1. Can we see super-Planckian domains? Takehara Workshop June 6-8, 2011 Ken-ichi Nakao (Osaka City Unviersity) In collaboration with Tomohiro Harada and Umpei Miyamoto (Rikkyo University) Hirotada Okawa and Masaru Shibata (YITP)

2. If GR correctly describes gravitational phenomena… Concentration of mass Collapse due to self-gravity Large spacetime curvature, large energy density, large stress All know theories of physics including GR are not available → New Physics (Superstring? Brane World? …..) Generation of spacetime singularities Near spacetime singularities § Introduction by Penrose(1965), Hawking & Penrose (1970)…..

3. Are spacetime singularities observable? Cosmic censorship hypothesis by Penrose (1969) Weak version: spacetime singularities generated by gravitational collapses are hiden behind event horizons Strong version ： spacetime is globally hyperbolic ？ × in 4-dim spaecetime. § Cosmic censorship hypothesis

4. “Domain of super-Planckian (SP) scale = Border of spacetime” 4 Domain A is called the border or SP domain, if E P = fundamental Planck scale  = a positive consitant of  Practical spacetime singularity = Domain in which GR is not available Harada and Nakao, PRD70, 041501 (2004) where Visible super-Planckian domain ≈ Naked singularity

5. Large extra-dimension scenario 5 Arkani-Hammed, Dimopoulos and Dvali 1998 Relation between D(>4)-dimensional Planck energy and 4-dimensional one: (R: length scale of extra-dimansions)

6. Collisions of high energy particles in large extra-dimension scenario 6 Giddings & Thomas (2001), Dimopoulos and Landsberg (2001) Gravitational radius of the center of mass energy E Production rate much larger than 4-dimansional theory Cross section of black hole production

7. Visible SP Domains generated by collisions of high energy particles 7 particle Domain into which two particles can enter at once If a black hole forms. If no black hole forms. Nakao, Harada and Miyamoto (2010) We call this the “collision domain”.

8. 8 Volume of the collision domain in (D-1)-dimensional space Area of the base of the cylinder Height of the cylinder Volume in extra-dimensions Average energy density in the collision domain (V N : Volume of an N-dim sphere)

9. 9 Condition of visible SP domain formation from 00-component of Einstein’s equations If max >1, then  is allowed. SP domain with no black hole can form ! 『 Gravitational radius=Compton length, if M = E P. 』  : max 2 where S N = Area of N dim sphere.

10. The value of max Generation rate of visible SP domains 10 Generation rate of visible SP domain >> Generation rate of BH Generation rate of black holes Practically, the weak version of cosmic censorship is not satisfied.

11. 13 High speed scattering of two black holes by higher dimensional numerical relativity Scattering or merger of high speed BH’s （ e.g., classical counterpart of high energy scattering of elementary particles ） Numerical relativity Method to study various phenomena with strong gravity by numeically integrating Einsteinn’s equations Evolution of binary composed of compact objects (NS, BH) Main target of GW physics New target of numerical relativity New techniques are necessary

12. 12 Scatter or merger of high speed BH’s BH b Rg Rg

13. 13 For large impact parameter b After scattering, two BH’s go apart to infinity Scatter or merger of high speed BH’s

14. 14 BH horizon For small impact parameter b Formation of one larger BH Scatter or merger of high speed BH’s

15. Initial data If the distance between two BH’s is large enough, each BH and its neighborhood is very similar to Schwarzschild-Tangherlini spacetime R g = E P (M/E P ) 1/2 ： gravitational radius Scattering of high speed BH’s in 5 dimensions (equal mass M ） 15 by Shibata, Okawa, Yamamoto (2008)

16. Boost transformation translation 16

17. Extrinsic curvature of t = const. hypersurface Other components vanish. 17

18. Initial data for two BH’s approaching to each other with velocity v  and  K ij are determined so that constraint equations are satidfied. 4-dim. spatial metric Extrinsic curvature However here, we have set  K ij. This is a good approximation for the case of large enough distance between the two BH’s. ここで、 In this initial data, there is almost no junk radiation. 18

19. on the horizon of a spherically symmetric BH with M= E P in 5-dim spacetime. SP domain:

20. is shown in the unit of (E P /M) 1/2. by Okawa, Shibata & KN (2011) BH’s 20 Example: b=3.38R g, v =0.7 Scattering of high speed BH’s in 5 dimensions (equal mass M ） Visible SP domain

21. 21 After scattering, two BH’s will go apart to infinity K2K2 [E P /M] 2

22. 22 K v [E P /M]

23. 23 If E P < M < 19E P, visible SP domain forms by the scattering of classical BH’s ! The largest value of K in our simulations is If finer simulations becomes possible, we may find larger K.

24. v Scatter Simulations break down. (Naked singularity?) Merger b [R g ] b C b B

25. Summary and discussion We can see super-Planckian physics if the spacetime dimension is larger than four (Naked singularity might form). We can see super-Planckian physics if the spacetime dimension is larger than four (Naked singularity might form). It is unclear why visible super-Planckian domain forms by the scattering of 5-dim BH’s. It is unclear why visible super-Planckian domain forms by the scattering of 5-dim BH’s. What is observed ? What is observed ? 25

26. 潮汐力による加速 L = 時空の曲率半径 m m X = 粒子間の距離 大雑把に 重力源 9 Geodesic deviation equations

27. m m 粒子間の距離がコンプトン長でも、プランク時間内で プランクエネルギーまで加速される 時空の曲率半径 L が プランク長さ l pl より短いとき 量子論的粒子生成で生まれた粒子は、 高エネルギー衝突によりブラックホールを形成？ 10