1. Horizons in de Sitter and Singularities in Anti-de Sitter spaces Akihiro Ishibashi (Kinki University) 2012 International workshop on String theory and cosmology 16 Jun 2012, Haeundae, Pusan joint work with: S. Hollands and S. Jager, Kengo Maeda

2. De Sitter and AdS De Sitter and Anti-de Sitter spaces play a central role in gravitational physics de Sitter space: for cosmology Inflationary universe Dark energy universe Anti-de Sitter space: for string theory Theoretical laboratory (Supergravity, AdS/CFT)

3. Main focus: global structure De Sitter space: Cosmic no-hair Cosmological horizon surrounds black hole horizon AdS space: can be viewed as a collapsing universe generically singular? I will attempt to give a brief overview of recent progress in understanding some basic properties of black hole horizons in dS and singularities in AdS spacetimes

4. DE SITTER SPACE AND HORIZONS

5. Asymptotically de Sitter black holes Black hole Cosmological horizon Just like a black hole horizon, cosmological horizon has thermodynamic feature with the entropy given by its cross-section area Gibbons – Hawking ’77

6. Black hole formation produces gravitational waves, radiated to de Sitter infinity how to define flux formula/Bondi energy at de Sitter infinity? Stationary black holes are characterized by its conserved charges (Classification of BHs in (anti-) de Sitter space is still under way) how to define conserved charges in asymptotically de Sitter space? BH is surrounded by Cosmological horizon (-- cannot be larger than Cosmo. Horizon) Is BH entropy always smaller than that of cosmological horizon?

7. Need to define Conserved charges when there is asymptotic symmetry Flux formula when there is no asymptotic symmetries (due to, e.g., gravitational waves) in any dimensions Entropy bound

8. The orbits of Static Killing vector become space-like outside the Cosmo. Horizon Static Killing vector field Question of how to define conserved quantities is non-trivial Conserved charges

9. Various definitions for conserved charges in asymptotically de Sitter space seminal work Abbott-Deser 82 defined at “SI” + vity of AD Shiromizu 94 Nakao -Shiromizu –Maeda 94 dS/CFT Weyl Shiromizu-Ida-Torii 01 Conformal Killing Spinor charge Kastor -Traschen 02

10. “Strongly”asymptotically de Sitter condition Asymptotic symmetry is that of exact de Sitter space

11. Strongly de Sitter boundary condition contracting expanding Future infinity Past infinity “Infinity” (viewed as a boundary) is Spatial 3 sphere

12. c.f. Scalar field energy wrt Killing symmetry Symplectic form ～ Klein-Gordon inner-product Killing vector Hamiltonian Covariant phase space method: Hamiltonian framework and at the same time, manifestly covariant

13. Covariant Hamiltonian Lagrangian density L: Symplectic current: Symplectic form: Defining equation for the Hamiltonian wrt symmetry ξ : c.f. Wald - Zoupas ’00

14. Integrability conditions If ω vanishes at the boundary, there exists a Hamiltonian. c.f. AdS case: Hollands-AI-Marolf ’06

15. “Taylor ” expand the boundary condition at infinity in terms of the conformal factor (for d>4) Metric at infinity “Electric” component of Weyl tensor

16. For asymptotic symmetry ξ Remarks: 1) If Constant mean curvature (CMC) time-slince can be taken, it coincides Weyl charge and Abbott-Deser charge 2) Cannot be positive definite, in particular, when matter fields exist… Nakao-Shiromizu-Maeda ‘94 Formula: Conserved :

17. Boundary conditions at infinity AdS infinity Impose symmetry Asympt. Flat spacetime Asympt. Symmetries (BMS group in 4-dim.) de Sitter infinity No symmetry Not strongly asympt. deSitter Flux formula

18. So that becomes smooth and well-defined Symplectic current is Gravitational flux formula where ： is (d-1)th-order coeff. of Ω expansion When no asymptotic symmetries… Symplectic potential is in general ill-defined Add “counter-term”

19. c.f. In 4-dimension Anninos-Ng-Strominger 10 Flux formula

20. Schwarzschild – de Sitter space Black Hole is surrounded by Cosmological (dS) horizon BH entropy has an upper-bound defined by cosmological horizon Gibbons – Hawking ’77 PRD15, 2738 Entropy bound

21. Previous work in 4-dimensions dS (cosmological) horizon entropy bound under time-symmetric slices Under maximal slices Total entropy bound under uniform expanding foliation D-bound: supposed to hold in the presence of matter Maeda-Koike-Narita-AI ‘98 Shiromizu-Nakao-Kodama-Maeda ‘93 Boucher-Gibbons-Horowitz ’84 Bousso ‘02

22. Event horizons for an observer Case (i) Case (ii) Case (iii) For any case, we can show that the horizon area never decreases under suitable energy conditions Case (iv)

23. Case (i) – Cosmic No Hair In general, there are no asymptotic symmetries at “infinity” (due to, e.g. the presence of gravitational radiations) The metric approaches locally the metric of exact De Sitter space --- Cosmic No-hair The area of cross-section of light cone approaches that of exact de Sitter horizon

24. Suppose w/ The area of any black hole in asymptotically de Sitter space is bounded by : Yamabe (topological) invariant Case (ii)-(iv) : Entropy bounds in dimensions If the metric of the MOTS (Marginally Outer Trapped Surface) approaches a metric of constant scalar curvature, there holds

25. MOTS: Marginally Outer Trapped Surface Black Hole: Cosmological horizon: Proof: (1)Convergence (2) By Rayhaudhuri equation + Property of MOTS (3) By Property of MOTS + Einstein equations + Dominant Energy Conditions (4) -- known for BH horizon Galloway & Schoen ‘06

26. Remarks BH apparent horizon is necessarily within BH Matter fields (w/ certain energy conditions) can be include -- need some “quantity” that monotonically increases from to. （ e.g. Hawking mass ） -- have not been able to get a formula like under a general setup As for a total entropy bound

27. Summary: Horizons in de Sitter space gave formulas for conserved charges/Flux formula for asymptotically de Sitter spaces using covariant Hamiltonian methods showed an entropy upper-bound for de Sitter black holes in terms of the topological invariant.

