Abstract
We perform a systematic study of the maximum Lyapunov exponent values λ for the motion of classical closed strings in Anti-de Sitter black hole geometries with spherical, planar and hyperbolic horizons. Analytical estimates from the linearized varia- tional equations together with numerical integrations predict the bulk Lyapunov exponent value as λ ≈ 2πTn, where n is the winding number of the string. The celebrated bound on chaos stating that λ ≤ 2πT is thus systematically modified for winding strings in the bulk. Within gauge/string duality, such strings apparently correspond to complicated operators which either do not move on Regge trajectories, or move on subleading trajectories with an unusual slope. Depending on the energy scale, the out-of-time-ordered correlation functions of these operators may still obey the bound 2πT, or they may violate it like the bulk exponent. We do not know exactly why the bound on chaos can be modified but the indication from the gauge/string dual viewpoint is that the correlation functions of the dual gauge operators never factorize and thus the original derivation of the bound on chaos does not apply.
Highlights
It is clear, as discussed in the original paper [1], that there are cases when the bound does not apply: mainly systems in which the correlation functions do not factorize even at arbitrarily long times, and systems without a clear separation of short timescales and long timescales
We perform a systematic study of the maximum Lyapunov exponent values λ for the motion of classical closed strings in Anti-de Sitter black hole geometries with spherical, planar and hyperbolic horizons
This makes dynamics in asymptotically anti-de Sitter (AdS) spacetimes with a black hole interesting: they have a field theory dual,1 and black holes are conjectured to be the fastest scramblers in nature [2, 3], i.e., they minimize the time for the overlap between the initial and current state to drop by an order of magnitude
Summary
A constant curvature black hole in N + 1 spacetime dimensions is a geometry of constant curvature with the metric [34,35,36]. Where dσN2 −1 is the horizon manifold, which has curvature k, and m and q define the mass and charge of the black hole It is a vacuum solution of the Einstein equations with constant negative cosmological constant and interpolates to AdS space with radius 1. In our black hole backgrounds we always have Bμν = 0 so we can pick the gauge hab = ηab = diag(−1, 1) For the planar black hole we obviously have Px,y, the momenta, as the integrals of motion. Φ1 cos φ2, φ2 = φ1 sin φ2), and the string with the wrapping Φ2 = nσ would provide an integrable system, with the separable Hamiltonian H n2 2. But these are special and fine-tuned; we will consider these cases elsewhere as they seem peripheral for our main story on the chaos bound
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