Correspondence and Rigidity Results on Asymptotically Anti-de Sitter - PowerPoint PPT Presentation

. . . . . . . . . . . . . . Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes Arick Shao Queen Mary University of London International Conference on Nonlinear Waves and General Relativity 14 December, 2017 The Chinese University of Hong Kong Joint work with G. Holzegel (Imperial College London) Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 39

. . . . . . . . . . . . . . . . Introduction Section 1 Introduction Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . 2 / 39

. Physical Motivations . . . . . . . . . . Introduction Correspondence Principles . Outstanding problem in theoretical physics: Reconciling Einstein’s theory of gravity with quantum fjeld theories. Infmuential research direction: AdS/CFT correspondence (AdS: Anti-de Sitter) (CFT: Conformal fjeld theory) Gravitational theory on spacetime encoded in some theory on its boundary (of one less dimension). Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 39 AdS/CFT ⇒ holographic principle: Original paper ∗ : 12154 12201 12381 12869 13156 13278 citations. † ∗ J. Maldacena, The large N limit of superconformal fjeld theories and supergravity (1999) † Data from http://inspirehep.net/record/451647/citations .

. What’s Missing? . . . . . . . . . Introduction Physical Motivations In AdS context, little rigorous mathematics for: . Positive statements of this principle. Precise formulations of this principle. In particular, in dynamical (non-static) settings. Main (long-term) questions: 1 Can rigorous statements toward holographic correspondences be formulated? 2 Can these statements be proved? 3 Can one understand the mechanisms behind such a correspondence? Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 39

. . . . . . . . . . . . . . . Introduction Anti-de Sitter Spacetime General Relativity Gravity described by Einstein’s theory of general relativity. Gravity and matter coupled via the Einstein equations: Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . 5 / 39 . . . . . Spacetime: ( n + 1 ) -dimensional Lorentzian manifold ( M , g ) . g : Lorentzian metric, with signature (− , + , . . . , +) . Gravity modelled by curvature of ( M , g ) . Ric g − 1 2 Sc g g + Λ g = T . No matter ( T ≡ 0) ⇒ Einstein-vacuum equations (EVE): 2 Λ Ric g = n − 1 g .

. . . . . . . . . . . . . . Introduction Anti-de Sitter Spacetime Anti-de Sitter Spacetime Anti-de Sitter (AdS) spacetime: Maximally symmetric solution of EVE... 2 . Lorentzian analogue of hyperbolic space. Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . 6 / 39 . . . . . ... with negative cosmological constant Λ = − n ( n − 1 ) Globally represented as ( R t × R n x , g ) , with g := ( 1 + r 2 ) − 1 dr 2 − ( 1 + r 2 ) dt 2 + r 2 ˚ γ . r > 0 , ω ∈ S n − 1 : Polar coordinates on R n . γ : Round metric for unit sphere S n − 1 . ˚

. . . . . . . . . . . . . . Introduction Anti-de Sitter Spacetime The Conformal Boundary t r . Asymptotically AdS (aAdS): Spacetime with “similar conformal boundary”. Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . 7 / 39 . . . . . . . . . . . . Consider “inverted radius” ρ := r − 1 : ( 1 + ρ 2 ) − 1 d ρ 2 − ( 1 + ρ 2 ) dt 2 + ˚ g = ρ − 2 [ ] γ ρ 2 g smooth at ρ = 0 ( r = ∞ ). ρ = 0 ρ is a boundary defjning function. r = 0 r = ∞ Conformal boundary “ I := { ρ = 0 } ” of AdS: I g := − dt 2 + ˚ ( I ≃ R t × S n − 1 , ˚ γ ) . Compactifjed AdS, mod S n − 1 .

. Introduction . . . . . . . . . . The Main Question . The Correspondence Question Question (0’) Is there some correspondence between: aAdS solution of EVE (“gravitational dynamics”). (Ideally: boundary metric, stress-energy tensor.) Attempt 1: Formulate this in terms of PDEs: Question: Solve for unique solution of EVE in interior? Data Solve for Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 39 Data prescribed at conformal boundary I . g =? Given: “Cauchy” data on conformal boundary I . I EVE, with data on I .

. . . . . . . . . . . . . . . Introduction The Main Question Ill-Posedness Bad news: This problem is generally ill-posed. For EVE to be well-posed, one requires: Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . 9 / 39 . . . . . AdS ≈ cylinder in Minkowski spacetime: C := { ( t , x ) ∈ R 1 + n | | x | < 1 } . EVE ≈ wave equation. r = 1 r = 1 Wave equations ill-posed with Cauchy data on C . Initial data at t = 0. Dirichlet or Neumann data on I . The cylinder C .

