Black hole vacua and rotation

Recent developments suggest that the near-region of rotating black holes behaves like a CFT. To understand this better, I propose to study quantum fields in this region. An instructive approach for this might be to put a large black hole in AdS and to think of the entire geometry as a toy model for the “near-region”. Quantum field theory on rotating black holes in AdS can be well-defined (unlike in flat space), if fields are quantized in the co-rotating-with-the-horizon frame. First, some generalities of constructing Hartle–Hawking Green functions in this approach are discussed. Then as a specific example where the details are easy to handle, I turn to 2+1$2+1$ dimensions (BTZ), write down the Green functions explicitly starting with the co-rotating frame, and observe some structural similarities they have with the Kerr-CFT scattering amplitudes. Finally, in BTZ, there is also an alternate construction for the Green functions: we can start from the covering AdS₃${\mathit{AdS}}_3$ space and use the method of images. Using a 19th century integral formula, I show the equality between the boundary correlators arising via the two constructions.