Cosmological singularities, AdS/CFT and de Sitter deformations

Abstract

Abstract.We have reviewed aspects of certain time-dependent deformations of AdS/CF T , containing cosmological singularities and their gauge theory duals. Towards understanding these solutions better, we have explored similar singular deformations of de Sitter space and argued that these solutions are constrained, possibly corresponding to specific initial conditions. Cosmological singularities and AdS/CFTGeneral relativity breaks down at cosmological singularities, with curvatures and tidal forces typically diverging: notions of spacetime, thus, break down. There is a rich history of string theory explorations of such singularities [1,2]. We have focused on describing certain timedependent deformations of AdS/CFT [3,4,5,6], where the bulk gravity theory develops a cosmological singularity, and breaks down while the holographic dual field theory, a sensible Hamiltonian quantum system typically subjected to a time-dependent gauge coupling, can potentially be addressed in the vicinity of the singularity. The bulk string theory on AdS 5 × S 5 (in Poincare slicing) with constant dilaton (scalar) is deformed to:withg μν , Φ functions of x μ alone (Φ = Φ(t) or Φ(x + ) then gives time-dependence). This is a solution ifRthese include, e.g., AdS-Kasner, FRW, BKL (based on the Bianchi classification), etc. In many cases, it is possible to find new coordinates such that the boundary metric ds 2 4 = lim z→0 z 2 ds 2 5 is flat at least as an expansion about z = 0. This suggests that the dual is the N =4 super Yang-Mills theory with the gauge coupling g 2 Y M = e Φ , deformed to have external time-dependence. It is useful to focus on sources approaching e Φ → 0 at some finite point in time: For instance, a coupling of the form g 2 Y M → t p , p > 0, gives rise to R tt ∼ 1 2Φ 2 ∼ 1 t 2 , i.e., a bulk singularity with curvatures and tidal forces diverging near t = 0. Analyzing the gauge theory is possible in some cases. While at first sight, one might imagine that the dual in such cases to be weakly coupled, this is not the case and interactions are important in general [6]. For instance, the gauge kinetic termsT rF 2 can be transformed

AdS null deformations with inhomogeneities

“…These conditions on the g n µν , Φ n , n > 0, are non-generic and appear to be nontrivial constraints fine-tuning the CFT state after turning on the sources g (0) , Φ (0) . These arguments have also been discussed in [46] in the cosmological singularities context of [45,40].…”

“…This appears to have some bearing on the Lifshitz singularity due to diverging tidal forces as r → ∞ [52]. In the conformal coordinates (26), we see that the AdS deformation with g (0) µν =g µν alone could potentially lead to singularities on the Poincare horizon r → ∞, with the curvature invariants R ∼ r 2gµν ∂ µ Φ∂ ν Φ+O(r 0 ), R ABCD R ABCD ∼ r 4R µναβR µναβ +O(r 0 ) etc diverging (this can be seen by expanding out the curvature components and using the equationR µν = 1 2 ∂ µ Φ∂ ν Φ: see also [46] in the cosmological singularities context [45,40]). Equivalently, a metric that is regular everywhere, with an expansion of this form about r = 0, must have the coefficients g µν generically nonzero if a singularity is to be avoided at large r. For the null metric (25), these curvature invariants vanish, since the lightlike solutions admit no nonzero contraction.…”