Chaotic Information Processing by Extremal Black Holes

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1 1 Chaotic Information Processing by Extremal Black Holes Stam Nicolis CNRS Laboratoire de Mathématiques et Physique Théorique (UMR7350) Fédération de Recherche Denis Poisson (FR2964) Département de Physique, Université de Tours VIIth Black Hole Workshop, Aveiro, PT 19 December Work based on arxiv : and in progress,with M. Axenides and E. G. Floratos.

2 1 Extremal black holes 2 The two observers 3 The AdS 2 [N]/CFT 1 [N] correspondence 4 Conclusions

3 Classical vs. Quantum Geometry Classical geometry The near horizon geometry of extremal black holes is given by the product of two metrics : ds 2 = [ ds 2] AdS 2 + [ ds 2] K The AdS 2 part describes the radial excitations. Its geometry is given by the equation with x, y, z R. x 2 + y 2 z 2 = 1 Geomterically, it s a coset : AdS 2 = SL(2, R)/SO(1, 1). This is a key property.

5 Quantum geometry The quantum black hole has a finite entropy. The finite number of states, N, means that a finite number of points of AdS 2 are accessible. These are described by x 2 + y 2 z 2 1 mod N with x, y, z Z/NZ. This modular discretization preserves the isometries of the geometry.

6 This equation can be solved as follows : x 2 + y 2 (1 + z 2 ) mod N So, for any integer, z Z/NZ, such that 1 + z 2 is a quadratic residue, we may write and obtain x z 2 + y z 2 1 mod N

7 x z t t mod N, y z t2 1 t mod N with t Z/NZ, such that t is invertible mod N.

8 A view at fixed z = 0, at increasing N (N = 31, 67, 1031) These don t look like circles! But, in fact, they have all the right properties. The set of points, for all values of z,is the coset AdS 2 [N] SL(2, Z N )/SO(1, 1, Z N ). So the elements of this group, 2 2 matrices, A, are the stars of the show. From them we shall construct the quantum evolution operators, U(A), on the boundary and in the bulk.

9 The observers Two observers : Observer at infinity : Boundary observer Observer in free fall : Bulk observer Quantum mechanically : Both probe discrete spectrum. But in different ways!

10 On the boundary Stability group, the Borel subgroup, {( ) } λ b B N 0 λ 1 λ Z N, b Z N The vacuum state is given by k 0 boundary δ k,0 and the excitations are given by the coset elements h = U(h) 0 boundary where the the operator U(h) is given by [( )] a b U(h) = U b a We must, also, include the point at infinity. In that way we get N + 1 states.

11 In the bulk Stability group, the scaling group {( λ 0 D = 0 λ 1 ) } λ Z N The vacuum state is defined by 0 bulk 1 N (1, 1,..., 1) and the excitations are with h the coset element h U(h) 0 bulk h = R(φ + ) ( 1 µ 0 1 Along with the point at infinity, we, also, obtain N + 1 states and can construct the correspondence. )

12 The bulk/boundary correspondence It is possible to express the states in the bulk in terms of the states on the boundary, through the corresponding propagators. The expressions are explicit and their evaluation relies on results from algebraic number theory. What s relevant is that the correspondence, for finite N is exact and the semi-classical limit, N is highly non-trivial.

13 Chaotic information processing Chaotic information processing means that maps U(h) realize chaotic motion in the space of states. So an initial state, that is localized around a small number (wrt N) states, very rapidly has non-zero overlap with all of them. This property can be quantified by calculating the usual measures of chaotic behavior.

14 Conclusions and Outlook Extremal black holes have fixed finite entropy finite dimensional space of states, that doesn t change in time. Issue : construct the dynamics of probes from bulk and boundary. The bulk observer uses the coherent states, that correspond to one stabilizer group : the dilatations. The boundary observer uses the coherent states, that correspond to another satbilizer group : the rotations. The observables are related by a transformation that realizes the AdS 2 [N]/CFT 1 [N] correspondence and is consistent with the extended nature of the probes. The factorization algorithms lead to fast information processing that is chaotic. Time dependent aspects, that probe beyond extremality, imply varying the dimension of the space of states.

The intricate interplay between integrability and chaos for the dynamics of the probes of extremal black hole spacetimes

The intricate interplay between integrability and chaos for the dynamics of the probes of extremal black hole spacetimes Stam Nicolis CNRS Laboratoire de Mathématiques et Physique Théorique (UMR7350) Fédération