Multiway junction conditions I: booklets and webs

We explore the construction of gluing together an arbitrary number of spacetimes along a common interface that gives a geometric structure we refer to as the booklet geometry. We derive junction conditions across the interface, in both Einstein gravity and dilaton gravity, using two methods (1) by geometrically introducing an auxiliary reverse extension manifold, and (2) by varying the action of the theory under consideration, and the two methods give the same result. Our junction condition works for both spacelike and timelike interfaces. We also provide simple examples that illustrate applications of the junction condition. We further comment on the geometry of a web of spacetime in the presence of multiple interfaces and its resemblance to Feynman diagrams.