1 「 Solitonic generation of solutions including five-dimensional black rings and black holes 」 T. M. (CST Nihon Univ.) Hideo Iguchi ( 〃 ) We show some solitonic solution-generating technique which is used by the authors and applied to generate some new 5- dimensional axisymmetric stationary solutions including S^1 rotating black rings and S^2 rotating black rings. (This talk is mainly based on the works appeared in Phys Rev D) MG11 Berlin - July 25 ’06
Transcript
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「 Solitonic generation of solutions including five-dimensional black rings
and black holes 」
T. M. (CST Nihon Univ.)
Hideo Iguchi ( 〃 )
We show some solitonic solution-generating technique which is used by the authors and applied to generate some new 5-dimensional axisymmetric stationary solutions including S^1 rotating black rings and S^2 rotating black rings. (This talk is mainly based on the works appeared in Phys Rev D)
MG11 Berlin - July 25 ’06
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I. IntroductionExplosive developments of the study of higher dimensional spacetimes have been done. ( inspired by string theory, recent brane-world model)
Especially solitonic solution-generating techniques are well-established !
We have several techniques in 4-dim. case ⇒ Why not to use these methods ?
• Topology of Event Horizon (Cai & Galloway,…)• Uniqueness of black holes (Gibbons, Ida &Shiromizu, …)• axisymetricity of rotating black hols (Hollands,Ishibashi & Wald, …)
qualitative discussion
solutions
K-K black holes, Bubbles (Harmark, Ishihara & Matsuo, … ) charged black holes and black rings, dipole rings, … (Kunz et al, Ida & Uchida, Yazadjiev, …) asymptotic rotational black black holes, static and rotational black rings (Myers & Perry, Emparan & Reall, …) supersymmetric black rings etc. (Elvang et al, … ) Black string (Wiseman, …)
general vacuum solutions are not so many like in four dimensional case …
A lot of researches have clarified the interesting features of higher dimensional gravity !
The method we adopted here Pomeransky, Tomizawa et al.
Inverse Scattering Method （ ISM ） ( Belinsky-Zakharov technique )
The solutions obtained are restrictive. The corresponding seeds are easily guessed.
The most general and promising method To extract appropriate seeds is not so easy.
2N-solitons solutions : new spacetimes (N-Kerr black holes … )(Minkowski spacetimes … )
Seed solutions : known spacetimes
Adding ‘solitons’ To solve linear eqs instead of nonlinear eqs
（ schematic picture of solitonic method ）
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c1 （ 5 dimensions ） c2 （ the solutions of vacuum Einstein equations ） c3 （ three commuting Killing vectors including time-translational invariance ） c4 （ single non-zero angular momentum component ） c5 （ asymptotical flatness ）
＜ Spacetimes ＞The following conditions are imposed on the spacetimes considered here :
II. Metric Form and Basic Equations
U0, U1 ， ω ， γ are functions of ρand z
metric (restricted Levy- Papapetrou-Weyl form)
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Then introducing the following new functions
&
and further
& :
is defined.
＜ Basic Equations ＞
5-dim. vacuum Einstein equations are reduced to : （ cf. Mazur & Bombelli ’87 ）
( Laplace equation in 3-dim. Euclidean space)⇒
( Ernst equation)⇒
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＜ procedure of solution-generation ＞
（１） U2 =T from the Laplace eq. (ii) （２） from the Ernst eq. (iii) （３） S & Ф are given from real and imaginary parts of
. （４） U0 is (S-T )/2 by definition. （５） ωis given by contour integration of (i). （６） γT is given by contour integration of (iv). （７） γ ｓ is given by contour integration of (v). （８） γ is γT + γ ｓ . （９） U1 is -(S+T )/2 from (vii).
(ii)(iv)
(vii)
(iii)(v)
(i)
(vi)
☆ Step (2) is cruicial because of the non-linearity of the Ernst equation.
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☆ the new line-element described with S and T :
[ ・・・ ] is determined by solving Ernst eq. (iii) essentially. [ ・・・ ] is not a trivial 4-dim. Vacuum solution. ⇔ γ has dependence on T through γT .
If you know any 4-dim. solution of Ernst equation, automatically you can get some new five-dimensional vacuum solution. … ?
☆
（ This problem can be resolved only by trial and error generally. We shall give some convenient method in the following ）
The problem is how to choose suitable seeds.
（ non-Ricci flat ）
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The formulas in the work by Castejon-Amenedo & Manko ’90 is suitable.
