Read e-book online AdS/CFT Correspondence: Einstein Metrics and Their Conformal PDF
By Olivier Biquard
For the reason that its discovery in 1997 by way of Maldacena, AdS/CFT correspondence has develop into one of many leading topics of curiosity in string idea, in addition to one of many major assembly issues among theoretical physics and arithmetic. at the actual facet, it offers a duality among a conception of quantum gravity and a box idea. The mathematical counterpart is the relation among Einstein metrics and their conformal limitations. The correspondence has been intensively studied, and many development emerged from the disagreement of viewpoints among arithmetic and physics. Written via best specialists and directed at learn mathematicians and theoretical physicists in addition to graduate scholars, this quantity offers an summary of this crucial quarter either in theoretical physics and in arithmetic. It includes survey articles giving a vast review of the topic and of the most questions, in addition to extra really good articles supplying new perception either at the Riemannian part and at the Lorentzian aspect of the speculation. A e-book of the ecu Mathematical Society. disbursed in the Americas by way of the yankee Mathematical Society.
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Extra info for AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries
The vector field T = ∇ρ is now a unit time-like vector field, so g(T , T ) = −1. 7) then give ¯ρ ρ ¯ρ ρ + (R − n(n + 1))/ρ 2 ≥ −2n ¯ρ ρ . + (Ricg −ng)(T , T )/ρ 2 . 17) where again φ = − ¯ ρ/ρ. 8) also holds, and hence Rγ < 0 implies φ(0) < 0. 16). 3 to the situation where Rγ = 0. 1, (S, g) has + = ∅ and no time-like geodesic is futurecomplete unless (S, g) is isometric to g = −dt 2 + e2t gN n , where gN n is Ricci-flat, cf.  for further details. 15) cannot be geodesically complete if Rγ < 0, and have at most one component of conformal infinity.
15) for T time-like. Let γ be a representative for [γ ] with constant scalar curvature Rγ . 16) where ρ is the geodesic defining function associated to ( − , γ ). In particular, any time-like geodesic in S is future incomplete, and no Cauchy surface ρ exists, even partially, for ρ 2 > 4n(n − 1)/|Rγ |, so that + = ∅. Proof. 1. Let g¯ = ρ 2 g be the C 2 geodesic compactification determined by the data ( − , γ ); as before the computations below are with respect to g¯ . 3) holds for Lorentzian metrics (where it is known as the Raychaudhuri equation).
17) where again φ = − ¯ ρ/ρ. 8) also holds, and hence Rγ < 0 implies φ(0) < 0. 16). 3 to the situation where Rγ = 0. 1, (S, g) has + = ∅ and no time-like geodesic is futurecomplete unless (S, g) is isometric to g = −dt 2 + e2t gN n , where gN n is Ricci-flat, cf.  for further details. 15) cannot be geodesically complete if Rγ < 0, and have at most one component of conformal infinity. This exhibits the role of the hypothesis Rγ > 0 in a more drastic way than the AH case. 26 Michael T. 8) is also complete to the future, and has a smooth future conformal infinity + .
AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries by Olivier Biquard