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A spacelike surface with cylinder topology can be described by various setsof canonical variables within pure AdS3 gravity. Each is made of one realcoordinate and one real momentum. The Hamiltonian can be either $H=0$ or it canbe nonzero and we display the canonical transformations that map one into theother, in two relevant cases. In a choice of canonical coordinates, one of themis the cylinder aspect $q$, which evolves nontrivially in time. The timedependence of the aspect is an analytic function of time $t$ and an "angularmomentum" $J$. By analytic continuation in both $t$ and $J$ we obtain aEuclidean evolution that can be described geometrically as the motion of acylinder inside the region of the 3D hyperbolic space bounded by two "domes"(i.e. half spheres), which is topologically a solid torus. We find that for agiven $J$ the Euclidean evolution cannot connect an initial aspect to anarbitrary final aspect; moreover, there are infinitely many Euclideantrajectories that connect any two allowed initial and final aspects. We computethe transition amplitude in two independent ways; first by solving exactly thetime-dependent Schr\"odinger equation, then by summing in a sensible way allthe saddle contributions, and we discuss why both approaches are mutuallyconsistent.

Cylinder transition amplitudes in pure AdS$_3$ Gravity