# Averaged null energy condition (ANEC) The ANEC states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative., i.e., $\int d u\, T_{u u} \geq 0.$It is believed to hold for classical matter but can be violated quantum mechanically (quantum fields in curved spacetime). There is also a looser condition called **Achronal ANEC** (AANEC). The integrand looks the same as ANEC except that the integral needs only be non-negative along *achronal* null geodesics (no points on the geodesic are timelike related). Unlike ANEC, AANEC is usually assumed to hold in the bulk. This is because it is a consequence of [[0082 Generalised second law|GSL]], a condition that is expected to hold, and implies [[0091 Boundary causality|boundary causality]], a property that is desired in holography. In [[0001 AdS-CFT|AdS/CFT]], it is important to distinguish between bulk ANEC and bulk ANEC. Bulk ANEC is related to the [[0005 Black hole second law|BH second law]], [[0225 Singularity theorems|singularity theorems]], etc., while boundary ANEC is equivalent to [[0091 Boundary causality|boundary causality]] of the bulk theory. ## Comments - It is implied by [[0480 Null energy condition|NEC]] (trivially). - It is only possible to have [[0083 Traversable wormhole|traversable wormholes]] if ANEC is violated. - It is reasonable to assume AANEC for realistic spacetimes. This still allows for traversable wormholes because ANEC can still be violated. In other words, we can have the integral to be negative only along chronal null geodesics. Chronality means that there is a shorter path around the wormhole than the null geodesic which goes through the wormhole. Such traversable wormholes are called *long*. Short traversable wormholes are hence forbidden by AANEC. ## Refs - AANEC - [[2007#Graham, Olum]] - AANEC from [[0082 Generalised second law|GSL]] - [[2009#Wall (Jan)]] - AANEC from no [[0570 Time machine|time machines]] - [[1988#Morris, Thorne, Yurtsever]] - [[1993#Friedman, Schleich, Witt]] - higher-spin - [[2016#Hartman, Kundu, Tajdini]] - [[2018#Kravchuk, Simmons-Duffin]]: continuous spin - [[2018]] - connection to RG - [[2023#Hartman, Mathys (Sep)]]: 4d - [[2023#Hartman, Mathys (Oct)]]: 2d ## Proofs - proof in interacting QFT - [[2016#Faulkner, Leigh, Parrikar, Wang]]: using [[0416 Modular Hamiltonian|modular flow]] and assuming monotonicity of [[0199 Relative entropy|relative entropy]] (an information-theoretic approach) - [[2016#Hartman, Kundu, Tajdini]]: using causality - proof of boundary ANEC from bulk causality - [[2008#Hofman, Maldacena]] - [[2014#Kelly, Wall]] ## Higher-spin generalisation The higher-spin version for the ANEC operator is given by$\mathcal{E}_s=\int_{-\infty}^{\infty} d u X_{u u u \cdots u}(u, v=0, \vec{x}=0),$where $s$ is even, and $X$ is the lowest-dimension operator with spin $s$. ## Relation to other concepts - connection to [[0351 Irreversibility theorems|a-theorem]]: [[2023#Hartman, Mathys (Sep)]] - [[0493 Conformal collider bounds|Hofman-Maldacena bounds]] are ANEC for a special class of states ## Related - [[0247 Energy conditions]] - [[0480 Null energy condition]] - [[0405 Quantum null energy condition]] - [[0083 Traversable wormhole]] - [[0243 Quantum focusing conjecture]] - [[0420 Topology change in gravity]]