AdS##_3## Cylinder: Killing Vectors & Isometry Group

[highflyyer]

The isometry group of the anti-de Sitter spacetime is $SO(d-1,2)$, which has a total of $\frac{1}{2}d(d+1)$ isometries.

For the three-dimensional anti-de Sitter spacetime, these are $6$ isometries. These isometries have corresponding Killing vectors, which in global coordinates, are given in equation (9.8) of http://www.hartmanhep.net/topics2015/9-ads3symmetries.pdf.

For an AdS$_3$ cylinder (in global coordinates) that is radially cut-off at a finite radius, the number of isometries would decrease to $2$ because only $2$ of the Killing vectors in equation (9.8) of http://www.hartmanhep.net/topics2015/9-ads3symmetries.pdf are independent of the radial coordinate.

Now, any physical quantity on a maximally symmetric spacetime is a function of the geodesic distance. How does this fact change for the radially-cut-off AdS$_3$ cylinder?

 

[Ambitwistor]

The fact that any physical quantity on a maximally symmetric spacetime is a function of the geodesic distance remains unchanged for the radially-cut-off AdS$_3$ cylinder. The only difference is that the geodesic distance is now limited to a maximum value due to the radial cutoff. This means that any physical quantity will have a maximum value as well, corresponding to the maximum geodesic distance in the cylinder. Additionally, the isometries of the radially-cut-off AdS$_3$ cylinder will still act on the geodesic distance in the same way as they do in the full AdS$_3$ spacetime, but their effects will be limited by the radial cutoff.