1. A no-hair theorem for static black objects in higher dimensions
9th August(String/Cosmology)
Summer Institute
A no-hair theorem for static black objects in higher dimensions
Tetsuya Shiromizu
Department of Physics, Kyoto University
with Roberto Emparan, Seiju Ohashi
(arXiv: )

2. A no-hair theorem
Static black objects in n-dimensional asymptotically flat spacetime do not have non-trivial electric p-form field strengths when (n+1)/2≦p≦n-1.

3. Outline 1. Introduction 2. Static vacuum BH uniqueness in 4D
3. A no-hair theorem
4. Remaining issues

4. 1. Introduction

5. Black objects in higher dimensions
Comprehensive, but complicated…
Myers-Perry (higher dim. Kerr)
Black ring
Black saturn
Black di-ring
・Conventional uniqueness does not hold for stationary black holes.
・M, J , Q cannot fix the stationary black hole spacetimes.

6. Recent progress in stationary cases
・stationarity implies axisymmetry ・rod(~local) structure can determine the black object spacetime uniquely in five dimensions. ・Multipole moment is good parameter which uniquely specifty black object spacetime in five dimensions.
(Hollands, Ishibashi, Wald, 2007)
(・・・, Armas and Harmark 2010)
(Tanabe, Ohashi, Shiromizu, to appear, 2010)

7. Static cases
Gibbons, Ida, Shiromizu, 2002
Static black hole spacetimes are uniquely specified by M and Q.
Electrically coupled to 1-form(or its dual)
Question: Electrically coupled to 2-form?
・Then the static solutions will be black ring type if it exists.
This is because string is coupled to 2-form naturally.
・In five dimensions, stationary dipole ring solution with electric 3-form field strength
has been found (R.Emparan, 2006).
・Static solution has not been discovered.

8. Theorem
Emparan, Ohashi, Shiromizu, 2010
Static black objects in n-dimensional asymptotically flat spacetime do not have non-trivial electric p-form field strengths when (n+1)/2≦p≦n-1.

9. 2. Static BH uniqueness in 4D

10. Static BH uniqueness in 4D
Two ways ・Israel 1967 ・Bunting & Masood-ul Alam 1987

11. 2.1 Israel

12. Schwarzschild spacetime

13. Metric of static spacetime

14. Vacuum Einstein equation

15. Three key equations

16. To show the uniqueness spherical symmetric
it is easy to show that spherical symmetric spacetimes must be Schwarzschild spacetime.

17. Volume integral

18. Boundary conditions
Event horizon H(V=0)
Infinity (r=∞,V=1)

19. Finish using 4-dim speciality
spherical symmetric

20. Think of Higher dimensions little

21. 2. 2 Bunting & Masood-ul-Alam

22. Lindblom’s theorem (1981) Lindblom’s theorem (1981)
well-known fact in Math.
Lindblom’s theorem (1981)
In vacuum(or perfect fluid or Maxwell)

23. Conformally flatness
Positive mass theorem and a clever conformal transformation can show the conformally flatness　of t=const. surface.
Then Lindblom’s theorem completes the proof of BH uniqueness.

24. Emparan, Ohashi, Shiromizu 2010
3. A no-hair theorem
Emparan, Ohashi, Shiromizu 2010

25. Model

26. No-hair theorem
Static black objects in n-dimensional asymptotically flat spacetime do not have non-trivial electric p-form field strengths when (n+1)/2≦p≦n-1.

27. Assumption Staticity The metric form field Asymptotic conditions
asymptotically flat

28. Focus on static slice Σ

29. Some equations

30. Asymptotic condition at infinity

31. Regularity condition at horizon
metric

32. Two steps Then we can show H_p=0 etc.
same with vacuum case(-> Gibbons,Ida,Shiromizu)

33. Conformal transformation

34. after conformal transformation Asymptotic behaviors

35. Sawing on horizon
Sawing along V=0

36. Positive energy theorem tells us

37. Conformally flatness
Positive energy theorem

38. Harmonic function in flat space
We can find a harmonic function in
v is a harmonic function in flat space

39. Extrinsic curvature of V=0(v=2) in
constant
V=0 surface is spherical in Euclid space (Kobayashi and Nomizu, Theorem 5.1)

40. Almost finish
v=const. surfaces

41. Back to school for undergraduate
Let assume v_1 and v_2 to be solutions satisfying the same boundary condition
We know one solution: higher dimensional Schwarzshild solution.
Thus, Schwarzshild solution is unique.

42. No-hair theorem
Static black objects in n-dimensional asymptotically flat spacetime do not have non-trivial electric p-form field strengths when (n+1)/2≦p≦n-1.

43. No-dipole-hair Trivially
No-“monopole”-hair .
But,
is still possible.
However, our no-hair theorem excludes this case.
(No -“dipole”-hair theorem)

44. Comment
・dual magnetic (n-p)-form field strength hair does not exist. ・easy to extend to cases with several different rank form fields

45. 4. Remaining issues

46. p≦(n+1)/2? For example, in n=6, p=3 hair is possible or not?
If possible, find the exact solutions.
If not, we need a new way to prove no-hair.

47. Extend to supergravity?
Easy…? It depends on if energy condition is satisfied or not.

48. Stationary solution(Emparan,2004)

49. 0≦V≦1
Maximum principle implies that the function V does not have the maximum. V could have the maximum on the boundary. Then we can see that 0≦V≦1.

50. Black holes in higher dimensions
braneworld
Schwarzshild radius in higher dimensions

51. Schwarzshild radius
BH production on the earth is possible!?

52. The production rate Rate ～
The production rate～（Luminosity　L）×（cross section σ）
Cross section
Eardley and Giddings, Yoshino and Nambu, 2003
BHs are produced through classical gravitational collapse
（BH’s area）
Center of mass energy
Large Hadron Collider(2007?)
Rate　～