

The seed of the original generation appears as an intermediate state.

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－∞ a1 a2 a3 a4 a5 a6 a7

＜ Generation of the di-ring I by using PISM (3) ＞ ( Resultant rod diagram )

（ iii ） Recovering :

horizon

Φ -axis

Ψ - axis

（ iv ） Scaling back :

（ iii ） Adjusting : Elimination of flaws at a1 and a4 by arbitrariness of BZ -parameters

Only the soliton’s positions remains in the metrics to be free parameters.

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＜ difference of generations between di-ring I and di-ring II ＞

( Seed of diring II :E&K )

（ i ） Removing : （ ii ） Scaling :

a1 a2 a3 a4 a5 a6 a7

（ 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, a6 and a7 are the same!(soliton positions are connected in complicated way! )

III 　 Relation 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 .

If all the corresponding rod lengths and the Komar angular momentums are the same, They are isometric.

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.

For Multi-BH systems

(statement)

(remarks)

＜ Behavior of physical quantities of di-ring I and di-ring II ＞

( di-ring I )

a1 a2 a3 a4 a5 a6 a7

t s

( di-ring II )

a2 a3 a4’ a5 a6 a7 a1’

q p

1. Moduli-parameters

（ a ） Rod lengths for final states

（ b ） Soliton’s positions / hole’s lengths

or p , q II s , t I

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 )

BZ parameters :

difference from E&K’s expression for ADM mass

(I) (II)

(ADM j vs. ADM m)

Conical singularity is allowed.

Coincidence except upper-right part !

It seems that the set of diring II includes diring I.

t =16 (boundary)

This region cannot be excluded even if the t goes to infinity.

＜ Coincidence of regular solutions between di-ring I and di-ring II ＞

(di-ring I ) (di-ring II )

(di-ring I ) (di-ring II )

Branch 1

Branch 2

IV. Summary

More systematic analysis of the solution sets is needed.

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.