Ong Yen Chin 王元君 Tea Time Talk 20 May 2012

1. The Puzzles of Bang/Crunch Holography • Ong Yen Chin • 王元君 • Tea Time Talk • 20 May 2012

2. AdS/CFT Correspondence Black hole AdS Boundary at infinity. One-to-One Correspondence: The sum over all geometries with fixed boundary conditions is the same as the partition function of a (conformal) field theory living on the boundary. Anti-de Sitter space Drawing by Tom Brown

3. Seiberg-Witten Instability Asymptotically AdS space; Einstein manifold has metric proportional to Ricci curvature. Brane. Physically, Λ is an n-form field of the appropriate supergravity theory.

4. A Puzzle: What If There Are More Than One Boundary?

5. Well, that can’t happen… “Connectedness Of The Boundary In The AdS/CFT Correspondence”, Edward Witten, S.-T. Yau, Adv.Theor.Math.Phys.3:1635-1655,1999, arXiv:hep-th/9910245v1. The result can be generalized to include non-Einstein manifolds withboundary of non-negative scalar curvature [Ming-Liang Cai, Gregory J. Galloway, Boundaries of Zero Scalar Curvature in the AdS/CFT Correspondence, Adv.Theor.Math.Phys. 3 (1999) 1769, arXiv:hep-th/0003046].

6. Is Our Universe Asymptotically dS? • Folklore: From the fact that our universe is accelerating, we can conclude that our spacetime is asymptotically de Sitter. • Fact: It is not ruled out that our world could be unlike de Sitter spacetime. We could be in a temporary accelerating phase!

7. What if Our Universe is Asymptotically AdS? • Can we model cosmology on AdS? • One crucial difference between AdS and dS [and Minkowski]: AdS is not globally hyperbolic. [Cauchy surface does not uniquely determine what happens at a later time.] • AdS also violates DEC: Thus evade Hawking’s theorem on black hole topology: Black holes can be of non-trivial topology in AdS, e.g. flat torus.

8. What if Our Universe is Asymptotically AdS? However, by introducing scalar matter into AdS one can produce a globally hyperbolic spacetime which has a Big Bang and also a temporary burst of acceleration. [Rolando Cardenas, Tame Gonzalez, Yoelsy Leiva, Osmel Martin, Israel Quiros, A model of the Universe including Dark Energy accounted for by both a Quintessence Field and a (negative) Cosmological Constant, Phys.Rev. D67 (2003) 083501, arXiv:astro-ph/0206315] Remarks: Singularities are natural in AdS with matter fields. The introduction of anyamountof any form of matter [other than vacuum energy] satisfying the Strong Energy Condition causes singularities to develop, producing a spacetime which is not a small perturbation of AdS. C.f. Stability [small perturbation remains small] of Minkowski space [Christodoulou, Klainerman (1993)]; and that of de Sitter space [Friedrich (1986)].

9. Holography of “Crunchy” Cosmology • Maldacena-Maoz proposal: Singular Bang/Crunch cosmological spacetimes are described by a dual field theory defined on the non-singular boundary of the Euclidean version. This seems to contradict Witten-Yau’s theorem! Does AdS/CFT prohibit “Crunchy” AdS cosmology?

10. Could We Evade Witten-Yau’s Theorem? • Yes! Witten-Yau Theorem [even Cai-Galloway’s generalized version] might not hold if we turn on other fields! That is, allow the bulk matter to modify the bulk geometry so that it does not satisfy the premises of the theorem.Yang-Mills: Juan Maldacena and Liat Maoz [JHEP 0402 (2004) 053; arXiv:hep-th/0401024v2] However Yang-Mills field satisfies SEC, cannot produce temporary accelerating phase in AdS Cosmology. Quintessence: Brett McInnes [JHEP 0404:036,2004; arXiv:hep-th/0403104v3]

11. Evading the Cai-Galloway’s Theorem Ming-Liang Cai Gregory J. Galloway It turns out that in cosmology one is not always entitled to prescribe very rapid asymptotic rates of decay of matter fields towards infinity. Dark energy, in particular, dilutes with the cosmic expansion very slowly [if it even dilutes!], too slowly for the theorem to apply.

12. But Puzzles Remain • Thus “crunchy ” AdS-like cosmology proposed by Maldacena and Maoz is not prohibited. • Since there are no physical or mathematical objections to Maldacena-Maoz cosmologies, one has to confront the original issue identified by Witten and Yau: how can two apparently independent field theories [on the boundary] both be dual to the same bulk? It seems that either the two field theories, despite appearances, are not really independent, or the bulk, despite appearances, is not really connected. • Further Reading: Brett McInnes, Answering a Basic Objection to Bang/Crunch Holography, JHEP 0410:018,2004; arXiv:hep-th/0407189v3

13. Worth to think about? • What about the case with “two bulks but one boundary”? • m bulks and n boundaries?

14. More Puzzles • AdS has many other interesting features that are not yet completely understood: Many types of instabilities; implications for the dual field theory needs to be understood. • Turbulent Instability: Small perturbation of initial data does not stay small: singularities are generic. See e.g. Bizon and Rostworowski [1104.3702] and Dias, Horowitz, Santos [1109.1825]. • Perturbative Instability: Instability arises if there are scalars with mass m2 < −λ0, where λ0 is the eigenvalue of the lowest normalizable eigenmode of the Euclidean Laplace operator. If the Yamabe invariant is negative, this can occur even for “tachyons” which would ordinarily be allowed in AdS; that is, fields satisfying the Breitenlohner-Freedman bound. In particular, fields saturating the BF bound may be problematic. • Non-Perturbative Instability [Seiberg-Witten Instability]: If there are BPS branes, there could be an instability due to brane nucleation.

15. The End

16. Backup Slide: Energy Conditions Back to page 7, 10

17. Backup Slide: Comment on DEC Violation • Contrary to popular belief cosmologies which violate the Dominant Energy Condition [e.g. phantom with w<-1], do not necessarily violate causality. • Roughly speaking, DEC forbids anything to enter the region of spacetime controlled causally by S. However, the converse is false!