# Boundary causality Boundary causality is a requirement in [[0001 AdS-CFT|AdS/CFT]] that says that signals should not travel faster through the bulk than they possibly can on the boundary. ## Statement - $\left\langle\Psi\left|\left[T_{\mu \nu}(y), T_{\rho \sigma}(0)\right]\right| \Psi\right\rangle=0, \quad y^{2}>0$ for some bulk state $\Psi$ ## Refs - [[2016#Akers, Koeller, Leichenauer, Levine]] - comprehensive summary of a series of conditions ![[AkersKoellerLeichenauerLevine2016_relation9.png]] ## Condition - sufficient - [[2000#Gao, Wald]]: Einstein gravity & null energy condition - sufficient and necessary - [[2016#Engelhardt, Fischetti]]: a condition on how much the light cone can open (BCC) - necessary - [[2025#Fu, Izumi, Yoshida]]: spherically symmetric metrics ## Constraints using boundary causality - [[1994#Woolgar]] positive energy theorem - [[2002#Page, Surya, Woolgar]] positive energy theorem - [[2014#Camanho, Edelstein, Maldacena, Zhiboedov]] constrains higher derivative terms - using [[0117 Shockwave]] ## Conformally invariant ANEC - [[2019#Iizuka, Ishibashi, Maeda]]