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 BIRS-CMO Workshop Time-like Boundaries in General Relativistic Evolution Problems 1 August, 2019 Includes joint work with G. Holzegel (Imperial College London) Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 42

. . . . . . . . . . . . . . . . Introduction Section 1 Introduction Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . 2 / 42

. Physical Motivations . . . . . . . . . . Introduction Correspondence and Holography . 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 aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 42 AdS/CFT ⇒ holographic principle: Original paper ∗ : 12154 12201 12381 12869 13156 13278 13964 14577 14769 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-stationary) settings. Main questions: 1 Rigorous statements toward holographic correspondences? 2 Proofs of these statements? 3 Mechanisms behind such correspondences? Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 42

. . . . . . . . . . . . . . . . Introduction Some Background in Relativity General Relativity Gravity described by Einstein’s theory of general relativity: Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . 5 / 42 . . . . . Spacetime: ( n + 1 ) -dimensional Lorentzian manifold ( M , g ) . g : Lorentzian metric, with signature (− , + , . . . , +) . No matter fjelds ⇒ g satisfjes Einstein-vacuum equations (EVE): 2 Λ Ric g = n − 1 g . Ric g : Ricci curvature of g . Λ ∈ R : Cosmological constant.

. . . . . . . . . . . . . . Introduction Some Background in Relativity Anti-de Sitter Spacetime Anti-de Sitter (AdS) spacetime: 2 . Lorentzian analogue of hyperbolic space. Global representation of AdS spacetime: Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 42 Maximally symmetric solution of EVE, with Λ = − n ( n − 1 ) Λ < 0 analogue of Minkowski spacetime. g 0 := ( 1 + r 2 ) − 1 dr 2 − ( 1 + r 2 ) dt 2 + r 2 ˚ ( R t × R n x , g 0 ) , γ . γ : Round metric for unit sphere S n − 1 . ˚

. . . . . . . . . . . . . . Introduction Some Background in Relativity The Conformal Boundary t r . Asymptotically AdS (aAdS): Spacetime with “similar conformal boundary”. Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . 7 / 42 . . . Consider inverted radius ρ := r − 1 : ( 1 + ρ 2 ) − 1 d ρ 2 − ( 1 + ρ 2 ) dt 2 + ˚ g 0 = ρ − 2 [ ] γ ρ 2 g 0 : smooth at ρ = 0 ( r = ∞ ). ρ = 0 Formally attach boundary at “ ρ = 0”: r = ∞ r = 0 g := − dt 2 + ˚ ( I ≃ R t × S n − 1 , ˚ γ ) . I I : conformal boundary of AdS: ˚ g : (Lorentzian) boundary metric. Conformal AdS, mod S n − 1 .

. . . . . . . . . . . . Introduction . The Main Problem A Correspondence Question Question (Preliminary) Is there some one-to-one correspondence between: aAdS solution of EVE (“gravitational dynamics”). Attempt 1: Formulate in terms of PDEs. Goal: Solve EVE into interior? Data Solve for Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 42 Data prescribed at conformal boundary I . (Boundary metric ˚ g , boundary stress-energy tensor.) g =? ( ˚ g ,... ) Given: (Cauchy) data on conformal boundary I . I EVE, with data on I .

. . . . . . . . . . . . . . Introduction The Main Problem Ill-Posedness For wave equations on bounded domain, need: To solve EVE, need: (Friedrich, 1995), (Enciso–Kamran, 2019) t Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 42 Bad news: Problem is ill-posed. “Initial” hypersurface I is timelike! Initial data at t = 0. r = 1 r = 1 Dirichlet or Neumann data on C . Initial data at t = 0. The cylinder C . Dirichlet or Neumann data on I .

. . . . . . . . . . . . . . Introduction The Main Problem A Unique Continuation Problem Attempt 2: Formulate as unique continuation (UC) problem. Classical problem in PDEs. If a solution exists, then must it be unique? Question (Correspondence, Informal) ... then must these spacetimes be isometric? Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . 10 / 42 . . . . . Given two aAdS solutions g 1 , g 2 of EVE: Data ( g 1 ) g 1 = g 2 ? ∥ If g 1 , g 2 have same boundary data on I , ... Data ( g 2 ) I

. . . . . . . . . . . . . . . . Precise Formulations Section 2 Precise Formulations Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . 11 / 42

. . . . . . . . . . . . . . Precise Formulations Asymptotically AdS Spacetimes aAdS Manifolds Goal: Precise description of our aAdS spacetimes. Remark. Formulation allows for: General boundary topology/geometry. Example: AdS, planar AdS, toroidal AdS. t Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . 12 / 42 Step 1: Construction of aAdS manifold M . Conformal boundary: I n := R t × S n − 1 . M I S : cross-section of I . ρ Spacetime (near boundary): M := ( 0 , ρ 0 ] ρ × I . ρ = 0 aAdS manifold M , mod S .

. . . . . . . . . . . . . . . . Precise Formulations Asymptotically AdS Spacetimes aAdS Metrics Remark. No loss of generality from FG gauge. Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . 13 / 42 . . . . . Step 2: Construction of aAdS metric g . Assume g in Fefgerman–Graham (FG) gauge. g := ρ − 2 ( d ρ 2 + g ab dx a dx b ) . ρ trivial, decoupled from ( t , S ) -coordinates. g : vertical metric ( ρ -indexed family of Lorentzian metrics on I ). Assume g has (Lorentzian) boundary limit: ρ ↘ 0 g = ˚ lim g .

. . . . . . . . . . . . . . Precise Formulations Asymptotically AdS Spacetimes Einstein–Vacuum Spacetimes (Fefgerman–Graham, 1984, 2007) Ambient metric construction. Analytic conformal data on null cone. Spacetime given as series expansion in terms of conformal data. (Kichenassamy, 2004) Series converges for analytic data. Idea adapted to aAdS settings: Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . 14 / 42 Q. What if ( M , g ) also satisfjes EVE? What structure does this impose on g at I ? Given: Analytic conformal boundary data on I . Derive: Formal series expansion from I for g .

. . . . . . . . . . . . . . Precise Formulations Asymptotically AdS Spacetimes Fefgerman–Graham Expansions 2 n odd, 2 n even. Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . 15 / 42 . . . . . . . . . . . . Fefgerman–Graham expansion for vertical metric g :  ∑ n − 1  k = 0 ρ 2 k g ( 2 k ) + ρ n g ( n ) + . . . g = ∑ n k = 0 ρ 2 k g ( 2 k ) + ρ n ( log ρ )g ( ∗ ) + ρ n g ( n ) + . . .  g ( 0 ) = ˚ g : freely prescribed (boundary metric). g ( n ) : also partially free (related to boundary stress-energy tensor). Coeffjcients before g ( n ) : determined by g ( 0 ) . Coeffjcients after g ( n ) : determined by g ( 0 ) and g ( n ) . n even ⇒ anomalous ρ n ( log ρ ) -term. Expansion after g ( n ) is polyhomogeneous (includes ρ l ( log ρ ) m ).

. . . . . . . . . . . . Precise Formulations . Partial Boundary Expansions Non-Analytic Settings Full FG expansion only applicable to analytic settings. Too restrictive. Q. What about generic spacetimes? With fjnite regularity ( H s , C M ). Setting of well-posedness theories of EVE. Expect. Partial FG expansion (to fjnite order). But, do not wish to assume this a priori. Arick Shao (QMUL) Correspondence & Rigidity on aAdS . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 / 42