# The quotient method in AdS3 Since all [[0002 3D gravity|3D solutions in AdS]] are locally the same, they can all be obtained by taking quotients of the global AdS$_3$ solution. ## Refs - obtain [[0002 3D gravity]] - discussed in [[Rsc0004 CompereFiorucci Ch2 3d gravity]] ## Procedure - P -> $e^{s\xi}$P. $s=k\Delta s, k\in \mathbb{Z}$ - $\xi$ Killing -> after identification the metric is not multivalued - $\xi$ must be spacelike <- curves connects two points on the same orbit (of the identification group) become a closed loop ## Criteria on the isometries - do not have any *objectionable* singularities - see [[1999#Brill]] ## Examples - rotations - timelike fixed points -> timelike conical defects - represent particles - locally Lorentz boosts - spacelike fixed points - if hidden behind horizon -> black hole - some isometry has no fixed points, but still lead to singular regions in the quotient space - other type of black holes ## Examples in terms of $t=0$ surfaces for time-symmetric solutions - explain very nicely in [[1999#Brill]] - by first constructing coordinates corresponding to isometrics whose orbits are orthogonal planes (in the embedding space) - elliptic ![[0099_elliptic.png|250]] - parabolic ![[0099_parabolic.png|250]] - hyperbolic ![[0099_hyperbolic.png|250]] ## Fundamental domain - cut a strip and glue the edges - alternatively, can use the **doubling** method - any quotient space can be considered the double of a suitable region - because any isometry can be decomposed into two reflections