From accelerating and Poincar\'e coordinates to black holes in spacelike warped AdS${}_3$, and back

We first review spacelike stretched warped AdS${}_3$ and we describe its black hole quotients by using accelerating and Poincar'e coordinates. We then describe the maximal analytic extension of the black holes and present their causal diagrams. Finally, we calculate spacetime limits of the black hole phase space $(T_R,T_L)$. This is done by requiring that the identification vector ${\mathrm{\partial }}_{\theta }$ has a finite non-zero limit. The limits we obtain are the self-dual solution in accelerating or Poincar'e coordinates, depending respectively on whether the limiting spacetimes are non-extremal or extremal, and warped AdS with a periodic proper time identification.