「 Solitonic generation of solutions including five-dimensional black rings and black holes 」

1. MG11 Berlin - July 25 ’06 「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)

2. I.Introduction • Explosive developments of the study of higher dimensional spacetimes have been done. ( inspired by string theory, recent brane-world model) qualitative discussion • Topology of Event Horizon (Cai & Galloway,…) • Uniqueness of black holes (Gibbons, Ida &Shiromizu, …) • axisymetricity of rotating black hols (Hollands,Ishibashi & Wald, …) A lot of researches have clarified the interesting features of higher dimensional gravity ! 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 … ＜ constructing new solutions ＞ • We have several techniques in 4-dim. case⇒Why not to use these methods ? • Especially solitonic solution-generating techniques are well-established !

3. 2N-solitons solutions : new spacetimes (Minkowski spacetimes … ) (N-Kerr black holes … ) Seed solutions : known spacetimes （ Two ways ） The method we adopted here Pomeransky, Tomizawa et al. • Backrund transformation （ Neugebauer’s Method，… ） • Inverse Scattering Method（ISM） ( Belinsky-Zakharov technique ) • The most general and promising method • To extract appropriate seeds is not so easy. • The solutions obtained are restrictive. • The corresponding seeds are easily guessed. （ schematic picture of solitonic method ） Adding ‘solitons’ To solve linear eqs instead of nonlinear eqs

4. U0, U1，ω，γare functions of ρand z II. Metric Form and Basic Equations ＜ Spacetimes ＞ The following conditions are imposed on the spacetimes considered here : 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） metric (restricted Levy- Papapetrou-Weyl form)

5. ＜ Basic Equations ＞ • Then introducing the following new functions & : & and further is defined. • 5-dim. vacuum Einstein equations are reduced to : （cf. Mazur & Bombelli ’87） ( ⇒ Laplace equation in 3-dim. Euclidean space) ( ⇒ Ernst equation)

6. (iv) (ii) (vii) (vi) (v) (iii) (i) ＜ procedure of solution-generation ＞ （１）　 U2 =T from the Laplace eq. (ii) （２）　 from the Ernst eq. (iii) （３）　S & Фare given from real and imaginary parts of . （４）　U0 is (S-T )/2 by definition. （５）　ωis given by contour integration of (i). （６）　γTis given by contour integration of (iv). （７）　γｓis given by contour integration of (v). （８） γ isγT + γｓ. （９）　U1 is -(S+T )/2 from (vii). ☆ Step (2) is cruicial because of the non-linearity of the Ernst equation.

7. ☆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. （non-Ricci flat） ☆ If you know any 4-dim. solution of Ernst equation, automatically you can get some new five-dimensional vacuum solution. … ? The problem is how to choose suitable seeds. 　（ This problem can be resolved only by trial and error generally. We shall give some convenient method in the following ）

8. III. Method ＜ The method adopted here ＞ • 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 formulas in the work by Castejon-Amenedo & Manko ’90 is suitable. In the following, both the canonical coordinates and the prolate spheroidal coordinates are used. i ) Seed functions S(0)= 2U0(0)+U2(0) , T(0)= U2(0)are extracted. (generalized Weyl solution as a seed )

9. ii ) Solving the following Ricatti equations for S(0), GM potential and are determined. iii ) New functionsA，B and Care defined. ☆ To describe them in the canonical coordinates, the following relations can be used.

10. iv ) Ernst potential and the line-elment & • γ’can be given as γ function of the following generalized Weyl solution ： Static spacetime with a BH under some external gravitational field ☆thecontour-integral to determine γ’can be done easily.

11. IV. Some Applications ＜ Viewpoint of rod-structure analysis ＞ (Emparan & Reall ’02_1, Harmark ’04) To describe the procedure of our method visually, the viewpoint of the rod-structure is introduced, following the Harmark’s work: G has zero eigenvalues on z-axis(ρ=0) . • 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’. • Therod is divided into the some intervals whose edges are the isolated points above .

12. a1 a2 aN - 1 aN • The above rod on the z-axis are allotted to actual axes of ‘rotation’ where G has a zero-eigenvalue . G22～ 0 G00～0 G22 ～0 G11 ～0 horizons （time translation） axes （Φ-rotaion） axes （Ψ-rotaion） a3 a1 a2

13. To complete the rod-structure analysis, the direction-vectors of the rods are given . The directions show the type of fixed points. • 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: (rotational cases: Myers & Perry, Emparan & Reall) • Ωis angular velocity of horizons. • The time-like direction-vectors are null generator for horizons.

14. d ＜ Examples and the rod structures＞ As preparation, we give GM potentials for the following simple seed function (seed-element) (a)Seed-element Solving the linear equations introduced before for the seed,

15. 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. • It should be noticed that Φ–axis rods among the region between [－σ,σ] are cut and lifted with some rotation in our method. (b)S^2-rotating black ring( M & I ’05 ) The corresponding seed is 5-dim. Minkowski spacetime (λ > 1) : horizon Φ-axis Ψ - axis －σ σ －λσ

16. Seed functions and GM potentials : ★ 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. （steps : (i)～(iv) ） Full metric ! Regularity conditions : • Conical singularity can not be eliminated. • Closed time-like curves are removed when the parameters are adjusted as follows,

17. (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. The corresponding seed is 5-dim. static BR spacetime : －σ σ horizon Φ-axis Ψ - axis －∞ η1σ η2σ δ1σ δ2σ λσ

18. Seed functions and GM potentials : Correction part from the static ring 1 S^1 BR part Full metric !

19. Regularity conditions : In this case, we may adjust the parameters to delete CTC’s and conical singularity from the system. • Closed time-like curves are removed when the parameters are adjusted as follows, • Vanishing conditions of conical singularities: satisfies the above conditions. (ex.)

20. 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 !

21. V. Summary • 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.