Butterfly effect and T T ¯ -deformation

These notes present a comprehensive analysis of shock wave geometries in holographic settings, focusing on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">T</a:mi> <a:mover accent="true"> <a:mi mathvariant="normal">T</a:mi> <a:mo stretchy="false">¯</a:mo> </a:mover> </a:math> -deformed Bañados–Teitelboim–Zanelli black holes and their extensions. By constructing deformed metrics and employing Kruskal coordinates, we examine out-of-time-ordered correlators as probes of quantum chaos. We also study localized shock wave solutions and analyze their backreaction, highlighting regimes in which the Mezei-Stanford bound on the butterfly velocity is potentially violated. The results obtained via shock wave methods are corroborated with recent developments in pole skipping phenomena and the entanglement wedge approach, demonstrating consistency among distinct probes of chaos in holographic theories.