We must use non-trivial Weyl solutions as four-dimensional seeds generally. Asymptotic flatness must be held after adding solitons. Full metrics can be obtained easily.
（ The original work is Gutznaev & Manko ’87 ）
＜ The method adopted here ＞
i ) Seed functions S (0) = 2U0 (0) + U2 (0) , T (0) = U2 (0) are extracted.
In the following, both the canonical coordinates and the prolate spheroidal coordinates are used.
(generalized Weyl solution as a seed )
III. Method
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ii ) Solving the following Ricatti equations for S (0) , GM potential and are determined.
iii ) New functions A ， B and C are defined.
☆ To describe them in the canonical coordinates, the following relations can be used.
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iv ) Ernst potential and the line-elment
&
γ’ can be given as γ function of the following generalized Weyl solution ：
☆ the contour-integral to determine γ’ can be done easily. Static spacetime with a
To describe the procedure of our method visually, the viewpoint of the rod-structure is introduced, following the Harmark’s work:
Regular solutions must have just one eigenvalue except in isolated points on z-axis . The appropriate line source for Newton potential can be considered to exist on the z-axis in the hypothetical 3-dim. Euclidean space. We call this line source ‘rod’. The rod is divided into the some intervals whose edges are the isolated points above .
G has zero eigenvalues on z-axis (ρ=0) .
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The above rod on the z-axis are allotted to actual axes of ‘rotation’ where G has a zero-eigenvalue .
a1 a2 a3
axes（ Φ-rotaion ）
horizons（ time translation ）
G22 ～ 0 G00 ～ 0 G22 ～ 0 G11 ～ 0
axes（ Ψ-rotaion ）
a1
a2
aNaN -
1
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To complete the rod-structure analysis, the direction-vectors of the rods are given .
the direction-vector is unique as an element of RP3 space on each rod for the solution given.
(static cases : generalized Weyl solutions)Each rod has one of the following forms:
Ω is angular velocity of horizons. The time-like direction-vectors are null generator for horizons.
The directions show
the type of fixed points.
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＜ Examples and the rod structures ＞
As preparation, we give GM potentials for the following simple seed function
(a) Seed-element
(seed-element)
Solving the linear equations introduced before for the seed,
d
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(b) S^2-rotating black ring ( M & I ’05 )
－ λσ － σ σ
Ψ - axis
Φ -axis
horizon
cut, rotate and lift
Now we can automatically derive the GM potentials corresponding to any seed
which is constructed from element-seeds, because of the linearity of the equations.
The corresponding seed is 5-dim. Minkowski spacetime (λ > 1) :
It should be noticed that Φ –axis rods among the region between [ － σ, σ] are
cut and lifted with some rotation in our method.
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Seed functions and GM potentials :
Regularity conditions : Conical singularity can not be eliminated. Closed time-like curves are removed when the parameters are adjusted as follows,
In real fact using the following coordinate transformation, our solutions with adjusted parameters above can be transformed into the more convenient C-metric form of S^2-black ring derived by Figuears.
★
Full metric !
（ steps : (i) ～ (iv) ）
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(c) Superimposition of 1 S^1-rotating BR and 1 static BR
In principle, we can add a rotational ring to arbitrary static multi-black ring system.
In this place, as the simplest seed 1 static BR system is considered.
η2σ η1σ
－ σ σ
horizon
δ2σδ1σ λσ－∞
Ψ - axis
Φ -axis
The corresponding seed is 5-dim. static BR spacetime :
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Seed functions and GM potentials :
Full metric !
1 S^1 BR part Correction part from the static ring
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Regularity conditions :
Closed time-like curves are removed when the parameters are adjusted as follows,
Vanishing conditions of conical singularities:
(ex.) satisfies the above conditions.
In this case, we may adjust the parameters to delete CTC’s and conical singularity from the system.
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Direction vectors :
inside-ring :
outside-ring :
S^1-rotating BR by Emparan & Reall is realized in the case δ1 ＝ δ2 .
Whether the spacetimes are globally regular or not have not been fixed. Generally both rings have finite rotations and some ergo-regions !
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Using a solitonic method, we generated new solutions which correspond to five-dimensional axisymmetric, stationary and asymptotically flat spacetimes systematically.
This solutions-family includes two different types of single-rotational black ring, and also as more interesting solutions, the solutions corresponding to the superimposition of multi-rings can be generated.