. . . . . . . . . . . . Introduction . The Main Question A Unique Continuation Problem Attempt 2: Formulate as unique continuation problem for PDEs. If a solution exists, then must it be unique? Question (0) Suppose two aAdS solutions of the EVE have the same “boundary-Cauchy” data on their conformal boundaries. Then, must these solutions be isometric? Is there a one-to-one correspondence between aAdS solutions of EVE and some space of “boundary-Cauchy” data? Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 39

. . . . . . . . . . . . . . Introduction The Model Problem The Wave Equation Consider a model problem: Wave equation on fjxed AdS/aAdS spacetime. Question (1) The uniqueness problem. Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . 11 / 39 . . . . . I φ = 0? φ 1 , φ 2 : solutions on fjxed aAdS spacetime of φ = 0. ( □ g + σ ) φ = G ( φ, ∇ φ ) , σ ∈ R . If φ 1 , φ 2 have same Dirichlet and Neumann data on the boundary I , then is φ 1 = φ 2 locally near I ? G linear: φ = 0 at I ⇒ φ = 0 near I ?

. . . . . . . . . . . . . . . Introduction The Model Problem Why the Wave Equation? Question (1): essential step toward Question (0). Wave equation: fjrst linearization of EVE. Question (1) also has applications to rigidity results. Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . . 12 / 39 EVE ⇒ curvature satisfjes nonlinear wave equation. Remark. Why □ + σ , not □ ? σ determines asymptotics of φ near I .

. . . . . . . . . . . . . . . . Results on AdS Spacetime Section 2 Results on AdS Spacetime Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . 13 / 39

. . . . . . . . . . . . . . . Results on AdS Spacetime The Main Result Some Intuition n 2 (Breitenlohner–Freedman) Q. Is this condition suffjcient in general? Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . 14 / 39 . . . . . . . . . . . Consider: ( □ g + σ ) φ = 0. (Over)assume φ depends only on ρ ⇒ 2nd-order ODE for φ . Frobenius method ⇒ two branches of solutions: ∑ √ ∞ φ ± = ρ β ± a ± k ρ k , β ± = n 2 ± 4 − σ . k = 0 Thus, for φ to vanish, we must eliminate both branches: ρ − β + φ → 0, ρ ↘ 0. ( A. Almost , for physically relevant σ .)

. . . . . . . . . . . . Results on AdS Spacetime . The Main Result The Main Theorem Theorem (Holzegel–S.; 2015) 4 2 ), 4 2 ), Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . . 15 / 39 . . . Suppose φ is a C 2 -solution of | ( □ g + σ ) φ | ≤ ρ 2 + p ( | ∂ t φ | + | ∂ ρ φ | + | ∇ S 2 φ | ) + ρ p | φ | , I where σ ∈ R and p > 0 . Assume the vanishing condition if σ ≤ n 2 − 1 ( β + ≥ n + 1 φ = 0 | ρ − β + φ | + | ∇ t ,ρ, S 2 ( ρ − β + + 1 φ ) | → 0 , φ = 0 if σ ≥ n 2 − 1 ( β + ≤ n + 1 | ρ − n + 1 2 φ | + | ∇ t ,ρ, S 2 ( ρ − n − 1 2 φ ) | → 0 , as ρ ↘ 0 , on a suffjciently large time interval t ∈ [ 0 , t 0 ] , t 0 > π . Then, φ vanishes in the interior of AdS, near I ∩ { 0 < t < t 0 } .

. Some Remarks . . . . . . . . . Results on AdS Spacetime The Main Result 1 . First such correspondence result in dynamical, non-analytic setting. 2 The suffjciently large time interval assumption is new. Clearly necessary for global uniqueness. Surprisingly, seems necessary even for local uniqueness. 3 4 Result also holds for (appropriately defjned) tensor waves. Useful for future applications to EVE. Arick Shao (QMUL) Correspondence & Rigidity on AdS . . . . . . . . . . . . . . . . . . . . . . . 16 / 39 . . . . . . Vanishing condition optimal when σ ≤ n 2 − 1 4 . σ = n 2 − 1 4 : conformal mass. Q. (Open) Can we do better for σ > n 2 − 1 4 ?