An Observable in Classical Pure AdS3 Gravity: the Twist along a Geodesic

In this paper, we consider a little-studied observable in classical pure AdS3 gravity: the twist along a geodesic. The motivation is that the twist only supports on the geodesic so may be a candidate element in the center of the algebra in either entanglement wedge associated to the geodesic. We study the properties of the twist and get the following results. First, we get the system's evolution generated by the twist, which exhibits a relative shift along the geodesic. Second, we show that the twist commutes with the length of the same geodesic, which supports the proposal that the twist is a candidate element in the center.