Spacetime rigidity and positive mass

1. 7th July 2014, 名古屋大学　幾何学・数理物理学セミナー おりひめ星(Vega) Spacetime rigidity and positive mass 夏の大三角 Deneb Tetsuya Shiromizu, Department of Mathematics, Nagoya University ひこ星(Altair)

2. References -A. Lichnerowicz, Theories Relativistes de la Gravitation et de l’Electromagnetisme (Masson, Paris, 1955). -A. Komar, Physical Review, 934(1959). -W. Boucher, G. W. Gibbons and G. T. Horowitz, Physical Review D30, 2447(1984). -T. Shiromizu, S. Ohashiand R. Suzuki, Physical Review D86, 064041(2012). -G. L. Bunting and A. K. M. Masood-ul-Alam, General Relativity and Gravitation 19, 147(1987). -G. W. Gibbons, D. Ida and T. Shiromizu, Physical Review Letter 89, 041101(2002), Physical Review D66, 044010(2002), Progress of Theoretical Physics Supplement 148, 284(2003). -R. Emparan, S. Ohashi and T. Shiromizu, Physical Review D82, 084032(2010).

3. Content 1. General relativity 2. Positive mass theorem 3. Background of two topics 4. Strictly stationary spacetimes 5. Static black hole spacetimes 6. Future issues

4. 1. General relativity

5. Special Relativity Minkowski spacetime metric photon photon particle

6. Classification of vectors timelike null null specelike particle’s orbit is timelike curve

7. Physical Laws • Physical quantities are written in terms of scalar, vector and tensor fields. • Physical law is described by differential equations derivative

8. General Relativity Gravity is described by curved spacetime Manifold metric ・Physical quantities are written in terms of scalar, vector and tensor fields. ・Physical laws are often described by differential equations Covariant derivative It gives a natural parallel translation of tensor fields

9. Covariant derivative parallel transport

10. Einstein equation Matter(atoms,…) satisfies Einstein equation

11. Great predictions Expanding universe & Black holes

12. Expanding universe Metric of universe scale factor ~ spatial size of universe (dust: pressure P=0) universe is indeed expanding!

13. Singularity theorem In general spacetimes Some causality condition Universe is expanding Dominant energy condition (energy density is positive) Timelike geodesics is incomplete in the past General relativity is not valid at singularity New physics is needed

14. Black holes Non-rotating black hole Schwarzschild radius The surface of black hole: event horizon（事象の地平面） Penrose-Carter diagram

15. Extrinsic curvature Induced metric Parallel transport surface

16. 2. Positive mass theorem

17. Positive mass theorem I Schoen & Yau(1979) - Asymptotically flat - (4≦n≦8, spinfor n≧ 4) (i) Mass M is non-negative. (ii) M=0 iff spacetime is Minkowski

18. Positive mass theorem II Schoen & Yau(1981), Witten(1981) -Asymptotically flat -Einstein equation, energy condition - 4≦n≦8(4≦n for spin) (i) Mass M is non-negative. (ii) M=0 iff spacetime is Minkowski

19. “Applications” Lichinerowicz theorem Lichnerowicz 1955 Strictly stationary and vacuum spacetime is flat. Bunting & Masood-ul-Alam 1984 Gibbons, Ida & Shiromizu 2002 Non-rotating black hole uniqueness Static, asymptotically flat black hole is unique to be Schwarzschild solution in four and higher dimensions. Positive pressure theorem Shiromizu 1994 Black hole no hair theorem in higher dimensions Emparan, Ohashi & Shiromizu 2010 (Riemannian) Penrose inequality Bray 2001 Generalized Lichnerowicz theorem Shiromizu, Ohashi & Suzuki 2012 Einstein, electromagnetic field,…

20. 2. Background of two topics

21. Two • Strictly static/stationary spacetimes • Black hole spacetimes • Penrose inequality Lichinerowicz 1955, ..., Boucher, Gibbons & Horowitz 1984, …, Shiromizu, Ohashi & Suzuki(SOS) 2012 Israel1967, Bunting & Masood-ul Alam 1987, …, Gibbons, Ida & Shiromizu 2002,… Bray 2001

22. 2-1 Strictly stationary spacetimes Focus on the spacetimes without black holes

23. Strictly stationary spacetimes • Configurations of static/stationary spacetimes vacuum spacetime is Minkowski Lichnerowicz 1955 [for 4 dimensions]

24. Lichnerowicz 1 -stationary timelike Killing -twist the definition 　⇒ Einstein equation ⇒ volume integral static

25. Lichnerowicz2 static and vacuum flat

26. Modern and smart way Komar mass formula for stationary and asymptotically spacetimes spacetime is flat Positive mass theorem

27. Remarks • Non-vacuum cases? • Asymptotically deSitter/anti-deSitter spacetimes? Shiromizu, Ohashi & Suzuki 2012 “Strictly stationary spacetimes cannot have non-trivial configurations of form fields/complex scalar fields and then the spacetime should be exactly Minkowski or anti-deSitter spacetime”

28. 2-2 Black hole spacetimes Stationary: ∃timelike Killing vector fields Static: ∃hypersurface orthogonal timelike Killing vector

29. Black holes in 4 dimensions • Static vacuum black hole is unique to be Schwarzschild spacetime • Stationary vacuum black hole is unique to be Kerr spacetime Israel 1967, Bunting & Masood-ul -Alam1987 Carter 1971, 73

30. Black holes in higher dimensions • Static vacuum black hole is higher dimensional Schwarzschild spacetime • Stationary black holes are not unique! Higher dim. Kerr(Myers-Perry), Black ring(5-dim., Emparan & Reall, 2002),.. Gibbons, Ida & Shiromizu 2002

31. Remarks • Stationary black holes is axisymmetric. • 5-dim stationary and 2-rotational symmetric black holes are determined by mass, angular momentum and rod structure(~quasi local b.c. at horizon, axes). Hollands, Ishibashi & Wald 2007 Hollands & Yazadjiev 2007

32. Extensions • Static electrovacuum black hole is higher dimensional Reissner-Nordstrom spacetime. • 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. Gibbons, Ida & Shiromizu 2002 Emparan, Ohashi & Shiromizu 2010

33. 3. Strictly stationary spacetimes Assume that black holes do not exist

34. 3-1. 4 dimensions

35. Set-up Lagrangian Einstein equation Stationarity We cannot show that mass vanishes in the above set-up because it is supposed that matter fields exist. Definitions Twist Electromagnetic fields

36. Some equations Maxwell equation From the definition of ω From Einstein equation

37. Divergence “identity”

38. 4-2. Higher dimensions

39. Set-up Lagrangian Einstein equation Stationarity Definitions twist Field decomposition

40. “Identity” and

41. Summary and remarks “Strictly stationary spacetimes cannot have non-trivial configurations of form fields/complex scalar fields and then the spacetime should be exactly Minkowski or anti-deSitter spacetime” Remarks: We supposed that spacetime manifold is contractible. If not contractible, the current argument does not work. But, if one assume chronology condition, singularity appears [Gannon, 1975]. Our argument is relied on the presence of the divergence free identity. Accidental? Other systematic way?

42. 4. Static black hole spacetimes Static: ∃hypersurface orthogonal timelike Killing vector

43. Static black hole spacetimes metric Event horizon

44. 4-1. Vacuum

45. Black hole spacetimes metric n-dimensional Einstein equation

46. Asymptotically Flatness Example: Schwarzschild solution in isotropic coordinate

47. Proof

48. Regularity at V=0 V=const. surfaces

49. Conformal transformation

50. Paste Glue at V=0