28. ANTI-DE SITTER SPACE AND SINGULARITIES

29. Global structure of AdS AdS is maximally symmetric, geodesically complete, but non-globally hyperbolic. AdS conformal infinity plays a role of a confining box: Waves bounce off at AdS-infinity e.g. cosmological chart: AdS conformal infinity Points are conjugate, corresponding to Big-bang/crunch All timelike geodesics emanating from re-converge to

30. Does perturbation make the conjugate points true curvature singularities? ? Perturbations Curvature singularity ? AdS has stable dynamics at least for linear perturbations AI & Wald 04

31. --- but nonlinearly unstable Turbulent instability (numerical results) Bizon – Rostworowski 11 (massless scalar field, spherical system) Generic small perturbations of AdS become large and eventually form black holes Dias – Horowitz – Santos 11 (vacuum, gravitational d.o.f. excited) The energy cascades from low frequency to high frequency

32. So far only a few pieces of supporting evidence and the answer is still speculative Is AdS generically singular ?

33. One of their motivations: Singularity theorem for closed universe “The negative cosmological constant acts like a confining box for fields inside, that is, AdS is like a closed universe for fields inside. So, one expects that generic solutions will be singular.” Dias-Horowitz-Santos 11

34. Singularity theorems “Singularity” = “incomplete causal geodesics” Singularity theorems show geodesic incompleteness under conditions of 1. Convergence (generic & energy conditions) 2. Global structure (causality or Cauchy surface) 3. Strong-gravity (trapped set) There are several different versions of the singularity theorems depending upon different combinations of Conditions 1. 2. 3.

35. Penrose’s theorem for gravitational collapse 1. null convergence 2. non-compact Cauchy surface 3. closed trapped surface This version does not apply to asymptotic AdS case Because no Cauchy surface in asymptotically AdS case

36. Hawking’s theorem for cosmology 1. convergence for every causal curve 2. compact spacelike hypersurface (w/o edge) S 3. unit normals to S are everywhere converging This version removes the assumption of Cauchy surface, but still does not apply because no such a compact hypersurface in AdS

37. Hawking & Penrose’s theorem 1. convergence for causal curves holds 2. chronology condition holds 3. there exists a trapped set : (i) compact, achronal set w/o edge, or (ii) closed trapped surface, or (iii) point p s.t. null geodesics from p are focused and start to reconverge This version does apply to AdS case if (3)-(ii) or (iii) holds

38. The idea of proof Affine distance from any point to AdS conformal boundary along a null geodesic is infinite. Can find a globally hyperbolic subset that contains an endless timelike curve & Standard argument about maximum length geodesic in globally hyperbolic region and a consequence of the convergence conditions lead a contradiction that falsifies the assumption of geodesic completeness

39. Bizon – Rostworowski 11 Black hole forms even starting from arbitrarily small initial perturbations -- want to consider a singularity theorem without imposing a closed trapped surface Initial small perturbations grow by repeating bounce off by AdS infinity

40. Collapse in asymptotically flat space Scalar field collapse in asymptotically Flat spactime Critical phenomena: Choptuik 93 Choptuik, 93 Subcritical initial data: Wave packet bounces off the center and escapes to infinity Supercritical initial data: Wave packet collapses to form a black hole

41. Subcritical initial data gets amplified and becomes supercritical after reflecting several times at AdS infinity, and finally collapses to form a black hole Collapse in asymptotically AdS space Wave packet cannot disperse in AdS as it bounces off at AdS infinity To take into consideration the process of bouncing-off at AdS infinity several times, one has to deal with a non-globally hyperbolic region

42. Global method used in the standard proof of singularity theorems does not appear to work when we deal with non-globally hyperbolic region. e.g. Standard argument about the existence of maximum length of causal curves does not apply We attempt to prove a singularity theorem w/ no trapped condition by restricting to some specific cases.

43. Spherically systems w/ perfect fluid Fluid stress-tensor Metric Raychaudhuri equation for the fluid-lines

44. A singularity theorem If the following averaged convergence: and some other reasonable conditions hold, then asymptotically AdS space w/ perfect fluid must be singular. Maeda & AI to appear

45. Sketch of proof Averaged convergence condition leads grows and eventually forms a closed trapped surface Combined with Hawking-Penrose’s version of singularity theorem yields the geodesic incompleteness : Number of how many times the fluid lines bounce back and forth between the center and AdS infinity

46. Summary: AdS and singularities AdS is nonlinearly unstable: AdS apears generically singular We obtained singularity theorems for a specific case: Spherical symmetry and perfect fluid without imposing strong gravity condition. We needed to impose the averaged convergence conditions which seem too restrictive. We used the Dirichlet boundary conditions at AdS-infinity If one imposes dynamical boundary conditions, it might be possible to relax some of the conditions for fields inside What does this imply in terms of dual field theories in AdS conformal boundary?