Black Hole Interior in AdS/CFT

 Black holes in AdS/CFT are interesting things. One could ask if, following along with the usual bulk-boundary description, there is a way to find CFT operators dual to bulk fields in the exterior and interior of a large AdS-Schwarzschild black hole. The exterior is simple enough; it is our usual HKLL scheme that has to be used. This is following the extrapolate dictionary,

\[\mathcal{O}(t, \Omega )=\lim _{r\to \infty }r^{\Delta }\phi (r, t, \Omega )\;,\]

where $\Delta $ is the usual conformal weight. One could also sit in the Poincare setting for working with the expansion of these operators and modes. The primary idea here is that dual to a bulk field, one can either take a set of local operators that are ``smeared", or a family of nonlocal operators instead. For instance, solving $(\Box -m^{2})\phi =0$, and compressing each set of Bessel functions and denoting the normalizable mode by $\xi _{\omega , k}(t, x, z)$ as in 1211.6767, a nonlocal CFT operator in the Poincare patch looks like

\[\Phi _{\text{CFT}}(t, x, z)=\int \frac{d\omega d^{D-1}k}{(2\pi )^{D}}\;\mathcal{O}_{\omega , k}\xi _{\omega , k}(t, x, z)+\mathcal{O}^{\dagger }\xi ^{*}_{\omega , k}(t, x, z)\;.\]

Then, operators in region II of the Penrose diagram can be written as

\[\phi ^{\text{II}}_{\text{CFT}}(t, x, z)=\int _{\omega >0}\frac{d\omega d^{D-1}k}{(2\pi )^{D}}\; \mathcal{O}_{\omega , k}g^{(1)}_{\omega , k}(t, x, z)+\tilde{\mathcal{O}}_{\omega , k}g^{(2)}_{\omega , k}(t, x, z)+\dots \;.\]

This is obtained from interpolation between operators in the regions I and III. Read more on this in 1211.6767 and 1310.6334. More on this will be detailed in Part Two of my Bulk Physics, Algebras and All That notes, which will come out by tomorrow.